Operations Management

Forecasting ( 預測 ) Chapter 4

Outlinebull GLOBAL COMPANY PROFILE TUPPERWARE

CORPORATIONbull WHAT IS FORECASTING ( 何謂預測 ) - Forecasting Time Horizons ( 預測的期間 ) -The Influence of Product Life Cycle( 産品壽命週期的影響 )

bull TYPES OF FORECASTS ( 預測的類型 )bull THE STRATEGIC IMPORTANCE OF FORECAST

ING ( 預測策略的重要性 )ndash Human Resources ( 人力資源 )ndash Capacity ( 産能 )ndash Supply-Chain Management ( 供應鏈管理 )

bull SEVEN STEPS IN THE FORECASTING SYSTEM ( 預測的七個基本步驟 )

Outline - Continuedbull FORECASTING APPROACHES ( 預測方法 )

ndash Overview of Qualitative Methods ( 質性方法 )ndash Overview of Quantitative Methods ( 量化方法 )

bull TIME-SERIES FORECASTING ( 時間序列預測 )ndash Decomposition of Time Series ( 時間序列的分解 )ndash Naiumlve Approach ( 自然預測法 )ndash Moving Averages ( 移動平均法 )ndash Exponential Smoothing( 指數平滑法 )ndash Exponential Smoothing with Trend Adjustment ( 指數平滑法的趨勢調整 )ndash Trend Projections( 趨勢投影法 )ndash Seasonal Variations in Data( 季節變動 )ndash Cyclic Variations in Data ( 週期變動 )

Outline - Continuedbull ASSOCIATIVE FORECASTING METHODS REGRESSION AN

D CORRELATION ANALYSIS

( 關聯預測技術 廻歸與相關分析 )ndash Using Regression Analysis to Forecast ( 廻歸分析 )

ndash Standard Error of the Estimate ( 佑計標準差 )

ndash Correlation Coefficients for Regression Lines ( 廻歸線的相關關係係數 )

ndash Multiple-Regression Analysis ( 多元廻歸分析 )

bull MONITORING AND CONTROLLING FORECASTS ( 預測的管控 )ndash Adaptive Smoothing( 調適平滑法 )ndash Focus Forecasting ( 聚焦預測法 )

bull FORECASTING IN THE SERVICE SECTOR

( 服務領域的預測 )

Learning Objectives

When you complete this chapter you should be able to

Identify or Definendash Forecastingndash Types of forecastsndash Time horizonsndash Approaches to forecasts

Learning Objectives-continued

When you complete this chapter you should be able to

Describe or Explainndash Moving averagesndash Exponential smoothingndash Trend projectionsndash Regression and correlation analysisndash Measures of forecast accuracy

Forecasting at Tupperware

bull Each of 50 profit centers around the world is responsible for computerized monthly quarterly and 12-month sales projections

bull These projections are aggregated by region then globally at Tupperwarersquos World Headquarters

bull Tupperware uses all techniques discussed in text

Three Key Factors for Tupperware

bull The number of registered ldquoconsultantsrdquo or sales representatives

bull The percentage of currently ldquoactiverdquo dealers (this number changes each week and month)

bull Sales per active dealer on a weekly basis

Tupperware - Forecast by Consensus

bull Although inputs come from sales marketing finance and production final forecasts are the consensus of all participating managers

bull The final step is Tupperwarersquos version of the ldquojury of executive opinionrdquo

What is Forecasting

bull Process of predicting a future event

bull Underlying basis of all business decisions

ndash Production

ndash Inventory

ndash Personnel

ndash Facilities

Sales will be $200 Million

bull Short-range forecastndash Up to 1 year usually less than 3 monthsndash Job scheduling worker assignments

bull Medium-range forecastndash 3 months to 3 yearsndash Sales amp production planning budgeting

bull Long-range forecastndash 3+ yearsndash New product planning facility location

Types of Forecasts by Time Horizon

Short-term vs Longer-term Forecasting

bull Mediumlong range forecasts deal with more comprehensive issues and support management decisions regarding planning and products plants and processes

bull Short-term forecasting usually employs different methodologies than longer-term forecasting

bull Short-term forecasts tend to be more accurate than longer-term forecasts

Influence of Product Life Cycle

bull Stages of introduction and growth require longer forecasts than maturity and decline

bull Forecasts useful in projectingndash staffing levels

ndash inventory levels and

ndash factory capacity

as product passes through life cycle stages

Introduction Growth Maturity Decline

Strategy and Issues During a Productrsquos Life

Introduction Growth Maturity Decline

Standardization

Less rapid product changes - more minor changes

Optimum capacity

Increasing stability of process

Long production runs

Product improvement and cost cutting

Little product differentiation

Cost minimization

Over capacity in the industry

Prune line to eliminate items not returning good margin

Reduce capacity

Forecasting critical

Product and process reliability

Competitive product improvements and options

Increase capacity

Shift toward product focused

Enhance distribution

Product design and development critical

Frequent product and process design changes

Short production runs

High production costs

Limited models

Attention to quality

Best period to increase market share

RampD product engineering critical

Practical to change price or quality image

Strengthen niche

Cost control critical

Poor time to change image price or quality

Competitive costs become critical

Defend market position

OM

Str

ateg

yIs

sues

Com

pany

Str

ateg

yIs

sues

HDTV

CD-ROM

Color copiers

Drive-thru restaurants Fax machines

Station wagons

Sales

3 12rdquo Floppy disks

Internet

Types of Forecasts

bull Economic forecastsndash Address business cycle eg inflation rate

money supply etc

bull Technological forecastsndash Predict rate of technological progressndash Predict acceptance of new product

bull Demand forecastsndash Predict sales of existing product

Seven Steps in Forecasting ( 預測的七個基本步驟 )

bull Determine the use of the forecastbull Select the items to be forecastedbull Determine the time horizon of the forecastbull Select the forecasting model(s)bull Gather the databull Make the forecastbull Validate and implement results

Product Demand Charted over 4 Years with Trend and Seasonality

Year1

Year2

Year3

Year4

Seasonal peaks Trend component

Actual demand line

Average demand over four years

Dem

and

for p

rodu

ct o

r ser

vice

Random variation

Actual Demand Moving Average Weighted Moving Average

0

5

10

15

20

25

30

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Sal

es D

eman

d

Actual sales

Moving average

Weighted moving average

Realities of Forecasting

bull Forecasts are seldom perfect

bull Most forecasting methods assume that there is some underlying stability in the system

bull Both product family and aggregated product forecasts are more accurate than individual product forecasts

Forecasting Approaches

bull Used when situation is lsquostablersquo amp historical data existndash Existing productsndash Current technology

bull Involves mathematical techniquesndash eg forecasting sales of

color televisions

Quantitative Methodsbull Used when situation is

vague amp little data existndash New productsndash New technology

bull Involves intuition experiencendash eg forecasting sales on

Internet

Qualitative Methods

Overview of Qualitative Methods

bull Jury of executive opinionndash Pool opinions of high-level executives

sometimes augment by statistical models

bull Delphi methodndash Panel of experts queried iteratively

bull Sales force compositendash Estimates from individual salespersons are

reviewed for reasonableness then aggregated

bull Consumer Market Surveyndash Ask the customer

bull Involves small group of high-level managers

ndash Group estimates demand by working together

bull Combines managerial experience with statistical models

bull Relatively quick

bull lsquoGroup-thinkrsquodisadvantage

copy 1995 Corel Corp

Jury of Executive Opinion

Sales Force Composite

bull Each salesperson projects his or her sales

bull Combined at district amp national levels

bull Sales reps know customersrsquo wants

bull Tends to be overly optimistic

SalesSales

copy 1995 Corel Corp

Delphi Method

bull Iterative group process

bull 3 types of peoplendash Decision makers

ndash Staff

ndash Respondents

bull Reduces lsquogroup-thinkrsquo

Respondents Respondents

Staff Staff

Decision MakersDecision Makers(Sales)

(What will sales be survey)

(Sales will be 45 50 55)

(Sales will be 50)

Consumer Market Survey

bull Ask customers about purchasing plans

bull What consumers say and what they actually do are often different

bull Sometimes difficult to answer

How many hours will you use the Internet

next week

How many hours will you use the Internet

next week

copy 1995 Corel Corp

Overview of Quantitative Approaches

bull Naiumlve approach

bull Moving averages

bull Exponential smoothing

bull Trend projection

bull Linear regression

Time-series Models

Associative models

Quantitative Forecasting Methods (Non-Naive)

QuantitativeForecasting

LinearRegression

AssociativeModels

ExponentialSmoothing

MovingAverage

Time SeriesModels

TrendProjection

bull Set of evenly spaced numerical datandash Obtained by observing response variable at

regular time periods

bull Forecast based only on past valuesndash Assumes that factors influencing past and

present will continue influence in future

bull ExampleYear 1998 1999 2000 2001 2002Sales 787 635 897 932 921

What is a Time Series

TrendTrend

SeasonalSeasonal

CyclicalCyclical

RandomRandom

Time Series Components

bull Persistent overall upward or downward pattern

bull Due to population technology etcbull Several years duration

Mo Qtr Yr

Response

copy 1984-1994 TMaker Co

Trend Component

bull Regular pattern of up amp down fluctuations

bull Due to weather customs etcbull Occurs within 1 year

Mo Qtr

Response

Summer

copy 1984-1994 TMaker Co

Seasonal Component

Common Seasonal PatternsPeriod of Pattern

ldquoSeasonrdquo Length

Number of

ldquoSeasonsrdquo in

Pattern

Week Day 7

Month Week 4 ndash 4 frac12

Month Day 28 ndash 31

Year Quarter 4

Year Month 12

Year Week 52

bull Repeating up amp down movementsbull Due to interactions of factors

influencing economybull Usually 2-10 years duration

Mo Qtr YrMo Qtr Yr

ResponseResponseCycle

Cyclical Component

bull Erratic unsystematic lsquoresidualrsquo fluctuations

bull Due to random variation or unforeseen

events

ndash Union strike

ndash Tornado

bull Short duration amp

nonrepeating

copy 1984-1994 TMaker Co

Random Component

bull Any observed value in a time series is the product (or sum) of time series components

bull Multiplicative modelndash Yi = Ti middot Si middot Ci middot Ri (if quarterly or mo data)

bull Additive modelndash Yi = Ti + Si + Ci + Ri (if quarterly or mo

data)

General Time Series Models

Naive Approach

bull Assumes demand in next period is the same as demand in most recent periodndash eg If May sales were 48

then June sales will be 48

bull Sometimes cost effective amp efficient

copy 1995 Corel Corp

bull MA is a series of arithmetic means

bull Used if little or no trend

bull Used often for smoothingndash Provides overall impression of data over

timebull Equation

MAMAnn

nn Demand inDemand in PreviousPrevious PeriodsPeriods

Moving Average Method

Yoursquore manager of a museum store that sells historical replicas You want to forecast sales (000) for 2003 using a 3-period moving average

1998 41999 62000 52001 32002 7

copy 1995 Corel Corp

Moving Average Example

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 6+5+3=14 143=4 23 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153=50 2002 7 6+5+3=14 143=47 2003 NA 5+3+7=15 153=50

95 96 97 98 99 00Year

Sales

2

4

6

8 Actual

Forecast

Moving Average Graph

bull Used when trend is present ndash Older data usually less important

bull Weights based on intuitionndash Often lay between 0 amp 1 amp sum to 10

bull Equation

WMA =WMA =ΣΣ(Weight for period (Weight for period nn) (Demand in period ) (Demand in period nn))

ΣΣWeightsWeights

Weighted Moving Average Method

Actual Demand Moving Average Weighted Moving

Average

0

5

10

15

20

25

30

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Sal

es D

eman

d

Actual sales

Moving average

Weighted moving average

bull Increasing n makes forecast less sensitive to changes

bull Do not forecast trend wellbull Require much historical data

copy 1984-1994 TMaker Co

Disadvantages of Moving Average Methods

bull Form of weighted moving averagendash Weights decline exponentiallyndash Most recent data weighted most

bull Requires smoothing constant ()ndash Ranges from 0 to 1ndash Subjectively chosen

bull Involves little record keeping of past data

Exponential Smoothing Method

bull Ft = At - 1 + (1-)At - 2 + (1- )2middotAt - 3

+ (1- )3At - 4 + + (1- )t-1middotA0

ndash Ft = Forecast value

ndash At = Actual value = Smoothing constant

bull Ft = Ft-1 + (At-1 - Ft-1)ndash Use for computing forecast

Exponential Smoothing Equations

During the past 8 quarters the Port of Baltimore has unloaded large quantities of grain ( = 10) The first quarter forecast was 175 QuarterActual

1 180 2 168

3 1594 1755 190

6 2057 1808 1829

Exponential Smoothing Example

Find the forecast for the 9th quarter

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

11 180 17500 (Given)

22 168168

33 159159

44 175175

55 190190

66 205205

17500 +17500 +

Exponential Smoothing Solution

QuarterQuarter ActuaActualForecast F t

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010((

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - -

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Ft

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - 17500 - 17500))

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing SolutionFt = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα= = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 +17500 + 1010(180 (180 - 17500- 17500)) = 17550 = 17550

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

1 180 17500 (Given)

22 168168 17500 + 10(180 - 17500) = 1755017500 + 10(180 - 17500) = 17550

33 159159 1755017550 ++ 1010(168 -(168 - 1755017550)) = 17475= 17475

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1995 180 17500 (Given)

1996 168 17500 + 10(180 - 17500) = 17550

1997 159 17550 + 10(168 - 17550) = 17475

1998 175

1999 190

2000 205

17475 + 10(159 - 17475)= 17318

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9 17822 + 10(182 - 17822) = 17858 182 17802 + 10(180 - 17802) = 17822

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Outlinebull GLOBAL COMPANY PROFILE TUPPERWARE

CORPORATIONbull WHAT IS FORECASTING ( 何謂預測 ) - Forecasting Time Horizons ( 預測的期間 ) -The Influence of Product Life Cycle( 産品壽命週期的影響 )

bull TYPES OF FORECASTS ( 預測的類型 )bull THE STRATEGIC IMPORTANCE OF FORECAST

ING ( 預測策略的重要性 )ndash Human Resources ( 人力資源 )ndash Capacity ( 産能 )ndash Supply-Chain Management ( 供應鏈管理 )

bull SEVEN STEPS IN THE FORECASTING SYSTEM ( 預測的七個基本步驟 )

Outline - Continuedbull FORECASTING APPROACHES ( 預測方法 )

ndash Overview of Qualitative Methods ( 質性方法 )ndash Overview of Quantitative Methods ( 量化方法 )

bull TIME-SERIES FORECASTING ( 時間序列預測 )ndash Decomposition of Time Series ( 時間序列的分解 )ndash Naiumlve Approach ( 自然預測法 )ndash Moving Averages ( 移動平均法 )ndash Exponential Smoothing( 指數平滑法 )ndash Exponential Smoothing with Trend Adjustment ( 指數平滑法的趨勢調整 )ndash Trend Projections( 趨勢投影法 )ndash Seasonal Variations in Data( 季節變動 )ndash Cyclic Variations in Data ( 週期變動 )

Outline - Continuedbull ASSOCIATIVE FORECASTING METHODS REGRESSION AN

D CORRELATION ANALYSIS

( 關聯預測技術 廻歸與相關分析 )ndash Using Regression Analysis to Forecast ( 廻歸分析 )

ndash Standard Error of the Estimate ( 佑計標準差 )

ndash Correlation Coefficients for Regression Lines ( 廻歸線的相關關係係數 )

ndash Multiple-Regression Analysis ( 多元廻歸分析 )

bull MONITORING AND CONTROLLING FORECASTS ( 預測的管控 )ndash Adaptive Smoothing( 調適平滑法 )ndash Focus Forecasting ( 聚焦預測法 )

bull FORECASTING IN THE SERVICE SECTOR

( 服務領域的預測 )

Learning Objectives

When you complete this chapter you should be able to

Identify or Definendash Forecastingndash Types of forecastsndash Time horizonsndash Approaches to forecasts

Learning Objectives-continued

When you complete this chapter you should be able to

Describe or Explainndash Moving averagesndash Exponential smoothingndash Trend projectionsndash Regression and correlation analysisndash Measures of forecast accuracy

Forecasting at Tupperware

bull Each of 50 profit centers around the world is responsible for computerized monthly quarterly and 12-month sales projections

bull These projections are aggregated by region then globally at Tupperwarersquos World Headquarters

bull Tupperware uses all techniques discussed in text

Three Key Factors for Tupperware

bull The number of registered ldquoconsultantsrdquo or sales representatives

bull The percentage of currently ldquoactiverdquo dealers (this number changes each week and month)

bull Sales per active dealer on a weekly basis

Tupperware - Forecast by Consensus

bull Although inputs come from sales marketing finance and production final forecasts are the consensus of all participating managers

bull The final step is Tupperwarersquos version of the ldquojury of executive opinionrdquo

What is Forecasting

bull Process of predicting a future event

bull Underlying basis of all business decisions

ndash Production

ndash Inventory

ndash Personnel

ndash Facilities

Sales will be $200 Million

bull Short-range forecastndash Up to 1 year usually less than 3 monthsndash Job scheduling worker assignments

bull Medium-range forecastndash 3 months to 3 yearsndash Sales amp production planning budgeting

bull Long-range forecastndash 3+ yearsndash New product planning facility location

Types of Forecasts by Time Horizon

Short-term vs Longer-term Forecasting

bull Mediumlong range forecasts deal with more comprehensive issues and support management decisions regarding planning and products plants and processes

bull Short-term forecasting usually employs different methodologies than longer-term forecasting

bull Short-term forecasts tend to be more accurate than longer-term forecasts

Influence of Product Life Cycle

bull Stages of introduction and growth require longer forecasts than maturity and decline

bull Forecasts useful in projectingndash staffing levels

ndash inventory levels and

ndash factory capacity

as product passes through life cycle stages

Introduction Growth Maturity Decline

Strategy and Issues During a Productrsquos Life

Introduction Growth Maturity Decline

Standardization

Less rapid product changes - more minor changes

Optimum capacity

Increasing stability of process

Long production runs

Product improvement and cost cutting

Little product differentiation

Cost minimization

Over capacity in the industry

Prune line to eliminate items not returning good margin

Reduce capacity

Forecasting critical

Product and process reliability

Competitive product improvements and options

Increase capacity

Shift toward product focused

Enhance distribution

Product design and development critical

Frequent product and process design changes

Short production runs

High production costs

Limited models

Attention to quality

Best period to increase market share

RampD product engineering critical

Practical to change price or quality image

Strengthen niche

Cost control critical

Poor time to change image price or quality

Competitive costs become critical

Defend market position

OM

Str

ateg

yIs

sues

Com

pany

Str

ateg

yIs

sues

HDTV

CD-ROM

Color copiers

Drive-thru restaurants Fax machines

Station wagons

Sales

3 12rdquo Floppy disks

Internet

Types of Forecasts

bull Economic forecastsndash Address business cycle eg inflation rate

money supply etc

bull Technological forecastsndash Predict rate of technological progressndash Predict acceptance of new product

bull Demand forecastsndash Predict sales of existing product

Seven Steps in Forecasting ( 預測的七個基本步驟 )

bull Determine the use of the forecastbull Select the items to be forecastedbull Determine the time horizon of the forecastbull Select the forecasting model(s)bull Gather the databull Make the forecastbull Validate and implement results

Product Demand Charted over 4 Years with Trend and Seasonality

Year1

Year2

Year3

Year4

Seasonal peaks Trend component

Actual demand line

Average demand over four years

Dem

and

for p

rodu

ct o

r ser

vice

Random variation

Actual Demand Moving Average Weighted Moving Average

0

5

10

15

20

25

30

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Sal

es D

eman

d

Actual sales

Moving average

Weighted moving average

Realities of Forecasting

bull Forecasts are seldom perfect

bull Most forecasting methods assume that there is some underlying stability in the system

bull Both product family and aggregated product forecasts are more accurate than individual product forecasts

Forecasting Approaches

bull Used when situation is lsquostablersquo amp historical data existndash Existing productsndash Current technology

bull Involves mathematical techniquesndash eg forecasting sales of

color televisions

Quantitative Methodsbull Used when situation is

vague amp little data existndash New productsndash New technology

bull Involves intuition experiencendash eg forecasting sales on

Internet

Qualitative Methods

Overview of Qualitative Methods

bull Jury of executive opinionndash Pool opinions of high-level executives

sometimes augment by statistical models

bull Delphi methodndash Panel of experts queried iteratively

bull Sales force compositendash Estimates from individual salespersons are

reviewed for reasonableness then aggregated

bull Consumer Market Surveyndash Ask the customer

bull Involves small group of high-level managers

ndash Group estimates demand by working together

bull Combines managerial experience with statistical models

bull Relatively quick

bull lsquoGroup-thinkrsquodisadvantage

copy 1995 Corel Corp

Jury of Executive Opinion

Sales Force Composite

bull Each salesperson projects his or her sales

bull Combined at district amp national levels

bull Sales reps know customersrsquo wants

bull Tends to be overly optimistic

SalesSales

copy 1995 Corel Corp

Delphi Method

bull Iterative group process

bull 3 types of peoplendash Decision makers

ndash Staff

ndash Respondents

bull Reduces lsquogroup-thinkrsquo

Respondents Respondents

Staff Staff

Decision MakersDecision Makers(Sales)

(What will sales be survey)

(Sales will be 45 50 55)

(Sales will be 50)

Consumer Market Survey

bull Ask customers about purchasing plans

bull What consumers say and what they actually do are often different

bull Sometimes difficult to answer

How many hours will you use the Internet

next week

How many hours will you use the Internet

next week

copy 1995 Corel Corp

Overview of Quantitative Approaches

bull Naiumlve approach

bull Moving averages

bull Exponential smoothing

bull Trend projection

bull Linear regression

Time-series Models

Associative models

Quantitative Forecasting Methods (Non-Naive)

QuantitativeForecasting

LinearRegression

AssociativeModels

ExponentialSmoothing

MovingAverage

Time SeriesModels

TrendProjection

bull Set of evenly spaced numerical datandash Obtained by observing response variable at

regular time periods

bull Forecast based only on past valuesndash Assumes that factors influencing past and

present will continue influence in future

bull ExampleYear 1998 1999 2000 2001 2002Sales 787 635 897 932 921

What is a Time Series

TrendTrend

SeasonalSeasonal

CyclicalCyclical

RandomRandom

Time Series Components

bull Persistent overall upward or downward pattern

bull Due to population technology etcbull Several years duration

Mo Qtr Yr

Response

copy 1984-1994 TMaker Co

Trend Component

bull Regular pattern of up amp down fluctuations

bull Due to weather customs etcbull Occurs within 1 year

Mo Qtr

Response

Summer

copy 1984-1994 TMaker Co

Seasonal Component

Common Seasonal PatternsPeriod of Pattern

ldquoSeasonrdquo Length

Number of

ldquoSeasonsrdquo in

Pattern

Week Day 7

Month Week 4 ndash 4 frac12

Month Day 28 ndash 31

Year Quarter 4

Year Month 12

Year Week 52

bull Repeating up amp down movementsbull Due to interactions of factors

influencing economybull Usually 2-10 years duration

Mo Qtr YrMo Qtr Yr

ResponseResponseCycle

Cyclical Component

bull Erratic unsystematic lsquoresidualrsquo fluctuations

bull Due to random variation or unforeseen

events

ndash Union strike

ndash Tornado

bull Short duration amp

nonrepeating

copy 1984-1994 TMaker Co

Random Component

bull Any observed value in a time series is the product (or sum) of time series components

bull Multiplicative modelndash Yi = Ti middot Si middot Ci middot Ri (if quarterly or mo data)

bull Additive modelndash Yi = Ti + Si + Ci + Ri (if quarterly or mo

data)

General Time Series Models

Naive Approach

bull Assumes demand in next period is the same as demand in most recent periodndash eg If May sales were 48

then June sales will be 48

bull Sometimes cost effective amp efficient

copy 1995 Corel Corp

bull MA is a series of arithmetic means

bull Used if little or no trend

bull Used often for smoothingndash Provides overall impression of data over

timebull Equation

MAMAnn

nn Demand inDemand in PreviousPrevious PeriodsPeriods

Moving Average Method

Yoursquore manager of a museum store that sells historical replicas You want to forecast sales (000) for 2003 using a 3-period moving average

1998 41999 62000 52001 32002 7

copy 1995 Corel Corp

Moving Average Example

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 6+5+3=14 143=4 23 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153=50 2002 7 6+5+3=14 143=47 2003 NA 5+3+7=15 153=50

95 96 97 98 99 00Year

Sales

2

4

6

8 Actual

Forecast

Moving Average Graph

bull Used when trend is present ndash Older data usually less important

bull Weights based on intuitionndash Often lay between 0 amp 1 amp sum to 10

bull Equation

WMA =WMA =ΣΣ(Weight for period (Weight for period nn) (Demand in period ) (Demand in period nn))

ΣΣWeightsWeights

Weighted Moving Average Method

Actual Demand Moving Average Weighted Moving

Average

0

5

10

15

20

25

30

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Sal

es D

eman

d

Actual sales

Moving average

Weighted moving average

bull Increasing n makes forecast less sensitive to changes

bull Do not forecast trend wellbull Require much historical data

copy 1984-1994 TMaker Co

Disadvantages of Moving Average Methods

bull Form of weighted moving averagendash Weights decline exponentiallyndash Most recent data weighted most

bull Requires smoothing constant ()ndash Ranges from 0 to 1ndash Subjectively chosen

bull Involves little record keeping of past data

Exponential Smoothing Method

bull Ft = At - 1 + (1-)At - 2 + (1- )2middotAt - 3

+ (1- )3At - 4 + + (1- )t-1middotA0

ndash Ft = Forecast value

ndash At = Actual value = Smoothing constant

bull Ft = Ft-1 + (At-1 - Ft-1)ndash Use for computing forecast

Exponential Smoothing Equations

During the past 8 quarters the Port of Baltimore has unloaded large quantities of grain ( = 10) The first quarter forecast was 175 QuarterActual

1 180 2 168

3 1594 1755 190

6 2057 1808 1829

Exponential Smoothing Example

Find the forecast for the 9th quarter

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

11 180 17500 (Given)

22 168168

33 159159

44 175175

55 190190

66 205205

17500 +17500 +

Exponential Smoothing Solution

QuarterQuarter ActuaActualForecast F t

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010((

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - -

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Ft

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - 17500 - 17500))

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing SolutionFt = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα= = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 +17500 + 1010(180 (180 - 17500- 17500)) = 17550 = 17550

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

1 180 17500 (Given)

22 168168 17500 + 10(180 - 17500) = 1755017500 + 10(180 - 17500) = 17550

33 159159 1755017550 ++ 1010(168 -(168 - 1755017550)) = 17475= 17475

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1995 180 17500 (Given)

1996 168 17500 + 10(180 - 17500) = 17550

1997 159 17550 + 10(168 - 17550) = 17475

1998 175

1999 190

2000 205

17475 + 10(159 - 17475)= 17318

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9 17822 + 10(182 - 17822) = 17858 182 17802 + 10(180 - 17802) = 17822

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Outline - Continuedbull FORECASTING APPROACHES ( 預測方法 )

ndash Overview of Qualitative Methods ( 質性方法 )ndash Overview of Quantitative Methods ( 量化方法 )

bull TIME-SERIES FORECASTING ( 時間序列預測 )ndash Decomposition of Time Series ( 時間序列的分解 )ndash Naiumlve Approach ( 自然預測法 )ndash Moving Averages ( 移動平均法 )ndash Exponential Smoothing( 指數平滑法 )ndash Exponential Smoothing with Trend Adjustment ( 指數平滑法的趨勢調整 )ndash Trend Projections( 趨勢投影法 )ndash Seasonal Variations in Data( 季節變動 )ndash Cyclic Variations in Data ( 週期變動 )

Outline - Continuedbull ASSOCIATIVE FORECASTING METHODS REGRESSION AN

D CORRELATION ANALYSIS

( 關聯預測技術 廻歸與相關分析 )ndash Using Regression Analysis to Forecast ( 廻歸分析 )

ndash Standard Error of the Estimate ( 佑計標準差 )

ndash Correlation Coefficients for Regression Lines ( 廻歸線的相關關係係數 )

ndash Multiple-Regression Analysis ( 多元廻歸分析 )

bull MONITORING AND CONTROLLING FORECASTS ( 預測的管控 )ndash Adaptive Smoothing( 調適平滑法 )ndash Focus Forecasting ( 聚焦預測法 )

bull FORECASTING IN THE SERVICE SECTOR

( 服務領域的預測 )

Learning Objectives

When you complete this chapter you should be able to

Identify or Definendash Forecastingndash Types of forecastsndash Time horizonsndash Approaches to forecasts

Learning Objectives-continued

When you complete this chapter you should be able to

Describe or Explainndash Moving averagesndash Exponential smoothingndash Trend projectionsndash Regression and correlation analysisndash Measures of forecast accuracy

Forecasting at Tupperware

bull Each of 50 profit centers around the world is responsible for computerized monthly quarterly and 12-month sales projections

bull These projections are aggregated by region then globally at Tupperwarersquos World Headquarters

bull Tupperware uses all techniques discussed in text

Three Key Factors for Tupperware

bull The number of registered ldquoconsultantsrdquo or sales representatives

bull The percentage of currently ldquoactiverdquo dealers (this number changes each week and month)

bull Sales per active dealer on a weekly basis

Tupperware - Forecast by Consensus

bull Although inputs come from sales marketing finance and production final forecasts are the consensus of all participating managers

bull The final step is Tupperwarersquos version of the ldquojury of executive opinionrdquo

What is Forecasting

bull Process of predicting a future event

bull Underlying basis of all business decisions

ndash Production

ndash Inventory

ndash Personnel

ndash Facilities

Sales will be $200 Million

bull Short-range forecastndash Up to 1 year usually less than 3 monthsndash Job scheduling worker assignments

bull Medium-range forecastndash 3 months to 3 yearsndash Sales amp production planning budgeting

bull Long-range forecastndash 3+ yearsndash New product planning facility location

Types of Forecasts by Time Horizon

Short-term vs Longer-term Forecasting

bull Mediumlong range forecasts deal with more comprehensive issues and support management decisions regarding planning and products plants and processes

bull Short-term forecasting usually employs different methodologies than longer-term forecasting

bull Short-term forecasts tend to be more accurate than longer-term forecasts

Influence of Product Life Cycle

bull Stages of introduction and growth require longer forecasts than maturity and decline

bull Forecasts useful in projectingndash staffing levels

ndash inventory levels and

ndash factory capacity

as product passes through life cycle stages

Introduction Growth Maturity Decline

Strategy and Issues During a Productrsquos Life

Introduction Growth Maturity Decline

Standardization

Less rapid product changes - more minor changes

Optimum capacity

Increasing stability of process

Long production runs

Product improvement and cost cutting

Little product differentiation

Cost minimization

Over capacity in the industry

Prune line to eliminate items not returning good margin

Reduce capacity

Forecasting critical

Product and process reliability

Competitive product improvements and options

Increase capacity

Shift toward product focused

Enhance distribution

Product design and development critical

Frequent product and process design changes

Short production runs

High production costs

Limited models

Attention to quality

Best period to increase market share

RampD product engineering critical

Practical to change price or quality image

Strengthen niche

Cost control critical

Poor time to change image price or quality

Competitive costs become critical

Defend market position

OM

Str

ateg

yIs

sues

Com

pany

Str

ateg

yIs

sues

HDTV

CD-ROM

Color copiers

Drive-thru restaurants Fax machines

Station wagons

Sales

3 12rdquo Floppy disks

Internet

Types of Forecasts

bull Economic forecastsndash Address business cycle eg inflation rate

money supply etc

bull Technological forecastsndash Predict rate of technological progressndash Predict acceptance of new product

bull Demand forecastsndash Predict sales of existing product

Seven Steps in Forecasting ( 預測的七個基本步驟 )

bull Determine the use of the forecastbull Select the items to be forecastedbull Determine the time horizon of the forecastbull Select the forecasting model(s)bull Gather the databull Make the forecastbull Validate and implement results

Product Demand Charted over 4 Years with Trend and Seasonality

Year1

Year2

Year3

Year4

Seasonal peaks Trend component

Actual demand line

Average demand over four years

Dem

and

for p

rodu

ct o

r ser

vice

Random variation

Actual Demand Moving Average Weighted Moving Average

0

5

10

15

20

25

30

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Sal

es D

eman

d

Actual sales

Moving average

Weighted moving average

Realities of Forecasting

bull Forecasts are seldom perfect

bull Most forecasting methods assume that there is some underlying stability in the system

bull Both product family and aggregated product forecasts are more accurate than individual product forecasts

Forecasting Approaches

bull Used when situation is lsquostablersquo amp historical data existndash Existing productsndash Current technology

bull Involves mathematical techniquesndash eg forecasting sales of

color televisions

Quantitative Methodsbull Used when situation is

vague amp little data existndash New productsndash New technology

bull Involves intuition experiencendash eg forecasting sales on

Internet

Qualitative Methods

Overview of Qualitative Methods

bull Jury of executive opinionndash Pool opinions of high-level executives

sometimes augment by statistical models

bull Delphi methodndash Panel of experts queried iteratively

bull Sales force compositendash Estimates from individual salespersons are

reviewed for reasonableness then aggregated

bull Consumer Market Surveyndash Ask the customer

bull Involves small group of high-level managers

ndash Group estimates demand by working together

bull Combines managerial experience with statistical models

bull Relatively quick

bull lsquoGroup-thinkrsquodisadvantage

copy 1995 Corel Corp

Jury of Executive Opinion

Sales Force Composite

bull Each salesperson projects his or her sales

bull Combined at district amp national levels

bull Sales reps know customersrsquo wants

bull Tends to be overly optimistic

SalesSales

copy 1995 Corel Corp

Delphi Method

bull Iterative group process

bull 3 types of peoplendash Decision makers

ndash Staff

ndash Respondents

bull Reduces lsquogroup-thinkrsquo

Respondents Respondents

Staff Staff

Decision MakersDecision Makers(Sales)

(What will sales be survey)

(Sales will be 45 50 55)

(Sales will be 50)

Consumer Market Survey

bull Ask customers about purchasing plans

bull What consumers say and what they actually do are often different

bull Sometimes difficult to answer

How many hours will you use the Internet

next week

How many hours will you use the Internet

next week

copy 1995 Corel Corp

Overview of Quantitative Approaches

bull Naiumlve approach

bull Moving averages

bull Exponential smoothing

bull Trend projection

bull Linear regression

Time-series Models

Associative models

Quantitative Forecasting Methods (Non-Naive)

QuantitativeForecasting

LinearRegression

AssociativeModels

ExponentialSmoothing

MovingAverage

Time SeriesModels

TrendProjection

bull Set of evenly spaced numerical datandash Obtained by observing response variable at

regular time periods

bull Forecast based only on past valuesndash Assumes that factors influencing past and

present will continue influence in future

bull ExampleYear 1998 1999 2000 2001 2002Sales 787 635 897 932 921

What is a Time Series

TrendTrend

SeasonalSeasonal

CyclicalCyclical

RandomRandom

Time Series Components

bull Persistent overall upward or downward pattern

bull Due to population technology etcbull Several years duration

Mo Qtr Yr

Response

copy 1984-1994 TMaker Co

Trend Component

bull Regular pattern of up amp down fluctuations

bull Due to weather customs etcbull Occurs within 1 year

Mo Qtr

Response

Summer

copy 1984-1994 TMaker Co

Seasonal Component

Common Seasonal PatternsPeriod of Pattern

ldquoSeasonrdquo Length

Number of

ldquoSeasonsrdquo in

Pattern

Week Day 7

Month Week 4 ndash 4 frac12

Month Day 28 ndash 31

Year Quarter 4

Year Month 12

Year Week 52

bull Repeating up amp down movementsbull Due to interactions of factors

influencing economybull Usually 2-10 years duration

Mo Qtr YrMo Qtr Yr

ResponseResponseCycle

Cyclical Component

bull Erratic unsystematic lsquoresidualrsquo fluctuations

bull Due to random variation or unforeseen

events

ndash Union strike

ndash Tornado

bull Short duration amp

nonrepeating

copy 1984-1994 TMaker Co

Random Component

bull Any observed value in a time series is the product (or sum) of time series components

bull Multiplicative modelndash Yi = Ti middot Si middot Ci middot Ri (if quarterly or mo data)

bull Additive modelndash Yi = Ti + Si + Ci + Ri (if quarterly or mo

data)

General Time Series Models

Naive Approach

bull Assumes demand in next period is the same as demand in most recent periodndash eg If May sales were 48

then June sales will be 48

bull Sometimes cost effective amp efficient

copy 1995 Corel Corp

bull MA is a series of arithmetic means

bull Used if little or no trend

bull Used often for smoothingndash Provides overall impression of data over

timebull Equation

MAMAnn

nn Demand inDemand in PreviousPrevious PeriodsPeriods

Moving Average Method

Yoursquore manager of a museum store that sells historical replicas You want to forecast sales (000) for 2003 using a 3-period moving average

1998 41999 62000 52001 32002 7

copy 1995 Corel Corp

Moving Average Example

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 6+5+3=14 143=4 23 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153=50 2002 7 6+5+3=14 143=47 2003 NA 5+3+7=15 153=50

95 96 97 98 99 00Year

Sales

2

4

6

8 Actual

Forecast

Moving Average Graph

bull Used when trend is present ndash Older data usually less important

bull Weights based on intuitionndash Often lay between 0 amp 1 amp sum to 10

bull Equation

WMA =WMA =ΣΣ(Weight for period (Weight for period nn) (Demand in period ) (Demand in period nn))

ΣΣWeightsWeights

Weighted Moving Average Method

Actual Demand Moving Average Weighted Moving

Average

0

5

10

15

20

25

30

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Sal

es D

eman

d

Actual sales

Moving average

Weighted moving average

bull Increasing n makes forecast less sensitive to changes

bull Do not forecast trend wellbull Require much historical data

copy 1984-1994 TMaker Co

Disadvantages of Moving Average Methods

bull Form of weighted moving averagendash Weights decline exponentiallyndash Most recent data weighted most

bull Requires smoothing constant ()ndash Ranges from 0 to 1ndash Subjectively chosen

bull Involves little record keeping of past data

Exponential Smoothing Method

bull Ft = At - 1 + (1-)At - 2 + (1- )2middotAt - 3

+ (1- )3At - 4 + + (1- )t-1middotA0

ndash Ft = Forecast value

ndash At = Actual value = Smoothing constant

bull Ft = Ft-1 + (At-1 - Ft-1)ndash Use for computing forecast

Exponential Smoothing Equations

During the past 8 quarters the Port of Baltimore has unloaded large quantities of grain ( = 10) The first quarter forecast was 175 QuarterActual

1 180 2 168

3 1594 1755 190

6 2057 1808 1829

Exponential Smoothing Example

Find the forecast for the 9th quarter

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

11 180 17500 (Given)

22 168168

33 159159

44 175175

55 190190

66 205205

17500 +17500 +

Exponential Smoothing Solution

QuarterQuarter ActuaActualForecast F t

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010((

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - -

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Ft

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - 17500 - 17500))

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing SolutionFt = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα= = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 +17500 + 1010(180 (180 - 17500- 17500)) = 17550 = 17550

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

1 180 17500 (Given)

22 168168 17500 + 10(180 - 17500) = 1755017500 + 10(180 - 17500) = 17550

33 159159 1755017550 ++ 1010(168 -(168 - 1755017550)) = 17475= 17475

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1995 180 17500 (Given)

1996 168 17500 + 10(180 - 17500) = 17550

1997 159 17550 + 10(168 - 17550) = 17475

1998 175

1999 190

2000 205

17475 + 10(159 - 17475)= 17318

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9 17822 + 10(182 - 17822) = 17858 182 17802 + 10(180 - 17802) = 17822

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Outline - Continuedbull ASSOCIATIVE FORECASTING METHODS REGRESSION AN

D CORRELATION ANALYSIS

( 關聯預測技術 廻歸與相關分析 )ndash Using Regression Analysis to Forecast ( 廻歸分析 )

ndash Standard Error of the Estimate ( 佑計標準差 )

ndash Correlation Coefficients for Regression Lines ( 廻歸線的相關關係係數 )

ndash Multiple-Regression Analysis ( 多元廻歸分析 )

bull MONITORING AND CONTROLLING FORECASTS ( 預測的管控 )ndash Adaptive Smoothing( 調適平滑法 )ndash Focus Forecasting ( 聚焦預測法 )

bull FORECASTING IN THE SERVICE SECTOR

( 服務領域的預測 )

Learning Objectives

When you complete this chapter you should be able to

Identify or Definendash Forecastingndash Types of forecastsndash Time horizonsndash Approaches to forecasts

Learning Objectives-continued

When you complete this chapter you should be able to

Describe or Explainndash Moving averagesndash Exponential smoothingndash Trend projectionsndash Regression and correlation analysisndash Measures of forecast accuracy

Forecasting at Tupperware

bull Each of 50 profit centers around the world is responsible for computerized monthly quarterly and 12-month sales projections

bull These projections are aggregated by region then globally at Tupperwarersquos World Headquarters

bull Tupperware uses all techniques discussed in text

Three Key Factors for Tupperware

bull The number of registered ldquoconsultantsrdquo or sales representatives

bull The percentage of currently ldquoactiverdquo dealers (this number changes each week and month)

bull Sales per active dealer on a weekly basis

Tupperware - Forecast by Consensus

bull Although inputs come from sales marketing finance and production final forecasts are the consensus of all participating managers

bull The final step is Tupperwarersquos version of the ldquojury of executive opinionrdquo

What is Forecasting

bull Process of predicting a future event

bull Underlying basis of all business decisions

ndash Production

ndash Inventory

ndash Personnel

ndash Facilities

Sales will be $200 Million

bull Short-range forecastndash Up to 1 year usually less than 3 monthsndash Job scheduling worker assignments

bull Medium-range forecastndash 3 months to 3 yearsndash Sales amp production planning budgeting

bull Long-range forecastndash 3+ yearsndash New product planning facility location

Types of Forecasts by Time Horizon

Short-term vs Longer-term Forecasting

bull Mediumlong range forecasts deal with more comprehensive issues and support management decisions regarding planning and products plants and processes

bull Short-term forecasting usually employs different methodologies than longer-term forecasting

bull Short-term forecasts tend to be more accurate than longer-term forecasts

Influence of Product Life Cycle

bull Stages of introduction and growth require longer forecasts than maturity and decline

bull Forecasts useful in projectingndash staffing levels

ndash inventory levels and

ndash factory capacity

as product passes through life cycle stages

Introduction Growth Maturity Decline

Strategy and Issues During a Productrsquos Life

Introduction Growth Maturity Decline

Standardization

Less rapid product changes - more minor changes

Optimum capacity

Increasing stability of process

Long production runs

Product improvement and cost cutting

Little product differentiation

Cost minimization

Over capacity in the industry

Prune line to eliminate items not returning good margin

Reduce capacity

Forecasting critical

Product and process reliability

Competitive product improvements and options

Increase capacity

Shift toward product focused

Enhance distribution

Product design and development critical

Frequent product and process design changes

Short production runs

High production costs

Limited models

Attention to quality

Best period to increase market share

RampD product engineering critical

Practical to change price or quality image

Strengthen niche

Cost control critical

Poor time to change image price or quality

Competitive costs become critical

Defend market position

OM

Str

ateg

yIs

sues

Com

pany

Str

ateg

yIs

sues

HDTV

CD-ROM

Color copiers

Drive-thru restaurants Fax machines

Station wagons

Sales

3 12rdquo Floppy disks

Internet

Types of Forecasts

bull Economic forecastsndash Address business cycle eg inflation rate

money supply etc

bull Technological forecastsndash Predict rate of technological progressndash Predict acceptance of new product

bull Demand forecastsndash Predict sales of existing product

Seven Steps in Forecasting ( 預測的七個基本步驟 )

bull Determine the use of the forecastbull Select the items to be forecastedbull Determine the time horizon of the forecastbull Select the forecasting model(s)bull Gather the databull Make the forecastbull Validate and implement results

Product Demand Charted over 4 Years with Trend and Seasonality

Year1

Year2

Year3

Year4

Seasonal peaks Trend component

Actual demand line

Average demand over four years

Dem

and

for p

rodu

ct o

r ser

vice

Random variation

Actual Demand Moving Average Weighted Moving Average

0

5

10

15

20

25

30

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Sal

es D

eman

d

Actual sales

Moving average

Weighted moving average

Realities of Forecasting

bull Forecasts are seldom perfect

bull Most forecasting methods assume that there is some underlying stability in the system

bull Both product family and aggregated product forecasts are more accurate than individual product forecasts

Forecasting Approaches

bull Used when situation is lsquostablersquo amp historical data existndash Existing productsndash Current technology

bull Involves mathematical techniquesndash eg forecasting sales of

color televisions

Quantitative Methodsbull Used when situation is

vague amp little data existndash New productsndash New technology

bull Involves intuition experiencendash eg forecasting sales on

Internet

Qualitative Methods

Overview of Qualitative Methods

bull Jury of executive opinionndash Pool opinions of high-level executives

sometimes augment by statistical models

bull Delphi methodndash Panel of experts queried iteratively

bull Sales force compositendash Estimates from individual salespersons are

reviewed for reasonableness then aggregated

bull Consumer Market Surveyndash Ask the customer

bull Involves small group of high-level managers

ndash Group estimates demand by working together

bull Combines managerial experience with statistical models

bull Relatively quick

bull lsquoGroup-thinkrsquodisadvantage

copy 1995 Corel Corp

Jury of Executive Opinion

Sales Force Composite

bull Each salesperson projects his or her sales

bull Combined at district amp national levels

bull Sales reps know customersrsquo wants

bull Tends to be overly optimistic

SalesSales

copy 1995 Corel Corp

Delphi Method

bull Iterative group process

bull 3 types of peoplendash Decision makers

ndash Staff

ndash Respondents

bull Reduces lsquogroup-thinkrsquo

Respondents Respondents

Staff Staff

Decision MakersDecision Makers(Sales)

(What will sales be survey)

(Sales will be 45 50 55)

(Sales will be 50)

Consumer Market Survey

bull Ask customers about purchasing plans

bull What consumers say and what they actually do are often different

bull Sometimes difficult to answer

How many hours will you use the Internet

next week

How many hours will you use the Internet

next week

copy 1995 Corel Corp

Overview of Quantitative Approaches

bull Naiumlve approach

bull Moving averages

bull Exponential smoothing

bull Trend projection

bull Linear regression

Time-series Models

Associative models

Quantitative Forecasting Methods (Non-Naive)

QuantitativeForecasting

LinearRegression

AssociativeModels

ExponentialSmoothing

MovingAverage

Time SeriesModels

TrendProjection

bull Set of evenly spaced numerical datandash Obtained by observing response variable at

regular time periods

bull Forecast based only on past valuesndash Assumes that factors influencing past and

present will continue influence in future

bull ExampleYear 1998 1999 2000 2001 2002Sales 787 635 897 932 921

What is a Time Series

TrendTrend

SeasonalSeasonal

CyclicalCyclical

RandomRandom

Time Series Components

bull Persistent overall upward or downward pattern

bull Due to population technology etcbull Several years duration

Mo Qtr Yr

Response

copy 1984-1994 TMaker Co

Trend Component

bull Regular pattern of up amp down fluctuations

bull Due to weather customs etcbull Occurs within 1 year

Mo Qtr

Response

Summer

copy 1984-1994 TMaker Co

Seasonal Component

Common Seasonal PatternsPeriod of Pattern

ldquoSeasonrdquo Length

Number of

ldquoSeasonsrdquo in

Pattern

Week Day 7

Month Week 4 ndash 4 frac12

Month Day 28 ndash 31

Year Quarter 4

Year Month 12

Year Week 52

bull Repeating up amp down movementsbull Due to interactions of factors

influencing economybull Usually 2-10 years duration

Mo Qtr YrMo Qtr Yr

ResponseResponseCycle

Cyclical Component

bull Erratic unsystematic lsquoresidualrsquo fluctuations

bull Due to random variation or unforeseen

events

ndash Union strike

ndash Tornado

bull Short duration amp

nonrepeating

copy 1984-1994 TMaker Co

Random Component

bull Any observed value in a time series is the product (or sum) of time series components

bull Multiplicative modelndash Yi = Ti middot Si middot Ci middot Ri (if quarterly or mo data)

bull Additive modelndash Yi = Ti + Si + Ci + Ri (if quarterly or mo

data)

General Time Series Models

Naive Approach

bull Assumes demand in next period is the same as demand in most recent periodndash eg If May sales were 48

then June sales will be 48

bull Sometimes cost effective amp efficient

copy 1995 Corel Corp

bull MA is a series of arithmetic means

bull Used if little or no trend

bull Used often for smoothingndash Provides overall impression of data over

timebull Equation

MAMAnn

nn Demand inDemand in PreviousPrevious PeriodsPeriods

Moving Average Method

Yoursquore manager of a museum store that sells historical replicas You want to forecast sales (000) for 2003 using a 3-period moving average

1998 41999 62000 52001 32002 7

copy 1995 Corel Corp

Moving Average Example

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 6+5+3=14 143=4 23 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153=50 2002 7 6+5+3=14 143=47 2003 NA 5+3+7=15 153=50

95 96 97 98 99 00Year

Sales

2

4

6

8 Actual

Forecast

Moving Average Graph

bull Used when trend is present ndash Older data usually less important

bull Weights based on intuitionndash Often lay between 0 amp 1 amp sum to 10

bull Equation

WMA =WMA =ΣΣ(Weight for period (Weight for period nn) (Demand in period ) (Demand in period nn))

ΣΣWeightsWeights

Weighted Moving Average Method

Actual Demand Moving Average Weighted Moving

Average

0

5

10

15

20

25

30

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Sal

es D

eman

d

Actual sales

Moving average

Weighted moving average

bull Increasing n makes forecast less sensitive to changes

bull Do not forecast trend wellbull Require much historical data

copy 1984-1994 TMaker Co

Disadvantages of Moving Average Methods

bull Form of weighted moving averagendash Weights decline exponentiallyndash Most recent data weighted most

bull Requires smoothing constant ()ndash Ranges from 0 to 1ndash Subjectively chosen

bull Involves little record keeping of past data

Exponential Smoothing Method

bull Ft = At - 1 + (1-)At - 2 + (1- )2middotAt - 3

+ (1- )3At - 4 + + (1- )t-1middotA0

ndash Ft = Forecast value

ndash At = Actual value = Smoothing constant

bull Ft = Ft-1 + (At-1 - Ft-1)ndash Use for computing forecast

Exponential Smoothing Equations

During the past 8 quarters the Port of Baltimore has unloaded large quantities of grain ( = 10) The first quarter forecast was 175 QuarterActual

1 180 2 168

3 1594 1755 190

6 2057 1808 1829

Exponential Smoothing Example

Find the forecast for the 9th quarter

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

11 180 17500 (Given)

22 168168

33 159159

44 175175

55 190190

66 205205

17500 +17500 +

Exponential Smoothing Solution

QuarterQuarter ActuaActualForecast F t

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010((

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - -

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Ft

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - 17500 - 17500))

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing SolutionFt = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα= = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 +17500 + 1010(180 (180 - 17500- 17500)) = 17550 = 17550

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

1 180 17500 (Given)

22 168168 17500 + 10(180 - 17500) = 1755017500 + 10(180 - 17500) = 17550

33 159159 1755017550 ++ 1010(168 -(168 - 1755017550)) = 17475= 17475

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1995 180 17500 (Given)

1996 168 17500 + 10(180 - 17500) = 17550

1997 159 17550 + 10(168 - 17550) = 17475

1998 175

1999 190

2000 205

17475 + 10(159 - 17475)= 17318

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9 17822 + 10(182 - 17822) = 17858 182 17802 + 10(180 - 17802) = 17822

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Learning Objectives

When you complete this chapter you should be able to

Identify or Definendash Forecastingndash Types of forecastsndash Time horizonsndash Approaches to forecasts

Learning Objectives-continued

When you complete this chapter you should be able to

Describe or Explainndash Moving averagesndash Exponential smoothingndash Trend projectionsndash Regression and correlation analysisndash Measures of forecast accuracy

Forecasting at Tupperware

bull Each of 50 profit centers around the world is responsible for computerized monthly quarterly and 12-month sales projections

bull These projections are aggregated by region then globally at Tupperwarersquos World Headquarters

bull Tupperware uses all techniques discussed in text

Three Key Factors for Tupperware

bull The number of registered ldquoconsultantsrdquo or sales representatives

bull The percentage of currently ldquoactiverdquo dealers (this number changes each week and month)

bull Sales per active dealer on a weekly basis

Tupperware - Forecast by Consensus

bull Although inputs come from sales marketing finance and production final forecasts are the consensus of all participating managers

bull The final step is Tupperwarersquos version of the ldquojury of executive opinionrdquo

What is Forecasting

bull Process of predicting a future event

bull Underlying basis of all business decisions

ndash Production

ndash Inventory

ndash Personnel

ndash Facilities

Sales will be $200 Million

bull Short-range forecastndash Up to 1 year usually less than 3 monthsndash Job scheduling worker assignments

bull Medium-range forecastndash 3 months to 3 yearsndash Sales amp production planning budgeting

bull Long-range forecastndash 3+ yearsndash New product planning facility location

Types of Forecasts by Time Horizon

Short-term vs Longer-term Forecasting

bull Mediumlong range forecasts deal with more comprehensive issues and support management decisions regarding planning and products plants and processes

bull Short-term forecasting usually employs different methodologies than longer-term forecasting

bull Short-term forecasts tend to be more accurate than longer-term forecasts

Influence of Product Life Cycle

bull Stages of introduction and growth require longer forecasts than maturity and decline

bull Forecasts useful in projectingndash staffing levels

ndash inventory levels and

ndash factory capacity

as product passes through life cycle stages

Introduction Growth Maturity Decline

Strategy and Issues During a Productrsquos Life

Introduction Growth Maturity Decline

Standardization

Less rapid product changes - more minor changes

Optimum capacity

Increasing stability of process

Long production runs

Product improvement and cost cutting

Little product differentiation

Cost minimization

Over capacity in the industry

Prune line to eliminate items not returning good margin

Reduce capacity

Forecasting critical

Product and process reliability

Competitive product improvements and options

Increase capacity

Shift toward product focused

Enhance distribution

Product design and development critical

Frequent product and process design changes

Short production runs

High production costs

Limited models

Attention to quality

Best period to increase market share

RampD product engineering critical

Practical to change price or quality image

Strengthen niche

Cost control critical

Poor time to change image price or quality

Competitive costs become critical

Defend market position

OM

Str

ateg

yIs

sues

Com

pany

Str

ateg

yIs

sues

HDTV

CD-ROM

Color copiers

Drive-thru restaurants Fax machines

Station wagons

Sales

3 12rdquo Floppy disks

Internet

Types of Forecasts

bull Economic forecastsndash Address business cycle eg inflation rate

money supply etc

bull Technological forecastsndash Predict rate of technological progressndash Predict acceptance of new product

bull Demand forecastsndash Predict sales of existing product

Seven Steps in Forecasting ( 預測的七個基本步驟 )

bull Determine the use of the forecastbull Select the items to be forecastedbull Determine the time horizon of the forecastbull Select the forecasting model(s)bull Gather the databull Make the forecastbull Validate and implement results

Product Demand Charted over 4 Years with Trend and Seasonality

Year1

Year2

Year3

Year4

Seasonal peaks Trend component

Actual demand line

Average demand over four years

Dem

and

for p

rodu

ct o

r ser

vice

Random variation

Actual Demand Moving Average Weighted Moving Average

0

5

10

15

20

25

30

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Sal

es D

eman

d

Actual sales

Moving average

Weighted moving average

Realities of Forecasting

bull Forecasts are seldom perfect

bull Most forecasting methods assume that there is some underlying stability in the system

bull Both product family and aggregated product forecasts are more accurate than individual product forecasts

Forecasting Approaches

bull Used when situation is lsquostablersquo amp historical data existndash Existing productsndash Current technology

bull Involves mathematical techniquesndash eg forecasting sales of

color televisions

Quantitative Methodsbull Used when situation is

vague amp little data existndash New productsndash New technology

bull Involves intuition experiencendash eg forecasting sales on

Internet

Qualitative Methods

Overview of Qualitative Methods

bull Jury of executive opinionndash Pool opinions of high-level executives

sometimes augment by statistical models

bull Delphi methodndash Panel of experts queried iteratively

bull Sales force compositendash Estimates from individual salespersons are

reviewed for reasonableness then aggregated

bull Consumer Market Surveyndash Ask the customer

bull Involves small group of high-level managers

ndash Group estimates demand by working together

bull Combines managerial experience with statistical models

bull Relatively quick

bull lsquoGroup-thinkrsquodisadvantage

copy 1995 Corel Corp

Jury of Executive Opinion

Sales Force Composite

bull Each salesperson projects his or her sales

bull Combined at district amp national levels

bull Sales reps know customersrsquo wants

bull Tends to be overly optimistic

SalesSales

copy 1995 Corel Corp

Delphi Method

bull Iterative group process

bull 3 types of peoplendash Decision makers

ndash Staff

ndash Respondents

bull Reduces lsquogroup-thinkrsquo

Respondents Respondents

Staff Staff

Decision MakersDecision Makers(Sales)

(What will sales be survey)

(Sales will be 45 50 55)

(Sales will be 50)

Consumer Market Survey

bull Ask customers about purchasing plans

bull What consumers say and what they actually do are often different

bull Sometimes difficult to answer

How many hours will you use the Internet

next week

How many hours will you use the Internet

next week

copy 1995 Corel Corp

Overview of Quantitative Approaches

bull Naiumlve approach

bull Moving averages

bull Exponential smoothing

bull Trend projection

bull Linear regression

Time-series Models

Associative models

Quantitative Forecasting Methods (Non-Naive)

QuantitativeForecasting

LinearRegression

AssociativeModels

ExponentialSmoothing

MovingAverage

Time SeriesModels

TrendProjection

bull Set of evenly spaced numerical datandash Obtained by observing response variable at

regular time periods

bull Forecast based only on past valuesndash Assumes that factors influencing past and

present will continue influence in future

bull ExampleYear 1998 1999 2000 2001 2002Sales 787 635 897 932 921

What is a Time Series

TrendTrend

SeasonalSeasonal

CyclicalCyclical

RandomRandom

Time Series Components

bull Persistent overall upward or downward pattern

bull Due to population technology etcbull Several years duration

Mo Qtr Yr

Response

copy 1984-1994 TMaker Co

Trend Component

bull Regular pattern of up amp down fluctuations

bull Due to weather customs etcbull Occurs within 1 year

Mo Qtr

Response

Summer

copy 1984-1994 TMaker Co

Seasonal Component

Common Seasonal PatternsPeriod of Pattern

ldquoSeasonrdquo Length

Number of

ldquoSeasonsrdquo in

Pattern

Week Day 7

Month Week 4 ndash 4 frac12

Month Day 28 ndash 31

Year Quarter 4

Year Month 12

Year Week 52

bull Repeating up amp down movementsbull Due to interactions of factors

influencing economybull Usually 2-10 years duration

Mo Qtr YrMo Qtr Yr

ResponseResponseCycle

Cyclical Component

bull Erratic unsystematic lsquoresidualrsquo fluctuations

bull Due to random variation or unforeseen

events

ndash Union strike

ndash Tornado

bull Short duration amp

nonrepeating

copy 1984-1994 TMaker Co

Random Component

bull Any observed value in a time series is the product (or sum) of time series components

bull Multiplicative modelndash Yi = Ti middot Si middot Ci middot Ri (if quarterly or mo data)

bull Additive modelndash Yi = Ti + Si + Ci + Ri (if quarterly or mo

data)

General Time Series Models

Naive Approach

bull Assumes demand in next period is the same as demand in most recent periodndash eg If May sales were 48

then June sales will be 48

bull Sometimes cost effective amp efficient

copy 1995 Corel Corp

bull MA is a series of arithmetic means

bull Used if little or no trend

bull Used often for smoothingndash Provides overall impression of data over

timebull Equation

MAMAnn

nn Demand inDemand in PreviousPrevious PeriodsPeriods

Moving Average Method

Yoursquore manager of a museum store that sells historical replicas You want to forecast sales (000) for 2003 using a 3-period moving average

1998 41999 62000 52001 32002 7

copy 1995 Corel Corp

Moving Average Example

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 6+5+3=14 143=4 23 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153=50 2002 7 6+5+3=14 143=47 2003 NA 5+3+7=15 153=50

95 96 97 98 99 00Year

Sales

2

4

6

8 Actual

Forecast

Moving Average Graph

bull Used when trend is present ndash Older data usually less important

bull Weights based on intuitionndash Often lay between 0 amp 1 amp sum to 10

bull Equation

WMA =WMA =ΣΣ(Weight for period (Weight for period nn) (Demand in period ) (Demand in period nn))

ΣΣWeightsWeights

Weighted Moving Average Method

Actual Demand Moving Average Weighted Moving

Average

0

5

10

15

20

25

30

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Sal

es D

eman

d

Actual sales

Moving average

Weighted moving average

bull Increasing n makes forecast less sensitive to changes

bull Do not forecast trend wellbull Require much historical data

copy 1984-1994 TMaker Co

Disadvantages of Moving Average Methods

bull Form of weighted moving averagendash Weights decline exponentiallyndash Most recent data weighted most

bull Requires smoothing constant ()ndash Ranges from 0 to 1ndash Subjectively chosen

bull Involves little record keeping of past data

Exponential Smoothing Method

bull Ft = At - 1 + (1-)At - 2 + (1- )2middotAt - 3

+ (1- )3At - 4 + + (1- )t-1middotA0

ndash Ft = Forecast value

ndash At = Actual value = Smoothing constant

bull Ft = Ft-1 + (At-1 - Ft-1)ndash Use for computing forecast

Exponential Smoothing Equations

During the past 8 quarters the Port of Baltimore has unloaded large quantities of grain ( = 10) The first quarter forecast was 175 QuarterActual

1 180 2 168

3 1594 1755 190

6 2057 1808 1829

Exponential Smoothing Example

Find the forecast for the 9th quarter

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

11 180 17500 (Given)

22 168168

33 159159

44 175175

55 190190

66 205205

17500 +17500 +

Exponential Smoothing Solution

QuarterQuarter ActuaActualForecast F t

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010((

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - -

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Ft

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - 17500 - 17500))

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing SolutionFt = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα= = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 +17500 + 1010(180 (180 - 17500- 17500)) = 17550 = 17550

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

1 180 17500 (Given)

22 168168 17500 + 10(180 - 17500) = 1755017500 + 10(180 - 17500) = 17550

33 159159 1755017550 ++ 1010(168 -(168 - 1755017550)) = 17475= 17475

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1995 180 17500 (Given)

1996 168 17500 + 10(180 - 17500) = 17550

1997 159 17550 + 10(168 - 17550) = 17475

1998 175

1999 190

2000 205

17475 + 10(159 - 17475)= 17318

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9 17822 + 10(182 - 17822) = 17858 182 17802 + 10(180 - 17802) = 17822

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Learning Objectives-continued

When you complete this chapter you should be able to

Describe or Explainndash Moving averagesndash Exponential smoothingndash Trend projectionsndash Regression and correlation analysisndash Measures of forecast accuracy

Forecasting at Tupperware

bull Each of 50 profit centers around the world is responsible for computerized monthly quarterly and 12-month sales projections

bull These projections are aggregated by region then globally at Tupperwarersquos World Headquarters

bull Tupperware uses all techniques discussed in text

Three Key Factors for Tupperware

bull The number of registered ldquoconsultantsrdquo or sales representatives

bull The percentage of currently ldquoactiverdquo dealers (this number changes each week and month)

bull Sales per active dealer on a weekly basis

Tupperware - Forecast by Consensus

bull Although inputs come from sales marketing finance and production final forecasts are the consensus of all participating managers

bull The final step is Tupperwarersquos version of the ldquojury of executive opinionrdquo

What is Forecasting

bull Process of predicting a future event

bull Underlying basis of all business decisions

ndash Production

ndash Inventory

ndash Personnel

ndash Facilities

Sales will be $200 Million

bull Short-range forecastndash Up to 1 year usually less than 3 monthsndash Job scheduling worker assignments

bull Medium-range forecastndash 3 months to 3 yearsndash Sales amp production planning budgeting

bull Long-range forecastndash 3+ yearsndash New product planning facility location

Types of Forecasts by Time Horizon

Short-term vs Longer-term Forecasting

bull Mediumlong range forecasts deal with more comprehensive issues and support management decisions regarding planning and products plants and processes

bull Short-term forecasting usually employs different methodologies than longer-term forecasting

bull Short-term forecasts tend to be more accurate than longer-term forecasts

Influence of Product Life Cycle

bull Stages of introduction and growth require longer forecasts than maturity and decline

bull Forecasts useful in projectingndash staffing levels

ndash inventory levels and

ndash factory capacity

as product passes through life cycle stages

Introduction Growth Maturity Decline

Strategy and Issues During a Productrsquos Life

Introduction Growth Maturity Decline

Standardization

Less rapid product changes - more minor changes

Optimum capacity

Increasing stability of process

Long production runs

Product improvement and cost cutting

Little product differentiation

Cost minimization

Over capacity in the industry

Prune line to eliminate items not returning good margin

Reduce capacity

Forecasting critical

Product and process reliability

Competitive product improvements and options

Increase capacity

Shift toward product focused

Enhance distribution

Product design and development critical

Frequent product and process design changes

Short production runs

High production costs

Limited models

Attention to quality

Best period to increase market share

RampD product engineering critical

Practical to change price or quality image

Strengthen niche

Cost control critical

Poor time to change image price or quality

Competitive costs become critical

Defend market position

OM

Str

ateg

yIs

sues

Com

pany

Str

ateg

yIs

sues

HDTV

CD-ROM

Color copiers

Drive-thru restaurants Fax machines

Station wagons

Sales

3 12rdquo Floppy disks

Internet

Types of Forecasts

bull Economic forecastsndash Address business cycle eg inflation rate

money supply etc

bull Technological forecastsndash Predict rate of technological progressndash Predict acceptance of new product

bull Demand forecastsndash Predict sales of existing product

Seven Steps in Forecasting ( 預測的七個基本步驟 )

bull Determine the use of the forecastbull Select the items to be forecastedbull Determine the time horizon of the forecastbull Select the forecasting model(s)bull Gather the databull Make the forecastbull Validate and implement results

Product Demand Charted over 4 Years with Trend and Seasonality

Year1

Year2

Year3

Year4

Seasonal peaks Trend component

Actual demand line

Average demand over four years

Dem

and

for p

rodu

ct o

r ser

vice

Random variation

Actual Demand Moving Average Weighted Moving Average

0

5

10

15

20

25

30

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Sal

es D

eman

d

Actual sales

Moving average

Weighted moving average

Realities of Forecasting

bull Forecasts are seldom perfect

bull Most forecasting methods assume that there is some underlying stability in the system

bull Both product family and aggregated product forecasts are more accurate than individual product forecasts

Forecasting Approaches

bull Used when situation is lsquostablersquo amp historical data existndash Existing productsndash Current technology

bull Involves mathematical techniquesndash eg forecasting sales of

color televisions

Quantitative Methodsbull Used when situation is

vague amp little data existndash New productsndash New technology

bull Involves intuition experiencendash eg forecasting sales on

Internet

Qualitative Methods

Overview of Qualitative Methods

bull Jury of executive opinionndash Pool opinions of high-level executives

sometimes augment by statistical models

bull Delphi methodndash Panel of experts queried iteratively

bull Sales force compositendash Estimates from individual salespersons are

reviewed for reasonableness then aggregated

bull Consumer Market Surveyndash Ask the customer

bull Involves small group of high-level managers

ndash Group estimates demand by working together

bull Combines managerial experience with statistical models

bull Relatively quick

bull lsquoGroup-thinkrsquodisadvantage

copy 1995 Corel Corp

Jury of Executive Opinion

Sales Force Composite

bull Each salesperson projects his or her sales

bull Combined at district amp national levels

bull Sales reps know customersrsquo wants

bull Tends to be overly optimistic

SalesSales

copy 1995 Corel Corp

Delphi Method

bull Iterative group process

bull 3 types of peoplendash Decision makers

ndash Staff

ndash Respondents

bull Reduces lsquogroup-thinkrsquo

Respondents Respondents

Staff Staff

Decision MakersDecision Makers(Sales)

(What will sales be survey)

(Sales will be 45 50 55)

(Sales will be 50)

Consumer Market Survey

bull Ask customers about purchasing plans

bull What consumers say and what they actually do are often different

bull Sometimes difficult to answer

How many hours will you use the Internet

next week

How many hours will you use the Internet

next week

copy 1995 Corel Corp

Overview of Quantitative Approaches

bull Naiumlve approach

bull Moving averages

bull Exponential smoothing

bull Trend projection

bull Linear regression

Time-series Models

Associative models

Quantitative Forecasting Methods (Non-Naive)

QuantitativeForecasting

LinearRegression

AssociativeModels

ExponentialSmoothing

MovingAverage

Time SeriesModels

TrendProjection

bull Set of evenly spaced numerical datandash Obtained by observing response variable at

regular time periods

bull Forecast based only on past valuesndash Assumes that factors influencing past and

present will continue influence in future

bull ExampleYear 1998 1999 2000 2001 2002Sales 787 635 897 932 921

What is a Time Series

TrendTrend

SeasonalSeasonal

CyclicalCyclical

RandomRandom

Time Series Components

bull Persistent overall upward or downward pattern

bull Due to population technology etcbull Several years duration

Mo Qtr Yr

Response

copy 1984-1994 TMaker Co

Trend Component

bull Regular pattern of up amp down fluctuations

bull Due to weather customs etcbull Occurs within 1 year

Mo Qtr

Response

Summer

copy 1984-1994 TMaker Co

Seasonal Component

Common Seasonal PatternsPeriod of Pattern

ldquoSeasonrdquo Length

Number of

ldquoSeasonsrdquo in

Pattern

Week Day 7

Month Week 4 ndash 4 frac12

Month Day 28 ndash 31

Year Quarter 4

Year Month 12

Year Week 52

bull Repeating up amp down movementsbull Due to interactions of factors

influencing economybull Usually 2-10 years duration

Mo Qtr YrMo Qtr Yr

ResponseResponseCycle

Cyclical Component

bull Erratic unsystematic lsquoresidualrsquo fluctuations

bull Due to random variation or unforeseen

events

ndash Union strike

ndash Tornado

bull Short duration amp

nonrepeating

copy 1984-1994 TMaker Co

Random Component

bull Any observed value in a time series is the product (or sum) of time series components

bull Multiplicative modelndash Yi = Ti middot Si middot Ci middot Ri (if quarterly or mo data)

bull Additive modelndash Yi = Ti + Si + Ci + Ri (if quarterly or mo

data)

General Time Series Models

Naive Approach

bull Assumes demand in next period is the same as demand in most recent periodndash eg If May sales were 48

then June sales will be 48

bull Sometimes cost effective amp efficient

copy 1995 Corel Corp

bull MA is a series of arithmetic means

bull Used if little or no trend

bull Used often for smoothingndash Provides overall impression of data over

timebull Equation

MAMAnn

nn Demand inDemand in PreviousPrevious PeriodsPeriods

Moving Average Method

Yoursquore manager of a museum store that sells historical replicas You want to forecast sales (000) for 2003 using a 3-period moving average

1998 41999 62000 52001 32002 7

copy 1995 Corel Corp

Moving Average Example

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 6+5+3=14 143=4 23 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153=50 2002 7 6+5+3=14 143=47 2003 NA 5+3+7=15 153=50

95 96 97 98 99 00Year

Sales

2

4

6

8 Actual

Forecast

Moving Average Graph

bull Used when trend is present ndash Older data usually less important

bull Weights based on intuitionndash Often lay between 0 amp 1 amp sum to 10

bull Equation

WMA =WMA =ΣΣ(Weight for period (Weight for period nn) (Demand in period ) (Demand in period nn))

ΣΣWeightsWeights

Weighted Moving Average Method

Actual Demand Moving Average Weighted Moving

Average

0

5

10

15

20

25

30

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Sal

es D

eman

d

Actual sales

Moving average

Weighted moving average

bull Increasing n makes forecast less sensitive to changes

bull Do not forecast trend wellbull Require much historical data

copy 1984-1994 TMaker Co

Disadvantages of Moving Average Methods

bull Form of weighted moving averagendash Weights decline exponentiallyndash Most recent data weighted most

bull Requires smoothing constant ()ndash Ranges from 0 to 1ndash Subjectively chosen

bull Involves little record keeping of past data

Exponential Smoothing Method

bull Ft = At - 1 + (1-)At - 2 + (1- )2middotAt - 3

+ (1- )3At - 4 + + (1- )t-1middotA0

ndash Ft = Forecast value

ndash At = Actual value = Smoothing constant

bull Ft = Ft-1 + (At-1 - Ft-1)ndash Use for computing forecast

Exponential Smoothing Equations

During the past 8 quarters the Port of Baltimore has unloaded large quantities of grain ( = 10) The first quarter forecast was 175 QuarterActual

1 180 2 168

3 1594 1755 190

6 2057 1808 1829

Exponential Smoothing Example

Find the forecast for the 9th quarter

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

11 180 17500 (Given)

22 168168

33 159159

44 175175

55 190190

66 205205

17500 +17500 +

Exponential Smoothing Solution

QuarterQuarter ActuaActualForecast F t

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010((

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - -

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Ft

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - 17500 - 17500))

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing SolutionFt = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα= = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 +17500 + 1010(180 (180 - 17500- 17500)) = 17550 = 17550

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

1 180 17500 (Given)

22 168168 17500 + 10(180 - 17500) = 1755017500 + 10(180 - 17500) = 17550

33 159159 1755017550 ++ 1010(168 -(168 - 1755017550)) = 17475= 17475

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1995 180 17500 (Given)

1996 168 17500 + 10(180 - 17500) = 17550

1997 159 17550 + 10(168 - 17550) = 17475

1998 175

1999 190

2000 205

17475 + 10(159 - 17475)= 17318

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9 17822 + 10(182 - 17822) = 17858 182 17802 + 10(180 - 17802) = 17822

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Forecasting at Tupperware

bull Each of 50 profit centers around the world is responsible for computerized monthly quarterly and 12-month sales projections

bull These projections are aggregated by region then globally at Tupperwarersquos World Headquarters

bull Tupperware uses all techniques discussed in text

Three Key Factors for Tupperware

bull The number of registered ldquoconsultantsrdquo or sales representatives

bull The percentage of currently ldquoactiverdquo dealers (this number changes each week and month)

bull Sales per active dealer on a weekly basis

Tupperware - Forecast by Consensus

bull Although inputs come from sales marketing finance and production final forecasts are the consensus of all participating managers

bull The final step is Tupperwarersquos version of the ldquojury of executive opinionrdquo

What is Forecasting

bull Process of predicting a future event

bull Underlying basis of all business decisions

ndash Production

ndash Inventory

ndash Personnel

ndash Facilities

Sales will be $200 Million

bull Short-range forecastndash Up to 1 year usually less than 3 monthsndash Job scheduling worker assignments

bull Medium-range forecastndash 3 months to 3 yearsndash Sales amp production planning budgeting

bull Long-range forecastndash 3+ yearsndash New product planning facility location

Types of Forecasts by Time Horizon

Short-term vs Longer-term Forecasting

bull Mediumlong range forecasts deal with more comprehensive issues and support management decisions regarding planning and products plants and processes

bull Short-term forecasting usually employs different methodologies than longer-term forecasting

bull Short-term forecasts tend to be more accurate than longer-term forecasts

Influence of Product Life Cycle

bull Stages of introduction and growth require longer forecasts than maturity and decline

bull Forecasts useful in projectingndash staffing levels

ndash inventory levels and

ndash factory capacity

as product passes through life cycle stages

Introduction Growth Maturity Decline

Strategy and Issues During a Productrsquos Life

Introduction Growth Maturity Decline

Standardization

Less rapid product changes - more minor changes

Optimum capacity

Increasing stability of process

Long production runs

Product improvement and cost cutting

Little product differentiation

Cost minimization

Over capacity in the industry

Prune line to eliminate items not returning good margin

Reduce capacity

Forecasting critical

Product and process reliability

Competitive product improvements and options

Increase capacity

Shift toward product focused

Enhance distribution

Product design and development critical

Frequent product and process design changes

Short production runs

High production costs

Limited models

Attention to quality

Best period to increase market share

RampD product engineering critical

Practical to change price or quality image

Strengthen niche

Cost control critical

Poor time to change image price or quality

Competitive costs become critical

Defend market position

OM

Str

ateg

yIs

sues

Com

pany

Str

ateg

yIs

sues

HDTV

CD-ROM

Color copiers

Drive-thru restaurants Fax machines

Station wagons

Sales

3 12rdquo Floppy disks

Internet

Types of Forecasts

bull Economic forecastsndash Address business cycle eg inflation rate

money supply etc

bull Technological forecastsndash Predict rate of technological progressndash Predict acceptance of new product

bull Demand forecastsndash Predict sales of existing product

Seven Steps in Forecasting ( 預測的七個基本步驟 )

bull Determine the use of the forecastbull Select the items to be forecastedbull Determine the time horizon of the forecastbull Select the forecasting model(s)bull Gather the databull Make the forecastbull Validate and implement results

Product Demand Charted over 4 Years with Trend and Seasonality

Year1

Year2

Year3

Year4

Seasonal peaks Trend component

Actual demand line

Average demand over four years

Dem

and

for p

rodu

ct o

r ser

vice

Random variation

Actual Demand Moving Average Weighted Moving Average

0

5

10

15

20

25

30

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Sal

es D

eman

d

Actual sales

Moving average

Weighted moving average

Realities of Forecasting

bull Forecasts are seldom perfect

bull Most forecasting methods assume that there is some underlying stability in the system

bull Both product family and aggregated product forecasts are more accurate than individual product forecasts

Forecasting Approaches

bull Used when situation is lsquostablersquo amp historical data existndash Existing productsndash Current technology

bull Involves mathematical techniquesndash eg forecasting sales of

color televisions

Quantitative Methodsbull Used when situation is

vague amp little data existndash New productsndash New technology

bull Involves intuition experiencendash eg forecasting sales on

Internet

Qualitative Methods

Overview of Qualitative Methods

bull Jury of executive opinionndash Pool opinions of high-level executives

sometimes augment by statistical models

bull Delphi methodndash Panel of experts queried iteratively

bull Sales force compositendash Estimates from individual salespersons are

reviewed for reasonableness then aggregated

bull Consumer Market Surveyndash Ask the customer

bull Involves small group of high-level managers

ndash Group estimates demand by working together

bull Combines managerial experience with statistical models

bull Relatively quick

bull lsquoGroup-thinkrsquodisadvantage

copy 1995 Corel Corp

Jury of Executive Opinion

Sales Force Composite

bull Each salesperson projects his or her sales

bull Combined at district amp national levels

bull Sales reps know customersrsquo wants

bull Tends to be overly optimistic

SalesSales

copy 1995 Corel Corp

Delphi Method

bull Iterative group process

bull 3 types of peoplendash Decision makers

ndash Staff

ndash Respondents

bull Reduces lsquogroup-thinkrsquo

Respondents Respondents

Staff Staff

Decision MakersDecision Makers(Sales)

(What will sales be survey)

(Sales will be 45 50 55)

(Sales will be 50)

Consumer Market Survey

bull Ask customers about purchasing plans

bull What consumers say and what they actually do are often different

bull Sometimes difficult to answer

How many hours will you use the Internet

next week

How many hours will you use the Internet

next week

copy 1995 Corel Corp

Overview of Quantitative Approaches

bull Naiumlve approach

bull Moving averages

bull Exponential smoothing

bull Trend projection

bull Linear regression

Time-series Models

Associative models

Quantitative Forecasting Methods (Non-Naive)

QuantitativeForecasting

LinearRegression

AssociativeModels

ExponentialSmoothing

MovingAverage

Time SeriesModels

TrendProjection

bull Set of evenly spaced numerical datandash Obtained by observing response variable at

regular time periods

bull Forecast based only on past valuesndash Assumes that factors influencing past and

present will continue influence in future

bull ExampleYear 1998 1999 2000 2001 2002Sales 787 635 897 932 921

What is a Time Series

TrendTrend

SeasonalSeasonal

CyclicalCyclical

RandomRandom

Time Series Components

bull Persistent overall upward or downward pattern

bull Due to population technology etcbull Several years duration

Mo Qtr Yr

Response

copy 1984-1994 TMaker Co

Trend Component

bull Regular pattern of up amp down fluctuations

bull Due to weather customs etcbull Occurs within 1 year

Mo Qtr

Response

Summer

copy 1984-1994 TMaker Co

Seasonal Component

Common Seasonal PatternsPeriod of Pattern

ldquoSeasonrdquo Length

Number of

ldquoSeasonsrdquo in

Pattern

Week Day 7

Month Week 4 ndash 4 frac12

Month Day 28 ndash 31

Year Quarter 4

Year Month 12

Year Week 52

bull Repeating up amp down movementsbull Due to interactions of factors

influencing economybull Usually 2-10 years duration

Mo Qtr YrMo Qtr Yr

ResponseResponseCycle

Cyclical Component

bull Erratic unsystematic lsquoresidualrsquo fluctuations

bull Due to random variation or unforeseen

events

ndash Union strike

ndash Tornado

bull Short duration amp

nonrepeating

copy 1984-1994 TMaker Co

Random Component

bull Any observed value in a time series is the product (or sum) of time series components

bull Multiplicative modelndash Yi = Ti middot Si middot Ci middot Ri (if quarterly or mo data)

bull Additive modelndash Yi = Ti + Si + Ci + Ri (if quarterly or mo

data)

General Time Series Models

Naive Approach

bull Assumes demand in next period is the same as demand in most recent periodndash eg If May sales were 48

then June sales will be 48

bull Sometimes cost effective amp efficient

copy 1995 Corel Corp

bull MA is a series of arithmetic means

bull Used if little or no trend

bull Used often for smoothingndash Provides overall impression of data over

timebull Equation

MAMAnn

nn Demand inDemand in PreviousPrevious PeriodsPeriods

Moving Average Method

Yoursquore manager of a museum store that sells historical replicas You want to forecast sales (000) for 2003 using a 3-period moving average

1998 41999 62000 52001 32002 7

copy 1995 Corel Corp

Moving Average Example

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 6+5+3=14 143=4 23 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153=50 2002 7 6+5+3=14 143=47 2003 NA 5+3+7=15 153=50

95 96 97 98 99 00Year

Sales

2

4

6

8 Actual

Forecast

Moving Average Graph

bull Used when trend is present ndash Older data usually less important

bull Weights based on intuitionndash Often lay between 0 amp 1 amp sum to 10

bull Equation

WMA =WMA =ΣΣ(Weight for period (Weight for period nn) (Demand in period ) (Demand in period nn))

ΣΣWeightsWeights

Weighted Moving Average Method

Actual Demand Moving Average Weighted Moving

Average

0

5

10

15

20

25

30

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Sal

es D

eman

d

Actual sales

Moving average

Weighted moving average

bull Increasing n makes forecast less sensitive to changes

bull Do not forecast trend wellbull Require much historical data

copy 1984-1994 TMaker Co

Disadvantages of Moving Average Methods

bull Form of weighted moving averagendash Weights decline exponentiallyndash Most recent data weighted most

bull Requires smoothing constant ()ndash Ranges from 0 to 1ndash Subjectively chosen

bull Involves little record keeping of past data

Exponential Smoothing Method

bull Ft = At - 1 + (1-)At - 2 + (1- )2middotAt - 3

+ (1- )3At - 4 + + (1- )t-1middotA0

ndash Ft = Forecast value

ndash At = Actual value = Smoothing constant

bull Ft = Ft-1 + (At-1 - Ft-1)ndash Use for computing forecast

Exponential Smoothing Equations

During the past 8 quarters the Port of Baltimore has unloaded large quantities of grain ( = 10) The first quarter forecast was 175 QuarterActual

1 180 2 168

3 1594 1755 190

6 2057 1808 1829

Exponential Smoothing Example

Find the forecast for the 9th quarter

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

11 180 17500 (Given)

22 168168

33 159159

44 175175

55 190190

66 205205

17500 +17500 +

Exponential Smoothing Solution

QuarterQuarter ActuaActualForecast F t

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010((

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - -

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Ft

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - 17500 - 17500))

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing SolutionFt = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα= = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 +17500 + 1010(180 (180 - 17500- 17500)) = 17550 = 17550

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

1 180 17500 (Given)

22 168168 17500 + 10(180 - 17500) = 1755017500 + 10(180 - 17500) = 17550

33 159159 1755017550 ++ 1010(168 -(168 - 1755017550)) = 17475= 17475

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1995 180 17500 (Given)

1996 168 17500 + 10(180 - 17500) = 17550

1997 159 17550 + 10(168 - 17550) = 17475

1998 175

1999 190

2000 205

17475 + 10(159 - 17475)= 17318

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9 17822 + 10(182 - 17822) = 17858 182 17802 + 10(180 - 17802) = 17822

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Three Key Factors for Tupperware

bull The number of registered ldquoconsultantsrdquo or sales representatives

bull The percentage of currently ldquoactiverdquo dealers (this number changes each week and month)

bull Sales per active dealer on a weekly basis

Tupperware - Forecast by Consensus

bull Although inputs come from sales marketing finance and production final forecasts are the consensus of all participating managers

bull The final step is Tupperwarersquos version of the ldquojury of executive opinionrdquo

What is Forecasting

bull Process of predicting a future event

bull Underlying basis of all business decisions

ndash Production

ndash Inventory

ndash Personnel

ndash Facilities

Sales will be $200 Million

bull Short-range forecastndash Up to 1 year usually less than 3 monthsndash Job scheduling worker assignments

bull Medium-range forecastndash 3 months to 3 yearsndash Sales amp production planning budgeting

bull Long-range forecastndash 3+ yearsndash New product planning facility location

Types of Forecasts by Time Horizon

Short-term vs Longer-term Forecasting

bull Mediumlong range forecasts deal with more comprehensive issues and support management decisions regarding planning and products plants and processes

bull Short-term forecasting usually employs different methodologies than longer-term forecasting

bull Short-term forecasts tend to be more accurate than longer-term forecasts

Influence of Product Life Cycle

bull Stages of introduction and growth require longer forecasts than maturity and decline

bull Forecasts useful in projectingndash staffing levels

ndash inventory levels and

ndash factory capacity

as product passes through life cycle stages

Introduction Growth Maturity Decline

Strategy and Issues During a Productrsquos Life

Introduction Growth Maturity Decline

Standardization

Less rapid product changes - more minor changes

Optimum capacity

Increasing stability of process

Long production runs

Product improvement and cost cutting

Little product differentiation

Cost minimization

Over capacity in the industry

Prune line to eliminate items not returning good margin

Reduce capacity

Forecasting critical

Product and process reliability

Competitive product improvements and options

Increase capacity

Shift toward product focused

Enhance distribution

Product design and development critical

Frequent product and process design changes

Short production runs

High production costs

Limited models

Attention to quality

Best period to increase market share

RampD product engineering critical

Practical to change price or quality image

Strengthen niche

Cost control critical

Poor time to change image price or quality

Competitive costs become critical

Defend market position

OM

Str

ateg

yIs

sues

Com

pany

Str

ateg

yIs

sues

HDTV

CD-ROM

Color copiers

Drive-thru restaurants Fax machines

Station wagons

Sales

3 12rdquo Floppy disks

Internet

Types of Forecasts

bull Economic forecastsndash Address business cycle eg inflation rate

money supply etc

bull Technological forecastsndash Predict rate of technological progressndash Predict acceptance of new product

bull Demand forecastsndash Predict sales of existing product

Seven Steps in Forecasting ( 預測的七個基本步驟 )

bull Determine the use of the forecastbull Select the items to be forecastedbull Determine the time horizon of the forecastbull Select the forecasting model(s)bull Gather the databull Make the forecastbull Validate and implement results

Product Demand Charted over 4 Years with Trend and Seasonality

Year1

Year2

Year3

Year4

Seasonal peaks Trend component

Actual demand line

Average demand over four years

Dem

and

for p

rodu

ct o

r ser

vice

Random variation

Actual Demand Moving Average Weighted Moving Average

0

5

10

15

20

25

30

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Sal

es D

eman

d

Actual sales

Moving average

Weighted moving average

Realities of Forecasting

bull Forecasts are seldom perfect

bull Most forecasting methods assume that there is some underlying stability in the system

bull Both product family and aggregated product forecasts are more accurate than individual product forecasts

Forecasting Approaches

bull Used when situation is lsquostablersquo amp historical data existndash Existing productsndash Current technology

bull Involves mathematical techniquesndash eg forecasting sales of

color televisions

Quantitative Methodsbull Used when situation is

vague amp little data existndash New productsndash New technology

bull Involves intuition experiencendash eg forecasting sales on

Internet

Qualitative Methods

Overview of Qualitative Methods

bull Jury of executive opinionndash Pool opinions of high-level executives

sometimes augment by statistical models

bull Delphi methodndash Panel of experts queried iteratively

bull Sales force compositendash Estimates from individual salespersons are

reviewed for reasonableness then aggregated

bull Consumer Market Surveyndash Ask the customer

bull Involves small group of high-level managers

ndash Group estimates demand by working together

bull Combines managerial experience with statistical models

bull Relatively quick

bull lsquoGroup-thinkrsquodisadvantage

copy 1995 Corel Corp

Jury of Executive Opinion

Sales Force Composite

bull Each salesperson projects his or her sales

bull Combined at district amp national levels

bull Sales reps know customersrsquo wants

bull Tends to be overly optimistic

SalesSales

copy 1995 Corel Corp

Delphi Method

bull Iterative group process

bull 3 types of peoplendash Decision makers

ndash Staff

ndash Respondents

bull Reduces lsquogroup-thinkrsquo

Respondents Respondents

Staff Staff

Decision MakersDecision Makers(Sales)

(What will sales be survey)

(Sales will be 45 50 55)

(Sales will be 50)

Consumer Market Survey

bull Ask customers about purchasing plans

bull What consumers say and what they actually do are often different

bull Sometimes difficult to answer

How many hours will you use the Internet

next week

How many hours will you use the Internet

next week

copy 1995 Corel Corp

Overview of Quantitative Approaches

bull Naiumlve approach

bull Moving averages

bull Exponential smoothing

bull Trend projection

bull Linear regression

Time-series Models

Associative models

Quantitative Forecasting Methods (Non-Naive)

QuantitativeForecasting

LinearRegression

AssociativeModels

ExponentialSmoothing

MovingAverage

Time SeriesModels

TrendProjection

bull Set of evenly spaced numerical datandash Obtained by observing response variable at

regular time periods

bull Forecast based only on past valuesndash Assumes that factors influencing past and

present will continue influence in future

bull ExampleYear 1998 1999 2000 2001 2002Sales 787 635 897 932 921

What is a Time Series

TrendTrend

SeasonalSeasonal

CyclicalCyclical

RandomRandom

Time Series Components

bull Persistent overall upward or downward pattern

bull Due to population technology etcbull Several years duration

Mo Qtr Yr

Response

copy 1984-1994 TMaker Co

Trend Component

bull Regular pattern of up amp down fluctuations

bull Due to weather customs etcbull Occurs within 1 year

Mo Qtr

Response

Summer

copy 1984-1994 TMaker Co

Seasonal Component

Common Seasonal PatternsPeriod of Pattern

ldquoSeasonrdquo Length

Number of

ldquoSeasonsrdquo in

Pattern

Week Day 7

Month Week 4 ndash 4 frac12

Month Day 28 ndash 31

Year Quarter 4

Year Month 12

Year Week 52

bull Repeating up amp down movementsbull Due to interactions of factors

influencing economybull Usually 2-10 years duration

Mo Qtr YrMo Qtr Yr

ResponseResponseCycle

Cyclical Component

bull Erratic unsystematic lsquoresidualrsquo fluctuations

bull Due to random variation or unforeseen

events

ndash Union strike

ndash Tornado

bull Short duration amp

nonrepeating

copy 1984-1994 TMaker Co

Random Component

bull Any observed value in a time series is the product (or sum) of time series components

bull Multiplicative modelndash Yi = Ti middot Si middot Ci middot Ri (if quarterly or mo data)

bull Additive modelndash Yi = Ti + Si + Ci + Ri (if quarterly or mo

data)

General Time Series Models

Naive Approach

bull Assumes demand in next period is the same as demand in most recent periodndash eg If May sales were 48

then June sales will be 48

bull Sometimes cost effective amp efficient

copy 1995 Corel Corp

bull MA is a series of arithmetic means

bull Used if little or no trend

bull Used often for smoothingndash Provides overall impression of data over

timebull Equation

MAMAnn

nn Demand inDemand in PreviousPrevious PeriodsPeriods

Moving Average Method

Yoursquore manager of a museum store that sells historical replicas You want to forecast sales (000) for 2003 using a 3-period moving average

1998 41999 62000 52001 32002 7

copy 1995 Corel Corp

Moving Average Example

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 6+5+3=14 143=4 23 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153=50 2002 7 6+5+3=14 143=47 2003 NA 5+3+7=15 153=50

95 96 97 98 99 00Year

Sales

2

4

6

8 Actual

Forecast

Moving Average Graph

bull Used when trend is present ndash Older data usually less important

bull Weights based on intuitionndash Often lay between 0 amp 1 amp sum to 10

bull Equation

WMA =WMA =ΣΣ(Weight for period (Weight for period nn) (Demand in period ) (Demand in period nn))

ΣΣWeightsWeights

Weighted Moving Average Method

Actual Demand Moving Average Weighted Moving

Average

0

5

10

15

20

25

30

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Sal

es D

eman

d

Actual sales

Moving average

Weighted moving average

bull Increasing n makes forecast less sensitive to changes

bull Do not forecast trend wellbull Require much historical data

copy 1984-1994 TMaker Co

Disadvantages of Moving Average Methods

bull Form of weighted moving averagendash Weights decline exponentiallyndash Most recent data weighted most

bull Requires smoothing constant ()ndash Ranges from 0 to 1ndash Subjectively chosen

bull Involves little record keeping of past data

Exponential Smoothing Method

bull Ft = At - 1 + (1-)At - 2 + (1- )2middotAt - 3

+ (1- )3At - 4 + + (1- )t-1middotA0

ndash Ft = Forecast value

ndash At = Actual value = Smoothing constant

bull Ft = Ft-1 + (At-1 - Ft-1)ndash Use for computing forecast

Exponential Smoothing Equations

During the past 8 quarters the Port of Baltimore has unloaded large quantities of grain ( = 10) The first quarter forecast was 175 QuarterActual

1 180 2 168

3 1594 1755 190

6 2057 1808 1829

Exponential Smoothing Example

Find the forecast for the 9th quarter

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

11 180 17500 (Given)

22 168168

33 159159

44 175175

55 190190

66 205205

17500 +17500 +

Exponential Smoothing Solution

QuarterQuarter ActuaActualForecast F t

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010((

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - -

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Ft

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - 17500 - 17500))

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing SolutionFt = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα= = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 +17500 + 1010(180 (180 - 17500- 17500)) = 17550 = 17550

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

1 180 17500 (Given)

22 168168 17500 + 10(180 - 17500) = 1755017500 + 10(180 - 17500) = 17550

33 159159 1755017550 ++ 1010(168 -(168 - 1755017550)) = 17475= 17475

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1995 180 17500 (Given)

1996 168 17500 + 10(180 - 17500) = 17550

1997 159 17550 + 10(168 - 17550) = 17475

1998 175

1999 190

2000 205

17475 + 10(159 - 17475)= 17318

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9 17822 + 10(182 - 17822) = 17858 182 17802 + 10(180 - 17802) = 17822

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Tupperware - Forecast by Consensus

bull Although inputs come from sales marketing finance and production final forecasts are the consensus of all participating managers

bull The final step is Tupperwarersquos version of the ldquojury of executive opinionrdquo

What is Forecasting

bull Process of predicting a future event

bull Underlying basis of all business decisions

ndash Production

ndash Inventory

ndash Personnel

ndash Facilities

Sales will be $200 Million

bull Short-range forecastndash Up to 1 year usually less than 3 monthsndash Job scheduling worker assignments

bull Medium-range forecastndash 3 months to 3 yearsndash Sales amp production planning budgeting

bull Long-range forecastndash 3+ yearsndash New product planning facility location

Types of Forecasts by Time Horizon

Short-term vs Longer-term Forecasting

bull Mediumlong range forecasts deal with more comprehensive issues and support management decisions regarding planning and products plants and processes

bull Short-term forecasting usually employs different methodologies than longer-term forecasting

bull Short-term forecasts tend to be more accurate than longer-term forecasts

Influence of Product Life Cycle

bull Stages of introduction and growth require longer forecasts than maturity and decline

bull Forecasts useful in projectingndash staffing levels

ndash inventory levels and

ndash factory capacity

as product passes through life cycle stages

Introduction Growth Maturity Decline

Strategy and Issues During a Productrsquos Life

Introduction Growth Maturity Decline

Standardization

Less rapid product changes - more minor changes

Optimum capacity

Increasing stability of process

Long production runs

Product improvement and cost cutting

Little product differentiation

Cost minimization

Over capacity in the industry

Prune line to eliminate items not returning good margin

Reduce capacity

Forecasting critical

Product and process reliability

Competitive product improvements and options

Increase capacity

Shift toward product focused

Enhance distribution

Product design and development critical

Frequent product and process design changes

Short production runs

High production costs

Limited models

Attention to quality

Best period to increase market share

RampD product engineering critical

Practical to change price or quality image

Strengthen niche

Cost control critical

Poor time to change image price or quality

Competitive costs become critical

Defend market position

OM

Str

ateg

yIs

sues

Com

pany

Str

ateg

yIs

sues

HDTV

CD-ROM

Color copiers

Drive-thru restaurants Fax machines

Station wagons

Sales

3 12rdquo Floppy disks

Internet

Types of Forecasts

bull Economic forecastsndash Address business cycle eg inflation rate

money supply etc

bull Technological forecastsndash Predict rate of technological progressndash Predict acceptance of new product

bull Demand forecastsndash Predict sales of existing product

Seven Steps in Forecasting ( 預測的七個基本步驟 )

bull Determine the use of the forecastbull Select the items to be forecastedbull Determine the time horizon of the forecastbull Select the forecasting model(s)bull Gather the databull Make the forecastbull Validate and implement results

Product Demand Charted over 4 Years with Trend and Seasonality

Year1

Year2

Year3

Year4

Seasonal peaks Trend component

Actual demand line

Average demand over four years

Dem

and

for p

rodu

ct o

r ser

vice

Random variation

Actual Demand Moving Average Weighted Moving Average

0

5

10

15

20

25

30

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Sal

es D

eman

d

Actual sales

Moving average

Weighted moving average

Realities of Forecasting

bull Forecasts are seldom perfect

bull Most forecasting methods assume that there is some underlying stability in the system

bull Both product family and aggregated product forecasts are more accurate than individual product forecasts

Forecasting Approaches

bull Used when situation is lsquostablersquo amp historical data existndash Existing productsndash Current technology

bull Involves mathematical techniquesndash eg forecasting sales of

color televisions

Quantitative Methodsbull Used when situation is

vague amp little data existndash New productsndash New technology

bull Involves intuition experiencendash eg forecasting sales on

Internet

Qualitative Methods

Overview of Qualitative Methods

bull Jury of executive opinionndash Pool opinions of high-level executives

sometimes augment by statistical models

bull Delphi methodndash Panel of experts queried iteratively

bull Sales force compositendash Estimates from individual salespersons are

reviewed for reasonableness then aggregated

bull Consumer Market Surveyndash Ask the customer

bull Involves small group of high-level managers

ndash Group estimates demand by working together

bull Combines managerial experience with statistical models

bull Relatively quick

bull lsquoGroup-thinkrsquodisadvantage

copy 1995 Corel Corp

Jury of Executive Opinion

Sales Force Composite

bull Each salesperson projects his or her sales

bull Combined at district amp national levels

bull Sales reps know customersrsquo wants

bull Tends to be overly optimistic

SalesSales

copy 1995 Corel Corp

Delphi Method

bull Iterative group process

bull 3 types of peoplendash Decision makers

ndash Staff

ndash Respondents

bull Reduces lsquogroup-thinkrsquo

Respondents Respondents

Staff Staff

Decision MakersDecision Makers(Sales)

(What will sales be survey)

(Sales will be 45 50 55)

(Sales will be 50)

Consumer Market Survey

bull Ask customers about purchasing plans

bull What consumers say and what they actually do are often different

bull Sometimes difficult to answer

How many hours will you use the Internet

next week

How many hours will you use the Internet

next week

copy 1995 Corel Corp

Overview of Quantitative Approaches

bull Naiumlve approach

bull Moving averages

bull Exponential smoothing

bull Trend projection

bull Linear regression

Time-series Models

Associative models

Quantitative Forecasting Methods (Non-Naive)

QuantitativeForecasting

LinearRegression

AssociativeModels

ExponentialSmoothing

MovingAverage

Time SeriesModels

TrendProjection

bull Set of evenly spaced numerical datandash Obtained by observing response variable at

regular time periods

bull Forecast based only on past valuesndash Assumes that factors influencing past and

present will continue influence in future

bull ExampleYear 1998 1999 2000 2001 2002Sales 787 635 897 932 921

What is a Time Series

TrendTrend

SeasonalSeasonal

CyclicalCyclical

RandomRandom

Time Series Components

bull Persistent overall upward or downward pattern

bull Due to population technology etcbull Several years duration

Mo Qtr Yr

Response

copy 1984-1994 TMaker Co

Trend Component

bull Regular pattern of up amp down fluctuations

bull Due to weather customs etcbull Occurs within 1 year

Mo Qtr

Response

Summer

copy 1984-1994 TMaker Co

Seasonal Component

Common Seasonal PatternsPeriod of Pattern

ldquoSeasonrdquo Length

Number of

ldquoSeasonsrdquo in

Pattern

Week Day 7

Month Week 4 ndash 4 frac12

Month Day 28 ndash 31

Year Quarter 4

Year Month 12

Year Week 52

bull Repeating up amp down movementsbull Due to interactions of factors

influencing economybull Usually 2-10 years duration

Mo Qtr YrMo Qtr Yr

ResponseResponseCycle

Cyclical Component

bull Erratic unsystematic lsquoresidualrsquo fluctuations

bull Due to random variation or unforeseen

events

ndash Union strike

ndash Tornado

bull Short duration amp

nonrepeating

copy 1984-1994 TMaker Co

Random Component

bull Any observed value in a time series is the product (or sum) of time series components

bull Multiplicative modelndash Yi = Ti middot Si middot Ci middot Ri (if quarterly or mo data)

bull Additive modelndash Yi = Ti + Si + Ci + Ri (if quarterly or mo

data)

General Time Series Models

Naive Approach

bull Assumes demand in next period is the same as demand in most recent periodndash eg If May sales were 48

then June sales will be 48

bull Sometimes cost effective amp efficient

copy 1995 Corel Corp

bull MA is a series of arithmetic means

bull Used if little or no trend

bull Used often for smoothingndash Provides overall impression of data over

timebull Equation

MAMAnn

nn Demand inDemand in PreviousPrevious PeriodsPeriods

Moving Average Method

Yoursquore manager of a museum store that sells historical replicas You want to forecast sales (000) for 2003 using a 3-period moving average

1998 41999 62000 52001 32002 7

copy 1995 Corel Corp

Moving Average Example

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 6+5+3=14 143=4 23 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153=50 2002 7 6+5+3=14 143=47 2003 NA 5+3+7=15 153=50

95 96 97 98 99 00Year

Sales

2

4

6

8 Actual

Forecast

Moving Average Graph

bull Used when trend is present ndash Older data usually less important

bull Weights based on intuitionndash Often lay between 0 amp 1 amp sum to 10

bull Equation

WMA =WMA =ΣΣ(Weight for period (Weight for period nn) (Demand in period ) (Demand in period nn))

ΣΣWeightsWeights

Weighted Moving Average Method

Actual Demand Moving Average Weighted Moving

Average

0

5

10

15

20

25

30

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Sal

es D

eman

d

Actual sales

Moving average

Weighted moving average

bull Increasing n makes forecast less sensitive to changes

bull Do not forecast trend wellbull Require much historical data

copy 1984-1994 TMaker Co

Disadvantages of Moving Average Methods

bull Form of weighted moving averagendash Weights decline exponentiallyndash Most recent data weighted most

bull Requires smoothing constant ()ndash Ranges from 0 to 1ndash Subjectively chosen

bull Involves little record keeping of past data

Exponential Smoothing Method

bull Ft = At - 1 + (1-)At - 2 + (1- )2middotAt - 3

+ (1- )3At - 4 + + (1- )t-1middotA0

ndash Ft = Forecast value

ndash At = Actual value = Smoothing constant

bull Ft = Ft-1 + (At-1 - Ft-1)ndash Use for computing forecast

Exponential Smoothing Equations

During the past 8 quarters the Port of Baltimore has unloaded large quantities of grain ( = 10) The first quarter forecast was 175 QuarterActual

1 180 2 168

3 1594 1755 190

6 2057 1808 1829

Exponential Smoothing Example

Find the forecast for the 9th quarter

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

11 180 17500 (Given)

22 168168

33 159159

44 175175

55 190190

66 205205

17500 +17500 +

Exponential Smoothing Solution

QuarterQuarter ActuaActualForecast F t

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010((

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - -

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Ft

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - 17500 - 17500))

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing SolutionFt = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα= = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 +17500 + 1010(180 (180 - 17500- 17500)) = 17550 = 17550

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

1 180 17500 (Given)

22 168168 17500 + 10(180 - 17500) = 1755017500 + 10(180 - 17500) = 17550

33 159159 1755017550 ++ 1010(168 -(168 - 1755017550)) = 17475= 17475

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1995 180 17500 (Given)

1996 168 17500 + 10(180 - 17500) = 17550

1997 159 17550 + 10(168 - 17550) = 17475

1998 175

1999 190

2000 205

17475 + 10(159 - 17475)= 17318

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9 17822 + 10(182 - 17822) = 17858 182 17802 + 10(180 - 17802) = 17822

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

What is Forecasting

bull Process of predicting a future event

bull Underlying basis of all business decisions

ndash Production

ndash Inventory

ndash Personnel

ndash Facilities

Sales will be $200 Million

bull Short-range forecastndash Up to 1 year usually less than 3 monthsndash Job scheduling worker assignments

bull Medium-range forecastndash 3 months to 3 yearsndash Sales amp production planning budgeting

bull Long-range forecastndash 3+ yearsndash New product planning facility location

Types of Forecasts by Time Horizon

Short-term vs Longer-term Forecasting

bull Mediumlong range forecasts deal with more comprehensive issues and support management decisions regarding planning and products plants and processes

bull Short-term forecasting usually employs different methodologies than longer-term forecasting

bull Short-term forecasts tend to be more accurate than longer-term forecasts

Influence of Product Life Cycle

bull Stages of introduction and growth require longer forecasts than maturity and decline

bull Forecasts useful in projectingndash staffing levels

ndash inventory levels and

ndash factory capacity

as product passes through life cycle stages

Introduction Growth Maturity Decline

Strategy and Issues During a Productrsquos Life

Introduction Growth Maturity Decline

Standardization

Less rapid product changes - more minor changes

Optimum capacity

Increasing stability of process

Long production runs

Product improvement and cost cutting

Little product differentiation

Cost minimization

Over capacity in the industry

Prune line to eliminate items not returning good margin

Reduce capacity

Forecasting critical

Product and process reliability

Competitive product improvements and options

Increase capacity

Shift toward product focused

Enhance distribution

Product design and development critical

Frequent product and process design changes

Short production runs

High production costs

Limited models

Attention to quality

Best period to increase market share

RampD product engineering critical

Practical to change price or quality image

Strengthen niche

Cost control critical

Poor time to change image price or quality

Competitive costs become critical

Defend market position

OM

Str

ateg

yIs

sues

Com

pany

Str

ateg

yIs

sues

HDTV

CD-ROM

Color copiers

Drive-thru restaurants Fax machines

Station wagons

Sales

3 12rdquo Floppy disks

Internet

Types of Forecasts

bull Economic forecastsndash Address business cycle eg inflation rate

money supply etc

bull Technological forecastsndash Predict rate of technological progressndash Predict acceptance of new product

bull Demand forecastsndash Predict sales of existing product

Seven Steps in Forecasting ( 預測的七個基本步驟 )

bull Determine the use of the forecastbull Select the items to be forecastedbull Determine the time horizon of the forecastbull Select the forecasting model(s)bull Gather the databull Make the forecastbull Validate and implement results

Product Demand Charted over 4 Years with Trend and Seasonality

Year1

Year2

Year3

Year4

Seasonal peaks Trend component

Actual demand line

Average demand over four years

Dem

and

for p

rodu

ct o

r ser

vice

Random variation

Actual Demand Moving Average Weighted Moving Average

0

5

10

15

20

25

30

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Sal

es D

eman

d

Actual sales

Moving average

Weighted moving average

Realities of Forecasting

bull Forecasts are seldom perfect

bull Most forecasting methods assume that there is some underlying stability in the system

bull Both product family and aggregated product forecasts are more accurate than individual product forecasts

Forecasting Approaches

bull Used when situation is lsquostablersquo amp historical data existndash Existing productsndash Current technology

bull Involves mathematical techniquesndash eg forecasting sales of

color televisions

Quantitative Methodsbull Used when situation is

vague amp little data existndash New productsndash New technology

bull Involves intuition experiencendash eg forecasting sales on

Internet

Qualitative Methods

Overview of Qualitative Methods

bull Jury of executive opinionndash Pool opinions of high-level executives

sometimes augment by statistical models

bull Delphi methodndash Panel of experts queried iteratively

bull Sales force compositendash Estimates from individual salespersons are

reviewed for reasonableness then aggregated

bull Consumer Market Surveyndash Ask the customer

bull Involves small group of high-level managers

ndash Group estimates demand by working together

bull Combines managerial experience with statistical models

bull Relatively quick

bull lsquoGroup-thinkrsquodisadvantage

copy 1995 Corel Corp

Jury of Executive Opinion

Sales Force Composite

bull Each salesperson projects his or her sales

bull Combined at district amp national levels

bull Sales reps know customersrsquo wants

bull Tends to be overly optimistic

SalesSales

copy 1995 Corel Corp

Delphi Method

bull Iterative group process

bull 3 types of peoplendash Decision makers

ndash Staff

ndash Respondents

bull Reduces lsquogroup-thinkrsquo

Respondents Respondents

Staff Staff

Decision MakersDecision Makers(Sales)

(What will sales be survey)

(Sales will be 45 50 55)

(Sales will be 50)

Consumer Market Survey

bull Ask customers about purchasing plans

bull What consumers say and what they actually do are often different

bull Sometimes difficult to answer

How many hours will you use the Internet

next week

How many hours will you use the Internet

next week

copy 1995 Corel Corp

Overview of Quantitative Approaches

bull Naiumlve approach

bull Moving averages

bull Exponential smoothing

bull Trend projection

bull Linear regression

Time-series Models

Associative models

Quantitative Forecasting Methods (Non-Naive)

QuantitativeForecasting

LinearRegression

AssociativeModels

ExponentialSmoothing

MovingAverage

Time SeriesModels

TrendProjection

bull Set of evenly spaced numerical datandash Obtained by observing response variable at

regular time periods

bull Forecast based only on past valuesndash Assumes that factors influencing past and

present will continue influence in future

bull ExampleYear 1998 1999 2000 2001 2002Sales 787 635 897 932 921

What is a Time Series

TrendTrend

SeasonalSeasonal

CyclicalCyclical

RandomRandom

Time Series Components

bull Persistent overall upward or downward pattern

bull Due to population technology etcbull Several years duration

Mo Qtr Yr

Response

copy 1984-1994 TMaker Co

Trend Component

bull Regular pattern of up amp down fluctuations

bull Due to weather customs etcbull Occurs within 1 year

Mo Qtr

Response

Summer

copy 1984-1994 TMaker Co

Seasonal Component

Common Seasonal PatternsPeriod of Pattern

ldquoSeasonrdquo Length

Number of

ldquoSeasonsrdquo in

Pattern

Week Day 7

Month Week 4 ndash 4 frac12

Month Day 28 ndash 31

Year Quarter 4

Year Month 12

Year Week 52

bull Repeating up amp down movementsbull Due to interactions of factors

influencing economybull Usually 2-10 years duration

Mo Qtr YrMo Qtr Yr

ResponseResponseCycle

Cyclical Component

bull Erratic unsystematic lsquoresidualrsquo fluctuations

bull Due to random variation or unforeseen

events

ndash Union strike

ndash Tornado

bull Short duration amp

nonrepeating

copy 1984-1994 TMaker Co

Random Component

bull Any observed value in a time series is the product (or sum) of time series components

bull Multiplicative modelndash Yi = Ti middot Si middot Ci middot Ri (if quarterly or mo data)

bull Additive modelndash Yi = Ti + Si + Ci + Ri (if quarterly or mo

data)

General Time Series Models

Naive Approach

bull Assumes demand in next period is the same as demand in most recent periodndash eg If May sales were 48

then June sales will be 48

bull Sometimes cost effective amp efficient

copy 1995 Corel Corp

bull MA is a series of arithmetic means

bull Used if little or no trend

bull Used often for smoothingndash Provides overall impression of data over

timebull Equation

MAMAnn

nn Demand inDemand in PreviousPrevious PeriodsPeriods

Moving Average Method

Yoursquore manager of a museum store that sells historical replicas You want to forecast sales (000) for 2003 using a 3-period moving average

1998 41999 62000 52001 32002 7

copy 1995 Corel Corp

Moving Average Example

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 6+5+3=14 143=4 23 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153=50 2002 7 6+5+3=14 143=47 2003 NA 5+3+7=15 153=50

95 96 97 98 99 00Year

Sales

2

4

6

8 Actual

Forecast

Moving Average Graph

bull Used when trend is present ndash Older data usually less important

bull Weights based on intuitionndash Often lay between 0 amp 1 amp sum to 10

bull Equation

WMA =WMA =ΣΣ(Weight for period (Weight for period nn) (Demand in period ) (Demand in period nn))

ΣΣWeightsWeights

Weighted Moving Average Method

Actual Demand Moving Average Weighted Moving

Average

0

5

10

15

20

25

30

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Sal

es D

eman

d

Actual sales

Moving average

Weighted moving average

bull Increasing n makes forecast less sensitive to changes

bull Do not forecast trend wellbull Require much historical data

copy 1984-1994 TMaker Co

Disadvantages of Moving Average Methods

bull Form of weighted moving averagendash Weights decline exponentiallyndash Most recent data weighted most

bull Requires smoothing constant ()ndash Ranges from 0 to 1ndash Subjectively chosen

bull Involves little record keeping of past data

Exponential Smoothing Method

bull Ft = At - 1 + (1-)At - 2 + (1- )2middotAt - 3

+ (1- )3At - 4 + + (1- )t-1middotA0

ndash Ft = Forecast value

ndash At = Actual value = Smoothing constant

bull Ft = Ft-1 + (At-1 - Ft-1)ndash Use for computing forecast

Exponential Smoothing Equations

During the past 8 quarters the Port of Baltimore has unloaded large quantities of grain ( = 10) The first quarter forecast was 175 QuarterActual

1 180 2 168

3 1594 1755 190

6 2057 1808 1829

Exponential Smoothing Example

Find the forecast for the 9th quarter

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

11 180 17500 (Given)

22 168168

33 159159

44 175175

55 190190

66 205205

17500 +17500 +

Exponential Smoothing Solution

QuarterQuarter ActuaActualForecast F t

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010((

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - -

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Ft

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - 17500 - 17500))

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing SolutionFt = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα= = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 +17500 + 1010(180 (180 - 17500- 17500)) = 17550 = 17550

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

1 180 17500 (Given)

22 168168 17500 + 10(180 - 17500) = 1755017500 + 10(180 - 17500) = 17550

33 159159 1755017550 ++ 1010(168 -(168 - 1755017550)) = 17475= 17475

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1995 180 17500 (Given)

1996 168 17500 + 10(180 - 17500) = 17550

1997 159 17550 + 10(168 - 17550) = 17475

1998 175

1999 190

2000 205

17475 + 10(159 - 17475)= 17318

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9 17822 + 10(182 - 17822) = 17858 182 17802 + 10(180 - 17802) = 17822

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

bull Short-range forecastndash Up to 1 year usually less than 3 monthsndash Job scheduling worker assignments

bull Medium-range forecastndash 3 months to 3 yearsndash Sales amp production planning budgeting

bull Long-range forecastndash 3+ yearsndash New product planning facility location

Types of Forecasts by Time Horizon

Short-term vs Longer-term Forecasting

bull Mediumlong range forecasts deal with more comprehensive issues and support management decisions regarding planning and products plants and processes

bull Short-term forecasting usually employs different methodologies than longer-term forecasting

bull Short-term forecasts tend to be more accurate than longer-term forecasts

Influence of Product Life Cycle

bull Stages of introduction and growth require longer forecasts than maturity and decline

bull Forecasts useful in projectingndash staffing levels

ndash inventory levels and

ndash factory capacity

as product passes through life cycle stages

Introduction Growth Maturity Decline

Strategy and Issues During a Productrsquos Life

Introduction Growth Maturity Decline

Standardization

Less rapid product changes - more minor changes

Optimum capacity

Increasing stability of process

Long production runs

Product improvement and cost cutting

Little product differentiation

Cost minimization

Over capacity in the industry

Prune line to eliminate items not returning good margin

Reduce capacity

Forecasting critical

Product and process reliability

Competitive product improvements and options

Increase capacity

Shift toward product focused

Enhance distribution

Product design and development critical

Frequent product and process design changes

Short production runs

High production costs

Limited models

Attention to quality

Best period to increase market share

RampD product engineering critical

Practical to change price or quality image

Strengthen niche

Cost control critical

Poor time to change image price or quality

Competitive costs become critical

Defend market position

OM

Str

ateg

yIs

sues

Com

pany

Str

ateg

yIs

sues

HDTV

CD-ROM

Color copiers

Drive-thru restaurants Fax machines

Station wagons

Sales

3 12rdquo Floppy disks

Internet

Types of Forecasts

bull Economic forecastsndash Address business cycle eg inflation rate

money supply etc

bull Technological forecastsndash Predict rate of technological progressndash Predict acceptance of new product

bull Demand forecastsndash Predict sales of existing product

Seven Steps in Forecasting ( 預測的七個基本步驟 )

bull Determine the use of the forecastbull Select the items to be forecastedbull Determine the time horizon of the forecastbull Select the forecasting model(s)bull Gather the databull Make the forecastbull Validate and implement results

Product Demand Charted over 4 Years with Trend and Seasonality

Year1

Year2

Year3

Year4

Seasonal peaks Trend component

Actual demand line

Average demand over four years

Dem

and

for p

rodu

ct o

r ser

vice

Random variation

Actual Demand Moving Average Weighted Moving Average

0

5

10

15

20

25

30

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Sal

es D

eman

d

Actual sales

Moving average

Weighted moving average

Realities of Forecasting

bull Forecasts are seldom perfect

bull Most forecasting methods assume that there is some underlying stability in the system

bull Both product family and aggregated product forecasts are more accurate than individual product forecasts

Forecasting Approaches

bull Used when situation is lsquostablersquo amp historical data existndash Existing productsndash Current technology

bull Involves mathematical techniquesndash eg forecasting sales of

color televisions

Quantitative Methodsbull Used when situation is

vague amp little data existndash New productsndash New technology

bull Involves intuition experiencendash eg forecasting sales on

Internet

Qualitative Methods

Overview of Qualitative Methods

bull Jury of executive opinionndash Pool opinions of high-level executives

sometimes augment by statistical models

bull Delphi methodndash Panel of experts queried iteratively

bull Sales force compositendash Estimates from individual salespersons are

reviewed for reasonableness then aggregated

bull Consumer Market Surveyndash Ask the customer

bull Involves small group of high-level managers

ndash Group estimates demand by working together

bull Combines managerial experience with statistical models

bull Relatively quick

bull lsquoGroup-thinkrsquodisadvantage

copy 1995 Corel Corp

Jury of Executive Opinion

Sales Force Composite

bull Each salesperson projects his or her sales

bull Combined at district amp national levels

bull Sales reps know customersrsquo wants

bull Tends to be overly optimistic

SalesSales

copy 1995 Corel Corp

Delphi Method

bull Iterative group process

bull 3 types of peoplendash Decision makers

ndash Staff

ndash Respondents

bull Reduces lsquogroup-thinkrsquo

Respondents Respondents

Staff Staff

Decision MakersDecision Makers(Sales)

(What will sales be survey)

(Sales will be 45 50 55)

(Sales will be 50)

Consumer Market Survey

bull Ask customers about purchasing plans

bull What consumers say and what they actually do are often different

bull Sometimes difficult to answer

How many hours will you use the Internet

next week

How many hours will you use the Internet

next week

copy 1995 Corel Corp

Overview of Quantitative Approaches

bull Naiumlve approach

bull Moving averages

bull Exponential smoothing

bull Trend projection

bull Linear regression

Time-series Models

Associative models

Quantitative Forecasting Methods (Non-Naive)

QuantitativeForecasting

LinearRegression

AssociativeModels

ExponentialSmoothing

MovingAverage

Time SeriesModels

TrendProjection

bull Set of evenly spaced numerical datandash Obtained by observing response variable at

regular time periods

bull Forecast based only on past valuesndash Assumes that factors influencing past and

present will continue influence in future

bull ExampleYear 1998 1999 2000 2001 2002Sales 787 635 897 932 921

What is a Time Series

TrendTrend

SeasonalSeasonal

CyclicalCyclical

RandomRandom

Time Series Components

bull Persistent overall upward or downward pattern

bull Due to population technology etcbull Several years duration

Mo Qtr Yr

Response

copy 1984-1994 TMaker Co

Trend Component

bull Regular pattern of up amp down fluctuations

bull Due to weather customs etcbull Occurs within 1 year

Mo Qtr

Response

Summer

copy 1984-1994 TMaker Co

Seasonal Component

Common Seasonal PatternsPeriod of Pattern

ldquoSeasonrdquo Length

Number of

ldquoSeasonsrdquo in

Pattern

Week Day 7

Month Week 4 ndash 4 frac12

Month Day 28 ndash 31

Year Quarter 4

Year Month 12

Year Week 52

bull Repeating up amp down movementsbull Due to interactions of factors

influencing economybull Usually 2-10 years duration

Mo Qtr YrMo Qtr Yr

ResponseResponseCycle

Cyclical Component

bull Erratic unsystematic lsquoresidualrsquo fluctuations

bull Due to random variation or unforeseen

events

ndash Union strike

ndash Tornado

bull Short duration amp

nonrepeating

copy 1984-1994 TMaker Co

Random Component

bull Any observed value in a time series is the product (or sum) of time series components

bull Multiplicative modelndash Yi = Ti middot Si middot Ci middot Ri (if quarterly or mo data)

bull Additive modelndash Yi = Ti + Si + Ci + Ri (if quarterly or mo

data)

General Time Series Models

Naive Approach

bull Assumes demand in next period is the same as demand in most recent periodndash eg If May sales were 48

then June sales will be 48

bull Sometimes cost effective amp efficient

copy 1995 Corel Corp

bull MA is a series of arithmetic means

bull Used if little or no trend

bull Used often for smoothingndash Provides overall impression of data over

timebull Equation

MAMAnn

nn Demand inDemand in PreviousPrevious PeriodsPeriods

Moving Average Method

Yoursquore manager of a museum store that sells historical replicas You want to forecast sales (000) for 2003 using a 3-period moving average

1998 41999 62000 52001 32002 7

copy 1995 Corel Corp

Moving Average Example

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 6+5+3=14 143=4 23 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153=50 2002 7 6+5+3=14 143=47 2003 NA 5+3+7=15 153=50

95 96 97 98 99 00Year

Sales

2

4

6

8 Actual

Forecast

Moving Average Graph

bull Used when trend is present ndash Older data usually less important

bull Weights based on intuitionndash Often lay between 0 amp 1 amp sum to 10

bull Equation

WMA =WMA =ΣΣ(Weight for period (Weight for period nn) (Demand in period ) (Demand in period nn))

ΣΣWeightsWeights

Weighted Moving Average Method

Actual Demand Moving Average Weighted Moving

Average

0

5

10

15

20

25

30

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Sal

es D

eman

d

Actual sales

Moving average

Weighted moving average

bull Increasing n makes forecast less sensitive to changes

bull Do not forecast trend wellbull Require much historical data

copy 1984-1994 TMaker Co

Disadvantages of Moving Average Methods

bull Form of weighted moving averagendash Weights decline exponentiallyndash Most recent data weighted most

bull Requires smoothing constant ()ndash Ranges from 0 to 1ndash Subjectively chosen

bull Involves little record keeping of past data

Exponential Smoothing Method

bull Ft = At - 1 + (1-)At - 2 + (1- )2middotAt - 3

+ (1- )3At - 4 + + (1- )t-1middotA0

ndash Ft = Forecast value

ndash At = Actual value = Smoothing constant

bull Ft = Ft-1 + (At-1 - Ft-1)ndash Use for computing forecast

Exponential Smoothing Equations

During the past 8 quarters the Port of Baltimore has unloaded large quantities of grain ( = 10) The first quarter forecast was 175 QuarterActual

1 180 2 168

3 1594 1755 190

6 2057 1808 1829

Exponential Smoothing Example

Find the forecast for the 9th quarter

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

11 180 17500 (Given)

22 168168

33 159159

44 175175

55 190190

66 205205

17500 +17500 +

Exponential Smoothing Solution

QuarterQuarter ActuaActualForecast F t

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010((

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - -

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Ft

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - 17500 - 17500))

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing SolutionFt = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα= = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 +17500 + 1010(180 (180 - 17500- 17500)) = 17550 = 17550

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

1 180 17500 (Given)

22 168168 17500 + 10(180 - 17500) = 1755017500 + 10(180 - 17500) = 17550

33 159159 1755017550 ++ 1010(168 -(168 - 1755017550)) = 17475= 17475

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1995 180 17500 (Given)

1996 168 17500 + 10(180 - 17500) = 17550

1997 159 17550 + 10(168 - 17550) = 17475

1998 175

1999 190

2000 205

17475 + 10(159 - 17475)= 17318

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9 17822 + 10(182 - 17822) = 17858 182 17802 + 10(180 - 17802) = 17822

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Short-term vs Longer-term Forecasting

bull Mediumlong range forecasts deal with more comprehensive issues and support management decisions regarding planning and products plants and processes

bull Short-term forecasting usually employs different methodologies than longer-term forecasting

bull Short-term forecasts tend to be more accurate than longer-term forecasts

Influence of Product Life Cycle

bull Stages of introduction and growth require longer forecasts than maturity and decline

bull Forecasts useful in projectingndash staffing levels

ndash inventory levels and

ndash factory capacity

as product passes through life cycle stages

Introduction Growth Maturity Decline

Strategy and Issues During a Productrsquos Life

Introduction Growth Maturity Decline

Standardization

Less rapid product changes - more minor changes

Optimum capacity

Increasing stability of process

Long production runs

Product improvement and cost cutting

Little product differentiation

Cost minimization

Over capacity in the industry

Prune line to eliminate items not returning good margin

Reduce capacity

Forecasting critical

Product and process reliability

Competitive product improvements and options

Increase capacity

Shift toward product focused

Enhance distribution

Product design and development critical

Frequent product and process design changes

Short production runs

High production costs

Limited models

Attention to quality

Best period to increase market share

RampD product engineering critical

Practical to change price or quality image

Strengthen niche

Cost control critical

Poor time to change image price or quality

Competitive costs become critical

Defend market position

OM

Str

ateg

yIs

sues

Com

pany

Str

ateg

yIs

sues

HDTV

CD-ROM

Color copiers

Drive-thru restaurants Fax machines

Station wagons

Sales

3 12rdquo Floppy disks

Internet

Types of Forecasts

bull Economic forecastsndash Address business cycle eg inflation rate

money supply etc

bull Technological forecastsndash Predict rate of technological progressndash Predict acceptance of new product

bull Demand forecastsndash Predict sales of existing product

Seven Steps in Forecasting ( 預測的七個基本步驟 )

bull Determine the use of the forecastbull Select the items to be forecastedbull Determine the time horizon of the forecastbull Select the forecasting model(s)bull Gather the databull Make the forecastbull Validate and implement results

Product Demand Charted over 4 Years with Trend and Seasonality

Year1

Year2

Year3

Year4

Seasonal peaks Trend component

Actual demand line

Average demand over four years

Dem

and

for p

rodu

ct o

r ser

vice

Random variation

Actual Demand Moving Average Weighted Moving Average

0

5

10

15

20

25

30

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Sal

es D

eman

d

Actual sales

Moving average

Weighted moving average

Realities of Forecasting

bull Forecasts are seldom perfect

bull Most forecasting methods assume that there is some underlying stability in the system

bull Both product family and aggregated product forecasts are more accurate than individual product forecasts

Forecasting Approaches

bull Used when situation is lsquostablersquo amp historical data existndash Existing productsndash Current technology

bull Involves mathematical techniquesndash eg forecasting sales of

color televisions

Quantitative Methodsbull Used when situation is

vague amp little data existndash New productsndash New technology

bull Involves intuition experiencendash eg forecasting sales on

Internet

Qualitative Methods

Overview of Qualitative Methods

bull Jury of executive opinionndash Pool opinions of high-level executives

sometimes augment by statistical models

bull Delphi methodndash Panel of experts queried iteratively

bull Sales force compositendash Estimates from individual salespersons are

reviewed for reasonableness then aggregated

bull Consumer Market Surveyndash Ask the customer

bull Involves small group of high-level managers

ndash Group estimates demand by working together

bull Combines managerial experience with statistical models

bull Relatively quick

bull lsquoGroup-thinkrsquodisadvantage

copy 1995 Corel Corp

Jury of Executive Opinion

Sales Force Composite

bull Each salesperson projects his or her sales

bull Combined at district amp national levels

bull Sales reps know customersrsquo wants

bull Tends to be overly optimistic

SalesSales

copy 1995 Corel Corp

Delphi Method

bull Iterative group process

bull 3 types of peoplendash Decision makers

ndash Staff

ndash Respondents

bull Reduces lsquogroup-thinkrsquo

Respondents Respondents

Staff Staff

Decision MakersDecision Makers(Sales)

(What will sales be survey)

(Sales will be 45 50 55)

(Sales will be 50)

Consumer Market Survey

bull Ask customers about purchasing plans

bull What consumers say and what they actually do are often different

bull Sometimes difficult to answer

How many hours will you use the Internet

next week

How many hours will you use the Internet

next week

copy 1995 Corel Corp

Overview of Quantitative Approaches

bull Naiumlve approach

bull Moving averages

bull Exponential smoothing

bull Trend projection

bull Linear regression

Time-series Models

Associative models

Quantitative Forecasting Methods (Non-Naive)

QuantitativeForecasting

LinearRegression

AssociativeModels

ExponentialSmoothing

MovingAverage

Time SeriesModels

TrendProjection

bull Set of evenly spaced numerical datandash Obtained by observing response variable at

regular time periods

bull Forecast based only on past valuesndash Assumes that factors influencing past and

present will continue influence in future

bull ExampleYear 1998 1999 2000 2001 2002Sales 787 635 897 932 921

What is a Time Series

TrendTrend

SeasonalSeasonal

CyclicalCyclical

RandomRandom

Time Series Components

bull Persistent overall upward or downward pattern

bull Due to population technology etcbull Several years duration

Mo Qtr Yr

Response

copy 1984-1994 TMaker Co

Trend Component

bull Regular pattern of up amp down fluctuations

bull Due to weather customs etcbull Occurs within 1 year

Mo Qtr

Response

Summer

copy 1984-1994 TMaker Co

Seasonal Component

Common Seasonal PatternsPeriod of Pattern

ldquoSeasonrdquo Length

Number of

ldquoSeasonsrdquo in

Pattern

Week Day 7

Month Week 4 ndash 4 frac12

Month Day 28 ndash 31

Year Quarter 4

Year Month 12

Year Week 52

bull Repeating up amp down movementsbull Due to interactions of factors

influencing economybull Usually 2-10 years duration

Mo Qtr YrMo Qtr Yr

ResponseResponseCycle

Cyclical Component

bull Erratic unsystematic lsquoresidualrsquo fluctuations

bull Due to random variation or unforeseen

events

ndash Union strike

ndash Tornado

bull Short duration amp

nonrepeating

copy 1984-1994 TMaker Co

Random Component

bull Any observed value in a time series is the product (or sum) of time series components

bull Multiplicative modelndash Yi = Ti middot Si middot Ci middot Ri (if quarterly or mo data)

bull Additive modelndash Yi = Ti + Si + Ci + Ri (if quarterly or mo

data)

General Time Series Models

Naive Approach

bull Assumes demand in next period is the same as demand in most recent periodndash eg If May sales were 48

then June sales will be 48

bull Sometimes cost effective amp efficient

copy 1995 Corel Corp

bull MA is a series of arithmetic means

bull Used if little or no trend

bull Used often for smoothingndash Provides overall impression of data over

timebull Equation

MAMAnn

nn Demand inDemand in PreviousPrevious PeriodsPeriods

Moving Average Method

Yoursquore manager of a museum store that sells historical replicas You want to forecast sales (000) for 2003 using a 3-period moving average

1998 41999 62000 52001 32002 7

copy 1995 Corel Corp

Moving Average Example

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 6+5+3=14 143=4 23 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153=50 2002 7 6+5+3=14 143=47 2003 NA 5+3+7=15 153=50

95 96 97 98 99 00Year

Sales

2

4

6

8 Actual

Forecast

Moving Average Graph

bull Used when trend is present ndash Older data usually less important

bull Weights based on intuitionndash Often lay between 0 amp 1 amp sum to 10

bull Equation

WMA =WMA =ΣΣ(Weight for period (Weight for period nn) (Demand in period ) (Demand in period nn))

ΣΣWeightsWeights

Weighted Moving Average Method

Actual Demand Moving Average Weighted Moving

Average

0

5

10

15

20

25

30

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Sal

es D

eman

d

Actual sales

Moving average

Weighted moving average

bull Increasing n makes forecast less sensitive to changes

bull Do not forecast trend wellbull Require much historical data

copy 1984-1994 TMaker Co

Disadvantages of Moving Average Methods

bull Form of weighted moving averagendash Weights decline exponentiallyndash Most recent data weighted most

bull Requires smoothing constant ()ndash Ranges from 0 to 1ndash Subjectively chosen

bull Involves little record keeping of past data

Exponential Smoothing Method

bull Ft = At - 1 + (1-)At - 2 + (1- )2middotAt - 3

+ (1- )3At - 4 + + (1- )t-1middotA0

ndash Ft = Forecast value

ndash At = Actual value = Smoothing constant

bull Ft = Ft-1 + (At-1 - Ft-1)ndash Use for computing forecast

Exponential Smoothing Equations

During the past 8 quarters the Port of Baltimore has unloaded large quantities of grain ( = 10) The first quarter forecast was 175 QuarterActual

1 180 2 168

3 1594 1755 190

6 2057 1808 1829

Exponential Smoothing Example

Find the forecast for the 9th quarter

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

11 180 17500 (Given)

22 168168

33 159159

44 175175

55 190190

66 205205

17500 +17500 +

Exponential Smoothing Solution

QuarterQuarter ActuaActualForecast F t

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010((

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - -

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Ft

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - 17500 - 17500))

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing SolutionFt = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα= = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 +17500 + 1010(180 (180 - 17500- 17500)) = 17550 = 17550

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

1 180 17500 (Given)

22 168168 17500 + 10(180 - 17500) = 1755017500 + 10(180 - 17500) = 17550

33 159159 1755017550 ++ 1010(168 -(168 - 1755017550)) = 17475= 17475

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1995 180 17500 (Given)

1996 168 17500 + 10(180 - 17500) = 17550

1997 159 17550 + 10(168 - 17550) = 17475

1998 175

1999 190

2000 205

17475 + 10(159 - 17475)= 17318

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9 17822 + 10(182 - 17822) = 17858 182 17802 + 10(180 - 17802) = 17822

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Influence of Product Life Cycle

bull Stages of introduction and growth require longer forecasts than maturity and decline

bull Forecasts useful in projectingndash staffing levels

ndash inventory levels and

ndash factory capacity

as product passes through life cycle stages

Introduction Growth Maturity Decline

Strategy and Issues During a Productrsquos Life

Introduction Growth Maturity Decline

Standardization

Less rapid product changes - more minor changes

Optimum capacity

Increasing stability of process

Long production runs

Product improvement and cost cutting

Little product differentiation

Cost minimization

Over capacity in the industry

Prune line to eliminate items not returning good margin

Reduce capacity

Forecasting critical

Product and process reliability

Competitive product improvements and options

Increase capacity

Shift toward product focused

Enhance distribution

Product design and development critical

Frequent product and process design changes

Short production runs

High production costs

Limited models

Attention to quality

Best period to increase market share

RampD product engineering critical

Practical to change price or quality image

Strengthen niche

Cost control critical

Poor time to change image price or quality

Competitive costs become critical

Defend market position

OM

Str

ateg

yIs

sues

Com

pany

Str

ateg

yIs

sues

HDTV

CD-ROM

Color copiers

Drive-thru restaurants Fax machines

Station wagons

Sales

3 12rdquo Floppy disks

Internet

Types of Forecasts

bull Economic forecastsndash Address business cycle eg inflation rate

money supply etc

bull Technological forecastsndash Predict rate of technological progressndash Predict acceptance of new product

bull Demand forecastsndash Predict sales of existing product

Seven Steps in Forecasting ( 預測的七個基本步驟 )

bull Determine the use of the forecastbull Select the items to be forecastedbull Determine the time horizon of the forecastbull Select the forecasting model(s)bull Gather the databull Make the forecastbull Validate and implement results

Product Demand Charted over 4 Years with Trend and Seasonality

Year1

Year2

Year3

Year4

Seasonal peaks Trend component

Actual demand line

Average demand over four years

Dem

and

for p

rodu

ct o

r ser

vice

Random variation

Actual Demand Moving Average Weighted Moving Average

0

5

10

15

20

25

30

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Sal

es D

eman

d

Actual sales

Moving average

Weighted moving average

Realities of Forecasting

bull Forecasts are seldom perfect

bull Most forecasting methods assume that there is some underlying stability in the system

bull Both product family and aggregated product forecasts are more accurate than individual product forecasts

Forecasting Approaches

bull Used when situation is lsquostablersquo amp historical data existndash Existing productsndash Current technology

bull Involves mathematical techniquesndash eg forecasting sales of

color televisions

Quantitative Methodsbull Used when situation is

vague amp little data existndash New productsndash New technology

bull Involves intuition experiencendash eg forecasting sales on

Internet

Qualitative Methods

Overview of Qualitative Methods

bull Jury of executive opinionndash Pool opinions of high-level executives

sometimes augment by statistical models

bull Delphi methodndash Panel of experts queried iteratively

bull Sales force compositendash Estimates from individual salespersons are

reviewed for reasonableness then aggregated

bull Consumer Market Surveyndash Ask the customer

bull Involves small group of high-level managers

ndash Group estimates demand by working together

bull Combines managerial experience with statistical models

bull Relatively quick

bull lsquoGroup-thinkrsquodisadvantage

copy 1995 Corel Corp

Jury of Executive Opinion

Sales Force Composite

bull Each salesperson projects his or her sales

bull Combined at district amp national levels

bull Sales reps know customersrsquo wants

bull Tends to be overly optimistic

SalesSales

copy 1995 Corel Corp

Delphi Method

bull Iterative group process

bull 3 types of peoplendash Decision makers

ndash Staff

ndash Respondents

bull Reduces lsquogroup-thinkrsquo

Respondents Respondents

Staff Staff

Decision MakersDecision Makers(Sales)

(What will sales be survey)

(Sales will be 45 50 55)

(Sales will be 50)

Consumer Market Survey

bull Ask customers about purchasing plans

bull What consumers say and what they actually do are often different

bull Sometimes difficult to answer

How many hours will you use the Internet

next week

How many hours will you use the Internet

next week

copy 1995 Corel Corp

Overview of Quantitative Approaches

bull Naiumlve approach

bull Moving averages

bull Exponential smoothing

bull Trend projection

bull Linear regression

Time-series Models

Associative models

Quantitative Forecasting Methods (Non-Naive)

QuantitativeForecasting

LinearRegression

AssociativeModels

ExponentialSmoothing

MovingAverage

Time SeriesModels

TrendProjection

bull Set of evenly spaced numerical datandash Obtained by observing response variable at

regular time periods

bull Forecast based only on past valuesndash Assumes that factors influencing past and

present will continue influence in future

bull ExampleYear 1998 1999 2000 2001 2002Sales 787 635 897 932 921

What is a Time Series

TrendTrend

SeasonalSeasonal

CyclicalCyclical

RandomRandom

Time Series Components

bull Persistent overall upward or downward pattern

bull Due to population technology etcbull Several years duration

Mo Qtr Yr

Response

copy 1984-1994 TMaker Co

Trend Component

bull Regular pattern of up amp down fluctuations

bull Due to weather customs etcbull Occurs within 1 year

Mo Qtr

Response

Summer

copy 1984-1994 TMaker Co

Seasonal Component

Common Seasonal PatternsPeriod of Pattern

ldquoSeasonrdquo Length

Number of

ldquoSeasonsrdquo in

Pattern

Week Day 7

Month Week 4 ndash 4 frac12

Month Day 28 ndash 31

Year Quarter 4

Year Month 12

Year Week 52

bull Repeating up amp down movementsbull Due to interactions of factors

influencing economybull Usually 2-10 years duration

Mo Qtr YrMo Qtr Yr

ResponseResponseCycle

Cyclical Component

bull Erratic unsystematic lsquoresidualrsquo fluctuations

bull Due to random variation or unforeseen

events

ndash Union strike

ndash Tornado

bull Short duration amp

nonrepeating

copy 1984-1994 TMaker Co

Random Component

bull Any observed value in a time series is the product (or sum) of time series components

bull Multiplicative modelndash Yi = Ti middot Si middot Ci middot Ri (if quarterly or mo data)

bull Additive modelndash Yi = Ti + Si + Ci + Ri (if quarterly or mo

data)

General Time Series Models

Naive Approach

bull Assumes demand in next period is the same as demand in most recent periodndash eg If May sales were 48

then June sales will be 48

bull Sometimes cost effective amp efficient

copy 1995 Corel Corp

bull MA is a series of arithmetic means

bull Used if little or no trend

bull Used often for smoothingndash Provides overall impression of data over

timebull Equation

MAMAnn

nn Demand inDemand in PreviousPrevious PeriodsPeriods

Moving Average Method

Yoursquore manager of a museum store that sells historical replicas You want to forecast sales (000) for 2003 using a 3-period moving average

1998 41999 62000 52001 32002 7

copy 1995 Corel Corp

Moving Average Example

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 6+5+3=14 143=4 23 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153=50 2002 7 6+5+3=14 143=47 2003 NA 5+3+7=15 153=50

95 96 97 98 99 00Year

Sales

2

4

6

8 Actual

Forecast

Moving Average Graph

bull Used when trend is present ndash Older data usually less important

bull Weights based on intuitionndash Often lay between 0 amp 1 amp sum to 10

bull Equation

WMA =WMA =ΣΣ(Weight for period (Weight for period nn) (Demand in period ) (Demand in period nn))

ΣΣWeightsWeights

Weighted Moving Average Method

Actual Demand Moving Average Weighted Moving

Average

0

5

10

15

20

25

30

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Sal

es D

eman

d

Actual sales

Moving average

Weighted moving average

bull Increasing n makes forecast less sensitive to changes

bull Do not forecast trend wellbull Require much historical data

copy 1984-1994 TMaker Co

Disadvantages of Moving Average Methods

bull Form of weighted moving averagendash Weights decline exponentiallyndash Most recent data weighted most

bull Requires smoothing constant ()ndash Ranges from 0 to 1ndash Subjectively chosen

bull Involves little record keeping of past data

Exponential Smoothing Method

bull Ft = At - 1 + (1-)At - 2 + (1- )2middotAt - 3

+ (1- )3At - 4 + + (1- )t-1middotA0

ndash Ft = Forecast value

ndash At = Actual value = Smoothing constant

bull Ft = Ft-1 + (At-1 - Ft-1)ndash Use for computing forecast

Exponential Smoothing Equations

During the past 8 quarters the Port of Baltimore has unloaded large quantities of grain ( = 10) The first quarter forecast was 175 QuarterActual

1 180 2 168

3 1594 1755 190

6 2057 1808 1829

Exponential Smoothing Example

Find the forecast for the 9th quarter

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

11 180 17500 (Given)

22 168168

33 159159

44 175175

55 190190

66 205205

17500 +17500 +

Exponential Smoothing Solution

QuarterQuarter ActuaActualForecast F t

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010((

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - -

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Ft

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - 17500 - 17500))

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing SolutionFt = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα= = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 +17500 + 1010(180 (180 - 17500- 17500)) = 17550 = 17550

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

1 180 17500 (Given)

22 168168 17500 + 10(180 - 17500) = 1755017500 + 10(180 - 17500) = 17550

33 159159 1755017550 ++ 1010(168 -(168 - 1755017550)) = 17475= 17475

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1995 180 17500 (Given)

1996 168 17500 + 10(180 - 17500) = 17550

1997 159 17550 + 10(168 - 17550) = 17475

1998 175

1999 190

2000 205

17475 + 10(159 - 17475)= 17318

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9 17822 + 10(182 - 17822) = 17858 182 17802 + 10(180 - 17802) = 17822

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Strategy and Issues During a Productrsquos Life

Introduction Growth Maturity Decline

Standardization

Less rapid product changes - more minor changes

Optimum capacity

Increasing stability of process

Long production runs

Product improvement and cost cutting

Little product differentiation

Cost minimization

Over capacity in the industry

Prune line to eliminate items not returning good margin

Reduce capacity

Forecasting critical

Product and process reliability

Competitive product improvements and options

Increase capacity

Shift toward product focused

Enhance distribution

Product design and development critical

Frequent product and process design changes

Short production runs

High production costs

Limited models

Attention to quality

Best period to increase market share

RampD product engineering critical

Practical to change price or quality image

Strengthen niche

Cost control critical

Poor time to change image price or quality

Competitive costs become critical

Defend market position

OM

Str

ateg

yIs

sues

Com

pany

Str

ateg

yIs

sues

HDTV

CD-ROM

Color copiers

Drive-thru restaurants Fax machines

Station wagons

Sales

3 12rdquo Floppy disks

Internet

Types of Forecasts

bull Economic forecastsndash Address business cycle eg inflation rate

money supply etc

bull Technological forecastsndash Predict rate of technological progressndash Predict acceptance of new product

bull Demand forecastsndash Predict sales of existing product

Seven Steps in Forecasting ( 預測的七個基本步驟 )

bull Determine the use of the forecastbull Select the items to be forecastedbull Determine the time horizon of the forecastbull Select the forecasting model(s)bull Gather the databull Make the forecastbull Validate and implement results

Product Demand Charted over 4 Years with Trend and Seasonality

Year1

Year2

Year3

Year4

Seasonal peaks Trend component

Actual demand line

Average demand over four years

Dem

and

for p

rodu

ct o

r ser

vice

Random variation

Actual Demand Moving Average Weighted Moving Average

0

5

10

15

20

25

30

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Sal

es D

eman

d

Actual sales

Moving average

Weighted moving average

Realities of Forecasting

bull Forecasts are seldom perfect

bull Most forecasting methods assume that there is some underlying stability in the system

bull Both product family and aggregated product forecasts are more accurate than individual product forecasts

Forecasting Approaches

bull Used when situation is lsquostablersquo amp historical data existndash Existing productsndash Current technology

bull Involves mathematical techniquesndash eg forecasting sales of

color televisions

Quantitative Methodsbull Used when situation is

vague amp little data existndash New productsndash New technology

bull Involves intuition experiencendash eg forecasting sales on

Internet

Qualitative Methods

Overview of Qualitative Methods

bull Jury of executive opinionndash Pool opinions of high-level executives

sometimes augment by statistical models

bull Delphi methodndash Panel of experts queried iteratively

bull Sales force compositendash Estimates from individual salespersons are

reviewed for reasonableness then aggregated

bull Consumer Market Surveyndash Ask the customer

bull Involves small group of high-level managers

ndash Group estimates demand by working together

bull Combines managerial experience with statistical models

bull Relatively quick

bull lsquoGroup-thinkrsquodisadvantage

copy 1995 Corel Corp

Jury of Executive Opinion

Sales Force Composite

bull Each salesperson projects his or her sales

bull Combined at district amp national levels

bull Sales reps know customersrsquo wants

bull Tends to be overly optimistic

SalesSales

copy 1995 Corel Corp

Delphi Method

bull Iterative group process

bull 3 types of peoplendash Decision makers

ndash Staff

ndash Respondents

bull Reduces lsquogroup-thinkrsquo

Respondents Respondents

Staff Staff

Decision MakersDecision Makers(Sales)

(What will sales be survey)

(Sales will be 45 50 55)

(Sales will be 50)

Consumer Market Survey

bull Ask customers about purchasing plans

bull What consumers say and what they actually do are often different

bull Sometimes difficult to answer

How many hours will you use the Internet

next week

How many hours will you use the Internet

next week

copy 1995 Corel Corp

Overview of Quantitative Approaches

bull Naiumlve approach

bull Moving averages

bull Exponential smoothing

bull Trend projection

bull Linear regression

Time-series Models

Associative models

Quantitative Forecasting Methods (Non-Naive)

QuantitativeForecasting

LinearRegression

AssociativeModels

ExponentialSmoothing

MovingAverage

Time SeriesModels

TrendProjection

bull Set of evenly spaced numerical datandash Obtained by observing response variable at

regular time periods

bull Forecast based only on past valuesndash Assumes that factors influencing past and

present will continue influence in future

bull ExampleYear 1998 1999 2000 2001 2002Sales 787 635 897 932 921

What is a Time Series

TrendTrend

SeasonalSeasonal

CyclicalCyclical

RandomRandom

Time Series Components

bull Persistent overall upward or downward pattern

bull Due to population technology etcbull Several years duration

Mo Qtr Yr

Response

copy 1984-1994 TMaker Co

Trend Component

bull Regular pattern of up amp down fluctuations

bull Due to weather customs etcbull Occurs within 1 year

Mo Qtr

Response

Summer

copy 1984-1994 TMaker Co

Seasonal Component

Common Seasonal PatternsPeriod of Pattern

ldquoSeasonrdquo Length

Number of

ldquoSeasonsrdquo in

Pattern

Week Day 7

Month Week 4 ndash 4 frac12

Month Day 28 ndash 31

Year Quarter 4

Year Month 12

Year Week 52

bull Repeating up amp down movementsbull Due to interactions of factors

influencing economybull Usually 2-10 years duration

Mo Qtr YrMo Qtr Yr

ResponseResponseCycle

Cyclical Component

bull Erratic unsystematic lsquoresidualrsquo fluctuations

bull Due to random variation or unforeseen

events

ndash Union strike

ndash Tornado

bull Short duration amp

nonrepeating

copy 1984-1994 TMaker Co

Random Component

bull Any observed value in a time series is the product (or sum) of time series components

bull Multiplicative modelndash Yi = Ti middot Si middot Ci middot Ri (if quarterly or mo data)

bull Additive modelndash Yi = Ti + Si + Ci + Ri (if quarterly or mo

data)

General Time Series Models

Naive Approach

bull Assumes demand in next period is the same as demand in most recent periodndash eg If May sales were 48

then June sales will be 48

bull Sometimes cost effective amp efficient

copy 1995 Corel Corp

bull MA is a series of arithmetic means

bull Used if little or no trend

bull Used often for smoothingndash Provides overall impression of data over

timebull Equation

MAMAnn

nn Demand inDemand in PreviousPrevious PeriodsPeriods

Moving Average Method

Yoursquore manager of a museum store that sells historical replicas You want to forecast sales (000) for 2003 using a 3-period moving average

1998 41999 62000 52001 32002 7

copy 1995 Corel Corp

Moving Average Example

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 6+5+3=14 143=4 23 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153=50 2002 7 6+5+3=14 143=47 2003 NA 5+3+7=15 153=50

95 96 97 98 99 00Year

Sales

2

4

6

8 Actual

Forecast

Moving Average Graph

bull Used when trend is present ndash Older data usually less important

bull Weights based on intuitionndash Often lay between 0 amp 1 amp sum to 10

bull Equation

WMA =WMA =ΣΣ(Weight for period (Weight for period nn) (Demand in period ) (Demand in period nn))

ΣΣWeightsWeights

Weighted Moving Average Method

Actual Demand Moving Average Weighted Moving

Average

0

5

10

15

20

25

30

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Sal

es D

eman

d

Actual sales

Moving average

Weighted moving average

bull Increasing n makes forecast less sensitive to changes

bull Do not forecast trend wellbull Require much historical data

copy 1984-1994 TMaker Co

Disadvantages of Moving Average Methods

bull Form of weighted moving averagendash Weights decline exponentiallyndash Most recent data weighted most

bull Requires smoothing constant ()ndash Ranges from 0 to 1ndash Subjectively chosen

bull Involves little record keeping of past data

Exponential Smoothing Method

bull Ft = At - 1 + (1-)At - 2 + (1- )2middotAt - 3

+ (1- )3At - 4 + + (1- )t-1middotA0

ndash Ft = Forecast value

ndash At = Actual value = Smoothing constant

bull Ft = Ft-1 + (At-1 - Ft-1)ndash Use for computing forecast

Exponential Smoothing Equations

During the past 8 quarters the Port of Baltimore has unloaded large quantities of grain ( = 10) The first quarter forecast was 175 QuarterActual

1 180 2 168

3 1594 1755 190

6 2057 1808 1829

Exponential Smoothing Example

Find the forecast for the 9th quarter

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

11 180 17500 (Given)

22 168168

33 159159

44 175175

55 190190

66 205205

17500 +17500 +

Exponential Smoothing Solution

QuarterQuarter ActuaActualForecast F t

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010((

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - -

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Ft

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - 17500 - 17500))

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing SolutionFt = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα= = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 +17500 + 1010(180 (180 - 17500- 17500)) = 17550 = 17550

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

1 180 17500 (Given)

22 168168 17500 + 10(180 - 17500) = 1755017500 + 10(180 - 17500) = 17550

33 159159 1755017550 ++ 1010(168 -(168 - 1755017550)) = 17475= 17475

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1995 180 17500 (Given)

1996 168 17500 + 10(180 - 17500) = 17550

1997 159 17550 + 10(168 - 17550) = 17475

1998 175

1999 190

2000 205

17475 + 10(159 - 17475)= 17318

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9 17822 + 10(182 - 17822) = 17858 182 17802 + 10(180 - 17802) = 17822

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Types of Forecasts

bull Economic forecastsndash Address business cycle eg inflation rate

money supply etc

bull Technological forecastsndash Predict rate of technological progressndash Predict acceptance of new product

bull Demand forecastsndash Predict sales of existing product

Seven Steps in Forecasting ( 預測的七個基本步驟 )

bull Determine the use of the forecastbull Select the items to be forecastedbull Determine the time horizon of the forecastbull Select the forecasting model(s)bull Gather the databull Make the forecastbull Validate and implement results

Product Demand Charted over 4 Years with Trend and Seasonality

Year1

Year2

Year3

Year4

Seasonal peaks Trend component

Actual demand line

Average demand over four years

Dem

and

for p

rodu

ct o

r ser

vice

Random variation

Actual Demand Moving Average Weighted Moving Average

0

5

10

15

20

25

30

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Sal

es D

eman

d

Actual sales

Moving average

Weighted moving average

Realities of Forecasting

bull Forecasts are seldom perfect

bull Most forecasting methods assume that there is some underlying stability in the system

bull Both product family and aggregated product forecasts are more accurate than individual product forecasts

Forecasting Approaches

bull Used when situation is lsquostablersquo amp historical data existndash Existing productsndash Current technology

bull Involves mathematical techniquesndash eg forecasting sales of

color televisions

Quantitative Methodsbull Used when situation is

vague amp little data existndash New productsndash New technology

bull Involves intuition experiencendash eg forecasting sales on

Internet

Qualitative Methods

Overview of Qualitative Methods

bull Jury of executive opinionndash Pool opinions of high-level executives

sometimes augment by statistical models

bull Delphi methodndash Panel of experts queried iteratively

bull Sales force compositendash Estimates from individual salespersons are

reviewed for reasonableness then aggregated

bull Consumer Market Surveyndash Ask the customer

bull Involves small group of high-level managers

ndash Group estimates demand by working together

bull Combines managerial experience with statistical models

bull Relatively quick

bull lsquoGroup-thinkrsquodisadvantage

copy 1995 Corel Corp

Jury of Executive Opinion

Sales Force Composite

bull Each salesperson projects his or her sales

bull Combined at district amp national levels

bull Sales reps know customersrsquo wants

bull Tends to be overly optimistic

SalesSales

copy 1995 Corel Corp

Delphi Method

bull Iterative group process

bull 3 types of peoplendash Decision makers

ndash Staff

ndash Respondents

bull Reduces lsquogroup-thinkrsquo

Respondents Respondents

Staff Staff

Decision MakersDecision Makers(Sales)

(What will sales be survey)

(Sales will be 45 50 55)

(Sales will be 50)

Consumer Market Survey

bull Ask customers about purchasing plans

bull What consumers say and what they actually do are often different

bull Sometimes difficult to answer

How many hours will you use the Internet

next week

How many hours will you use the Internet

next week

copy 1995 Corel Corp

Overview of Quantitative Approaches

bull Naiumlve approach

bull Moving averages

bull Exponential smoothing

bull Trend projection

bull Linear regression

Time-series Models

Associative models

Quantitative Forecasting Methods (Non-Naive)

QuantitativeForecasting

LinearRegression

AssociativeModels

ExponentialSmoothing

MovingAverage

Time SeriesModels

TrendProjection

bull Set of evenly spaced numerical datandash Obtained by observing response variable at

regular time periods

bull Forecast based only on past valuesndash Assumes that factors influencing past and

present will continue influence in future

bull ExampleYear 1998 1999 2000 2001 2002Sales 787 635 897 932 921

What is a Time Series

TrendTrend

SeasonalSeasonal

CyclicalCyclical

RandomRandom

Time Series Components

bull Persistent overall upward or downward pattern

bull Due to population technology etcbull Several years duration

Mo Qtr Yr

Response

copy 1984-1994 TMaker Co

Trend Component

bull Regular pattern of up amp down fluctuations

bull Due to weather customs etcbull Occurs within 1 year

Mo Qtr

Response

Summer

copy 1984-1994 TMaker Co

Seasonal Component

Common Seasonal PatternsPeriod of Pattern

ldquoSeasonrdquo Length

Number of

ldquoSeasonsrdquo in

Pattern

Week Day 7

Month Week 4 ndash 4 frac12

Month Day 28 ndash 31

Year Quarter 4

Year Month 12

Year Week 52

bull Repeating up amp down movementsbull Due to interactions of factors

influencing economybull Usually 2-10 years duration

Mo Qtr YrMo Qtr Yr

ResponseResponseCycle

Cyclical Component

bull Erratic unsystematic lsquoresidualrsquo fluctuations

bull Due to random variation or unforeseen

events

ndash Union strike

ndash Tornado

bull Short duration amp

nonrepeating

copy 1984-1994 TMaker Co

Random Component

bull Any observed value in a time series is the product (or sum) of time series components

bull Multiplicative modelndash Yi = Ti middot Si middot Ci middot Ri (if quarterly or mo data)

bull Additive modelndash Yi = Ti + Si + Ci + Ri (if quarterly or mo

data)

General Time Series Models

Naive Approach

bull Assumes demand in next period is the same as demand in most recent periodndash eg If May sales were 48

then June sales will be 48

bull Sometimes cost effective amp efficient

copy 1995 Corel Corp

bull MA is a series of arithmetic means

bull Used if little or no trend

bull Used often for smoothingndash Provides overall impression of data over

timebull Equation

MAMAnn

nn Demand inDemand in PreviousPrevious PeriodsPeriods

Moving Average Method

Yoursquore manager of a museum store that sells historical replicas You want to forecast sales (000) for 2003 using a 3-period moving average

1998 41999 62000 52001 32002 7

copy 1995 Corel Corp

Moving Average Example

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 6+5+3=14 143=4 23 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153=50 2002 7 6+5+3=14 143=47 2003 NA 5+3+7=15 153=50

95 96 97 98 99 00Year

Sales

2

4

6

8 Actual

Forecast

Moving Average Graph

bull Used when trend is present ndash Older data usually less important

bull Weights based on intuitionndash Often lay between 0 amp 1 amp sum to 10

bull Equation

WMA =WMA =ΣΣ(Weight for period (Weight for period nn) (Demand in period ) (Demand in period nn))

ΣΣWeightsWeights

Weighted Moving Average Method

Actual Demand Moving Average Weighted Moving

Average

0

5

10

15

20

25

30

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Sal

es D

eman

d

Actual sales

Moving average

Weighted moving average

bull Increasing n makes forecast less sensitive to changes

bull Do not forecast trend wellbull Require much historical data

copy 1984-1994 TMaker Co

Disadvantages of Moving Average Methods

bull Form of weighted moving averagendash Weights decline exponentiallyndash Most recent data weighted most

bull Requires smoothing constant ()ndash Ranges from 0 to 1ndash Subjectively chosen

bull Involves little record keeping of past data

Exponential Smoothing Method

bull Ft = At - 1 + (1-)At - 2 + (1- )2middotAt - 3

+ (1- )3At - 4 + + (1- )t-1middotA0

ndash Ft = Forecast value

ndash At = Actual value = Smoothing constant

bull Ft = Ft-1 + (At-1 - Ft-1)ndash Use for computing forecast

Exponential Smoothing Equations

During the past 8 quarters the Port of Baltimore has unloaded large quantities of grain ( = 10) The first quarter forecast was 175 QuarterActual

1 180 2 168

3 1594 1755 190

6 2057 1808 1829

Exponential Smoothing Example

Find the forecast for the 9th quarter

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

11 180 17500 (Given)

22 168168

33 159159

44 175175

55 190190

66 205205

17500 +17500 +

Exponential Smoothing Solution

QuarterQuarter ActuaActualForecast F t

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010((

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - -

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Ft

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - 17500 - 17500))

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing SolutionFt = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα= = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 +17500 + 1010(180 (180 - 17500- 17500)) = 17550 = 17550

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

1 180 17500 (Given)

22 168168 17500 + 10(180 - 17500) = 1755017500 + 10(180 - 17500) = 17550

33 159159 1755017550 ++ 1010(168 -(168 - 1755017550)) = 17475= 17475

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1995 180 17500 (Given)

1996 168 17500 + 10(180 - 17500) = 17550

1997 159 17550 + 10(168 - 17550) = 17475

1998 175

1999 190

2000 205

17475 + 10(159 - 17475)= 17318

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9 17822 + 10(182 - 17822) = 17858 182 17802 + 10(180 - 17802) = 17822

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Seven Steps in Forecasting ( 預測的七個基本步驟 )

bull Determine the use of the forecastbull Select the items to be forecastedbull Determine the time horizon of the forecastbull Select the forecasting model(s)bull Gather the databull Make the forecastbull Validate and implement results

Product Demand Charted over 4 Years with Trend and Seasonality

Year1

Year2

Year3

Year4

Seasonal peaks Trend component

Actual demand line

Average demand over four years

Dem

and

for p

rodu

ct o

r ser

vice

Random variation

Actual Demand Moving Average Weighted Moving Average

0

5

10

15

20

25

30

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Sal

es D

eman

d

Actual sales

Moving average

Weighted moving average

Realities of Forecasting

bull Forecasts are seldom perfect

bull Most forecasting methods assume that there is some underlying stability in the system

bull Both product family and aggregated product forecasts are more accurate than individual product forecasts

Forecasting Approaches

bull Used when situation is lsquostablersquo amp historical data existndash Existing productsndash Current technology

bull Involves mathematical techniquesndash eg forecasting sales of

color televisions

Quantitative Methodsbull Used when situation is

vague amp little data existndash New productsndash New technology

bull Involves intuition experiencendash eg forecasting sales on

Internet

Qualitative Methods

Overview of Qualitative Methods

bull Jury of executive opinionndash Pool opinions of high-level executives

sometimes augment by statistical models

bull Delphi methodndash Panel of experts queried iteratively

bull Sales force compositendash Estimates from individual salespersons are

reviewed for reasonableness then aggregated

bull Consumer Market Surveyndash Ask the customer

bull Involves small group of high-level managers

ndash Group estimates demand by working together

bull Combines managerial experience with statistical models

bull Relatively quick

bull lsquoGroup-thinkrsquodisadvantage

copy 1995 Corel Corp

Jury of Executive Opinion

Sales Force Composite

bull Each salesperson projects his or her sales

bull Combined at district amp national levels

bull Sales reps know customersrsquo wants

bull Tends to be overly optimistic

SalesSales

copy 1995 Corel Corp

Delphi Method

bull Iterative group process

bull 3 types of peoplendash Decision makers

ndash Staff

ndash Respondents

bull Reduces lsquogroup-thinkrsquo

Respondents Respondents

Staff Staff

Decision MakersDecision Makers(Sales)

(What will sales be survey)

(Sales will be 45 50 55)

(Sales will be 50)

Consumer Market Survey

bull Ask customers about purchasing plans

bull What consumers say and what they actually do are often different

bull Sometimes difficult to answer

How many hours will you use the Internet

next week

How many hours will you use the Internet

next week

copy 1995 Corel Corp

Overview of Quantitative Approaches

bull Naiumlve approach

bull Moving averages

bull Exponential smoothing

bull Trend projection

bull Linear regression

Time-series Models

Associative models

Quantitative Forecasting Methods (Non-Naive)

QuantitativeForecasting

LinearRegression

AssociativeModels

ExponentialSmoothing

MovingAverage

Time SeriesModels

TrendProjection

bull Set of evenly spaced numerical datandash Obtained by observing response variable at

regular time periods

bull Forecast based only on past valuesndash Assumes that factors influencing past and

present will continue influence in future

bull ExampleYear 1998 1999 2000 2001 2002Sales 787 635 897 932 921

What is a Time Series

TrendTrend

SeasonalSeasonal

CyclicalCyclical

RandomRandom

Time Series Components

bull Persistent overall upward or downward pattern

bull Due to population technology etcbull Several years duration

Mo Qtr Yr

Response

copy 1984-1994 TMaker Co

Trend Component

bull Regular pattern of up amp down fluctuations

bull Due to weather customs etcbull Occurs within 1 year

Mo Qtr

Response

Summer

copy 1984-1994 TMaker Co

Seasonal Component

Common Seasonal PatternsPeriod of Pattern

ldquoSeasonrdquo Length

Number of

ldquoSeasonsrdquo in

Pattern

Week Day 7

Month Week 4 ndash 4 frac12

Month Day 28 ndash 31

Year Quarter 4

Year Month 12

Year Week 52

bull Repeating up amp down movementsbull Due to interactions of factors

influencing economybull Usually 2-10 years duration

Mo Qtr YrMo Qtr Yr

ResponseResponseCycle

Cyclical Component

bull Erratic unsystematic lsquoresidualrsquo fluctuations

bull Due to random variation or unforeseen

events

ndash Union strike

ndash Tornado

bull Short duration amp

nonrepeating

copy 1984-1994 TMaker Co

Random Component

bull Any observed value in a time series is the product (or sum) of time series components

bull Multiplicative modelndash Yi = Ti middot Si middot Ci middot Ri (if quarterly or mo data)

bull Additive modelndash Yi = Ti + Si + Ci + Ri (if quarterly or mo

data)

General Time Series Models

Naive Approach

bull Assumes demand in next period is the same as demand in most recent periodndash eg If May sales were 48

then June sales will be 48

bull Sometimes cost effective amp efficient

copy 1995 Corel Corp

bull MA is a series of arithmetic means

bull Used if little or no trend

bull Used often for smoothingndash Provides overall impression of data over

timebull Equation

MAMAnn

nn Demand inDemand in PreviousPrevious PeriodsPeriods

Moving Average Method

Yoursquore manager of a museum store that sells historical replicas You want to forecast sales (000) for 2003 using a 3-period moving average

1998 41999 62000 52001 32002 7

copy 1995 Corel Corp

Moving Average Example

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 6+5+3=14 143=4 23 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153=50 2002 7 6+5+3=14 143=47 2003 NA 5+3+7=15 153=50

95 96 97 98 99 00Year

Sales

2

4

6

8 Actual

Forecast

Moving Average Graph

bull Used when trend is present ndash Older data usually less important

bull Weights based on intuitionndash Often lay between 0 amp 1 amp sum to 10

bull Equation

WMA =WMA =ΣΣ(Weight for period (Weight for period nn) (Demand in period ) (Demand in period nn))

ΣΣWeightsWeights

Weighted Moving Average Method

Actual Demand Moving Average Weighted Moving

Average

0

5

10

15

20

25

30

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Sal

es D

eman

d

Actual sales

Moving average

Weighted moving average

bull Increasing n makes forecast less sensitive to changes

bull Do not forecast trend wellbull Require much historical data

copy 1984-1994 TMaker Co

Disadvantages of Moving Average Methods

bull Form of weighted moving averagendash Weights decline exponentiallyndash Most recent data weighted most

bull Requires smoothing constant ()ndash Ranges from 0 to 1ndash Subjectively chosen

bull Involves little record keeping of past data

Exponential Smoothing Method

bull Ft = At - 1 + (1-)At - 2 + (1- )2middotAt - 3

+ (1- )3At - 4 + + (1- )t-1middotA0

ndash Ft = Forecast value

ndash At = Actual value = Smoothing constant

bull Ft = Ft-1 + (At-1 - Ft-1)ndash Use for computing forecast

Exponential Smoothing Equations

During the past 8 quarters the Port of Baltimore has unloaded large quantities of grain ( = 10) The first quarter forecast was 175 QuarterActual

1 180 2 168

3 1594 1755 190

6 2057 1808 1829

Exponential Smoothing Example

Find the forecast for the 9th quarter

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

11 180 17500 (Given)

22 168168

33 159159

44 175175

55 190190

66 205205

17500 +17500 +

Exponential Smoothing Solution

QuarterQuarter ActuaActualForecast F t

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010((

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - -

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Ft

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - 17500 - 17500))

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing SolutionFt = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα= = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 +17500 + 1010(180 (180 - 17500- 17500)) = 17550 = 17550

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

1 180 17500 (Given)

22 168168 17500 + 10(180 - 17500) = 1755017500 + 10(180 - 17500) = 17550

33 159159 1755017550 ++ 1010(168 -(168 - 1755017550)) = 17475= 17475

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1995 180 17500 (Given)

1996 168 17500 + 10(180 - 17500) = 17550

1997 159 17550 + 10(168 - 17550) = 17475

1998 175

1999 190

2000 205

17475 + 10(159 - 17475)= 17318

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9 17822 + 10(182 - 17822) = 17858 182 17802 + 10(180 - 17802) = 17822

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Product Demand Charted over 4 Years with Trend and Seasonality

Year1

Year2

Year3

Year4

Seasonal peaks Trend component

Actual demand line

Average demand over four years

Dem

and

for p

rodu

ct o

r ser

vice

Random variation

Actual Demand Moving Average Weighted Moving Average

0

5

10

15

20

25

30

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Sal

es D

eman

d

Actual sales

Moving average

Weighted moving average

Realities of Forecasting

bull Forecasts are seldom perfect

bull Most forecasting methods assume that there is some underlying stability in the system

bull Both product family and aggregated product forecasts are more accurate than individual product forecasts

Forecasting Approaches

bull Used when situation is lsquostablersquo amp historical data existndash Existing productsndash Current technology

bull Involves mathematical techniquesndash eg forecasting sales of

color televisions

Quantitative Methodsbull Used when situation is

vague amp little data existndash New productsndash New technology

bull Involves intuition experiencendash eg forecasting sales on

Internet

Qualitative Methods

Overview of Qualitative Methods

bull Jury of executive opinionndash Pool opinions of high-level executives

sometimes augment by statistical models

bull Delphi methodndash Panel of experts queried iteratively

bull Sales force compositendash Estimates from individual salespersons are

reviewed for reasonableness then aggregated

bull Consumer Market Surveyndash Ask the customer

bull Involves small group of high-level managers

ndash Group estimates demand by working together

bull Combines managerial experience with statistical models

bull Relatively quick

bull lsquoGroup-thinkrsquodisadvantage

copy 1995 Corel Corp

Jury of Executive Opinion

Sales Force Composite

bull Each salesperson projects his or her sales

bull Combined at district amp national levels

bull Sales reps know customersrsquo wants

bull Tends to be overly optimistic

SalesSales

copy 1995 Corel Corp

Delphi Method

bull Iterative group process

bull 3 types of peoplendash Decision makers

ndash Staff

ndash Respondents

bull Reduces lsquogroup-thinkrsquo

Respondents Respondents

Staff Staff

Decision MakersDecision Makers(Sales)

(What will sales be survey)

(Sales will be 45 50 55)

(Sales will be 50)

Consumer Market Survey

bull Ask customers about purchasing plans

bull What consumers say and what they actually do are often different

bull Sometimes difficult to answer

How many hours will you use the Internet

next week

How many hours will you use the Internet

next week

copy 1995 Corel Corp

Overview of Quantitative Approaches

bull Naiumlve approach

bull Moving averages

bull Exponential smoothing

bull Trend projection

bull Linear regression

Time-series Models

Associative models

Quantitative Forecasting Methods (Non-Naive)

QuantitativeForecasting

LinearRegression

AssociativeModels

ExponentialSmoothing

MovingAverage

Time SeriesModels

TrendProjection

bull Set of evenly spaced numerical datandash Obtained by observing response variable at

regular time periods

bull Forecast based only on past valuesndash Assumes that factors influencing past and

present will continue influence in future

bull ExampleYear 1998 1999 2000 2001 2002Sales 787 635 897 932 921

What is a Time Series

TrendTrend

SeasonalSeasonal

CyclicalCyclical

RandomRandom

Time Series Components

bull Persistent overall upward or downward pattern

bull Due to population technology etcbull Several years duration

Mo Qtr Yr

Response

copy 1984-1994 TMaker Co

Trend Component

bull Regular pattern of up amp down fluctuations

bull Due to weather customs etcbull Occurs within 1 year

Mo Qtr

Response

Summer

copy 1984-1994 TMaker Co

Seasonal Component

Common Seasonal PatternsPeriod of Pattern

ldquoSeasonrdquo Length

Number of

ldquoSeasonsrdquo in

Pattern

Week Day 7

Month Week 4 ndash 4 frac12

Month Day 28 ndash 31

Year Quarter 4

Year Month 12

Year Week 52

bull Repeating up amp down movementsbull Due to interactions of factors

influencing economybull Usually 2-10 years duration

Mo Qtr YrMo Qtr Yr

ResponseResponseCycle

Cyclical Component

bull Erratic unsystematic lsquoresidualrsquo fluctuations

bull Due to random variation or unforeseen

events

ndash Union strike

ndash Tornado

bull Short duration amp

nonrepeating

copy 1984-1994 TMaker Co

Random Component

bull Any observed value in a time series is the product (or sum) of time series components

bull Multiplicative modelndash Yi = Ti middot Si middot Ci middot Ri (if quarterly or mo data)

bull Additive modelndash Yi = Ti + Si + Ci + Ri (if quarterly or mo

data)

General Time Series Models

Naive Approach

bull Assumes demand in next period is the same as demand in most recent periodndash eg If May sales were 48

then June sales will be 48

bull Sometimes cost effective amp efficient

copy 1995 Corel Corp

bull MA is a series of arithmetic means

bull Used if little or no trend

bull Used often for smoothingndash Provides overall impression of data over

timebull Equation

MAMAnn

nn Demand inDemand in PreviousPrevious PeriodsPeriods

Moving Average Method

Yoursquore manager of a museum store that sells historical replicas You want to forecast sales (000) for 2003 using a 3-period moving average

1998 41999 62000 52001 32002 7

copy 1995 Corel Corp

Moving Average Example

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 6+5+3=14 143=4 23 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153=50 2002 7 6+5+3=14 143=47 2003 NA 5+3+7=15 153=50

95 96 97 98 99 00Year

Sales

2

4

6

8 Actual

Forecast

Moving Average Graph

bull Used when trend is present ndash Older data usually less important

bull Weights based on intuitionndash Often lay between 0 amp 1 amp sum to 10

bull Equation

WMA =WMA =ΣΣ(Weight for period (Weight for period nn) (Demand in period ) (Demand in period nn))

ΣΣWeightsWeights

Weighted Moving Average Method

Actual Demand Moving Average Weighted Moving

Average

0

5

10

15

20

25

30

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Sal

es D

eman

d

Actual sales

Moving average

Weighted moving average

bull Increasing n makes forecast less sensitive to changes

bull Do not forecast trend wellbull Require much historical data

copy 1984-1994 TMaker Co

Disadvantages of Moving Average Methods

bull Form of weighted moving averagendash Weights decline exponentiallyndash Most recent data weighted most

bull Requires smoothing constant ()ndash Ranges from 0 to 1ndash Subjectively chosen

bull Involves little record keeping of past data

Exponential Smoothing Method

bull Ft = At - 1 + (1-)At - 2 + (1- )2middotAt - 3

+ (1- )3At - 4 + + (1- )t-1middotA0

ndash Ft = Forecast value

ndash At = Actual value = Smoothing constant

bull Ft = Ft-1 + (At-1 - Ft-1)ndash Use for computing forecast

Exponential Smoothing Equations

During the past 8 quarters the Port of Baltimore has unloaded large quantities of grain ( = 10) The first quarter forecast was 175 QuarterActual

1 180 2 168

3 1594 1755 190

6 2057 1808 1829

Exponential Smoothing Example

Find the forecast for the 9th quarter

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

11 180 17500 (Given)

22 168168

33 159159

44 175175

55 190190

66 205205

17500 +17500 +

Exponential Smoothing Solution

QuarterQuarter ActuaActualForecast F t

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010((

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - -

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Ft

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - 17500 - 17500))

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing SolutionFt = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα= = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 +17500 + 1010(180 (180 - 17500- 17500)) = 17550 = 17550

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

1 180 17500 (Given)

22 168168 17500 + 10(180 - 17500) = 1755017500 + 10(180 - 17500) = 17550

33 159159 1755017550 ++ 1010(168 -(168 - 1755017550)) = 17475= 17475

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1995 180 17500 (Given)

1996 168 17500 + 10(180 - 17500) = 17550

1997 159 17550 + 10(168 - 17550) = 17475

1998 175

1999 190

2000 205

17475 + 10(159 - 17475)= 17318

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9 17822 + 10(182 - 17822) = 17858 182 17802 + 10(180 - 17802) = 17822

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Actual Demand Moving Average Weighted Moving Average

0

5

10

15

20

25

30

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Sal

es D

eman

d

Actual sales

Moving average

Weighted moving average

Realities of Forecasting

bull Forecasts are seldom perfect

bull Most forecasting methods assume that there is some underlying stability in the system

bull Both product family and aggregated product forecasts are more accurate than individual product forecasts

Forecasting Approaches

bull Used when situation is lsquostablersquo amp historical data existndash Existing productsndash Current technology

bull Involves mathematical techniquesndash eg forecasting sales of

color televisions

Quantitative Methodsbull Used when situation is

vague amp little data existndash New productsndash New technology

bull Involves intuition experiencendash eg forecasting sales on

Internet

Qualitative Methods

Overview of Qualitative Methods

bull Jury of executive opinionndash Pool opinions of high-level executives

sometimes augment by statistical models

bull Delphi methodndash Panel of experts queried iteratively

bull Sales force compositendash Estimates from individual salespersons are

reviewed for reasonableness then aggregated

bull Consumer Market Surveyndash Ask the customer

bull Involves small group of high-level managers

ndash Group estimates demand by working together

bull Combines managerial experience with statistical models

bull Relatively quick

bull lsquoGroup-thinkrsquodisadvantage

copy 1995 Corel Corp

Jury of Executive Opinion

Sales Force Composite

bull Each salesperson projects his or her sales

bull Combined at district amp national levels

bull Sales reps know customersrsquo wants

bull Tends to be overly optimistic

SalesSales

copy 1995 Corel Corp

Delphi Method

bull Iterative group process

bull 3 types of peoplendash Decision makers

ndash Staff

ndash Respondents

bull Reduces lsquogroup-thinkrsquo

Respondents Respondents

Staff Staff

Decision MakersDecision Makers(Sales)

(What will sales be survey)

(Sales will be 45 50 55)

(Sales will be 50)

Consumer Market Survey

bull Ask customers about purchasing plans

bull What consumers say and what they actually do are often different

bull Sometimes difficult to answer

How many hours will you use the Internet

next week

How many hours will you use the Internet

next week

copy 1995 Corel Corp

Overview of Quantitative Approaches

bull Naiumlve approach

bull Moving averages

bull Exponential smoothing

bull Trend projection

bull Linear regression

Time-series Models

Associative models

Quantitative Forecasting Methods (Non-Naive)

QuantitativeForecasting

LinearRegression

AssociativeModels

ExponentialSmoothing

MovingAverage

Time SeriesModels

TrendProjection

bull Set of evenly spaced numerical datandash Obtained by observing response variable at

regular time periods

bull Forecast based only on past valuesndash Assumes that factors influencing past and

present will continue influence in future

bull ExampleYear 1998 1999 2000 2001 2002Sales 787 635 897 932 921

What is a Time Series

TrendTrend

SeasonalSeasonal

CyclicalCyclical

RandomRandom

Time Series Components

bull Persistent overall upward or downward pattern

bull Due to population technology etcbull Several years duration

Mo Qtr Yr

Response

copy 1984-1994 TMaker Co

Trend Component

bull Regular pattern of up amp down fluctuations

bull Due to weather customs etcbull Occurs within 1 year

Mo Qtr

Response

Summer

copy 1984-1994 TMaker Co

Seasonal Component

Common Seasonal PatternsPeriod of Pattern

ldquoSeasonrdquo Length

Number of

ldquoSeasonsrdquo in

Pattern

Week Day 7

Month Week 4 ndash 4 frac12

Month Day 28 ndash 31

Year Quarter 4

Year Month 12

Year Week 52

bull Repeating up amp down movementsbull Due to interactions of factors

influencing economybull Usually 2-10 years duration

Mo Qtr YrMo Qtr Yr

ResponseResponseCycle

Cyclical Component

bull Erratic unsystematic lsquoresidualrsquo fluctuations

bull Due to random variation or unforeseen

events

ndash Union strike

ndash Tornado

bull Short duration amp

nonrepeating

copy 1984-1994 TMaker Co

Random Component

bull Any observed value in a time series is the product (or sum) of time series components

bull Multiplicative modelndash Yi = Ti middot Si middot Ci middot Ri (if quarterly or mo data)

bull Additive modelndash Yi = Ti + Si + Ci + Ri (if quarterly or mo

data)

General Time Series Models

Naive Approach

bull Assumes demand in next period is the same as demand in most recent periodndash eg If May sales were 48

then June sales will be 48

bull Sometimes cost effective amp efficient

copy 1995 Corel Corp

bull MA is a series of arithmetic means

bull Used if little or no trend

bull Used often for smoothingndash Provides overall impression of data over

timebull Equation

MAMAnn

nn Demand inDemand in PreviousPrevious PeriodsPeriods

Moving Average Method

Yoursquore manager of a museum store that sells historical replicas You want to forecast sales (000) for 2003 using a 3-period moving average

1998 41999 62000 52001 32002 7

copy 1995 Corel Corp

Moving Average Example

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 6+5+3=14 143=4 23 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153=50 2002 7 6+5+3=14 143=47 2003 NA 5+3+7=15 153=50

95 96 97 98 99 00Year

Sales

2

4

6

8 Actual

Forecast

Moving Average Graph

bull Used when trend is present ndash Older data usually less important

bull Weights based on intuitionndash Often lay between 0 amp 1 amp sum to 10

bull Equation

WMA =WMA =ΣΣ(Weight for period (Weight for period nn) (Demand in period ) (Demand in period nn))

ΣΣWeightsWeights

Weighted Moving Average Method

Actual Demand Moving Average Weighted Moving

Average

0

5

10

15

20

25

30

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Sal

es D

eman

d

Actual sales

Moving average

Weighted moving average

bull Increasing n makes forecast less sensitive to changes

bull Do not forecast trend wellbull Require much historical data

copy 1984-1994 TMaker Co

Disadvantages of Moving Average Methods

bull Form of weighted moving averagendash Weights decline exponentiallyndash Most recent data weighted most

bull Requires smoothing constant ()ndash Ranges from 0 to 1ndash Subjectively chosen

bull Involves little record keeping of past data

Exponential Smoothing Method

bull Ft = At - 1 + (1-)At - 2 + (1- )2middotAt - 3

+ (1- )3At - 4 + + (1- )t-1middotA0

ndash Ft = Forecast value

ndash At = Actual value = Smoothing constant

bull Ft = Ft-1 + (At-1 - Ft-1)ndash Use for computing forecast

Exponential Smoothing Equations

During the past 8 quarters the Port of Baltimore has unloaded large quantities of grain ( = 10) The first quarter forecast was 175 QuarterActual

1 180 2 168

3 1594 1755 190

6 2057 1808 1829

Exponential Smoothing Example

Find the forecast for the 9th quarter

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

11 180 17500 (Given)

22 168168

33 159159

44 175175

55 190190

66 205205

17500 +17500 +

Exponential Smoothing Solution

QuarterQuarter ActuaActualForecast F t

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010((

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - -

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Ft

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - 17500 - 17500))

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing SolutionFt = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα= = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 +17500 + 1010(180 (180 - 17500- 17500)) = 17550 = 17550

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

1 180 17500 (Given)

22 168168 17500 + 10(180 - 17500) = 1755017500 + 10(180 - 17500) = 17550

33 159159 1755017550 ++ 1010(168 -(168 - 1755017550)) = 17475= 17475

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1995 180 17500 (Given)

1996 168 17500 + 10(180 - 17500) = 17550

1997 159 17550 + 10(168 - 17550) = 17475

1998 175

1999 190

2000 205

17475 + 10(159 - 17475)= 17318

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9 17822 + 10(182 - 17822) = 17858 182 17802 + 10(180 - 17802) = 17822

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Realities of Forecasting

bull Forecasts are seldom perfect

bull Most forecasting methods assume that there is some underlying stability in the system

bull Both product family and aggregated product forecasts are more accurate than individual product forecasts

Forecasting Approaches

bull Used when situation is lsquostablersquo amp historical data existndash Existing productsndash Current technology

bull Involves mathematical techniquesndash eg forecasting sales of

color televisions

Quantitative Methodsbull Used when situation is

vague amp little data existndash New productsndash New technology

bull Involves intuition experiencendash eg forecasting sales on

Internet

Qualitative Methods

Overview of Qualitative Methods

bull Jury of executive opinionndash Pool opinions of high-level executives

sometimes augment by statistical models

bull Delphi methodndash Panel of experts queried iteratively

bull Sales force compositendash Estimates from individual salespersons are

reviewed for reasonableness then aggregated

bull Consumer Market Surveyndash Ask the customer

bull Involves small group of high-level managers

ndash Group estimates demand by working together

bull Combines managerial experience with statistical models

bull Relatively quick

bull lsquoGroup-thinkrsquodisadvantage

copy 1995 Corel Corp

Jury of Executive Opinion

Sales Force Composite

bull Each salesperson projects his or her sales

bull Combined at district amp national levels

bull Sales reps know customersrsquo wants

bull Tends to be overly optimistic

SalesSales

copy 1995 Corel Corp

Delphi Method

bull Iterative group process

bull 3 types of peoplendash Decision makers

ndash Staff

ndash Respondents

bull Reduces lsquogroup-thinkrsquo

Respondents Respondents

Staff Staff

Decision MakersDecision Makers(Sales)

(What will sales be survey)

(Sales will be 45 50 55)

(Sales will be 50)

Consumer Market Survey

bull Ask customers about purchasing plans

bull What consumers say and what they actually do are often different

bull Sometimes difficult to answer

How many hours will you use the Internet

next week

How many hours will you use the Internet

next week

copy 1995 Corel Corp

Overview of Quantitative Approaches

bull Naiumlve approach

bull Moving averages

bull Exponential smoothing

bull Trend projection

bull Linear regression

Time-series Models

Associative models

Quantitative Forecasting Methods (Non-Naive)

QuantitativeForecasting

LinearRegression

AssociativeModels

ExponentialSmoothing

MovingAverage

Time SeriesModels

TrendProjection

bull Set of evenly spaced numerical datandash Obtained by observing response variable at

regular time periods

bull Forecast based only on past valuesndash Assumes that factors influencing past and

present will continue influence in future

bull ExampleYear 1998 1999 2000 2001 2002Sales 787 635 897 932 921

What is a Time Series

TrendTrend

SeasonalSeasonal

CyclicalCyclical

RandomRandom

Time Series Components

bull Persistent overall upward or downward pattern

bull Due to population technology etcbull Several years duration

Mo Qtr Yr

Response

copy 1984-1994 TMaker Co

Trend Component

bull Regular pattern of up amp down fluctuations

bull Due to weather customs etcbull Occurs within 1 year

Mo Qtr

Response

Summer

copy 1984-1994 TMaker Co

Seasonal Component

Common Seasonal PatternsPeriod of Pattern

ldquoSeasonrdquo Length

Number of

ldquoSeasonsrdquo in

Pattern

Week Day 7

Month Week 4 ndash 4 frac12

Month Day 28 ndash 31

Year Quarter 4

Year Month 12

Year Week 52

bull Repeating up amp down movementsbull Due to interactions of factors

influencing economybull Usually 2-10 years duration

Mo Qtr YrMo Qtr Yr

ResponseResponseCycle

Cyclical Component

bull Erratic unsystematic lsquoresidualrsquo fluctuations

bull Due to random variation or unforeseen

events

ndash Union strike

ndash Tornado

bull Short duration amp

nonrepeating

copy 1984-1994 TMaker Co

Random Component

bull Any observed value in a time series is the product (or sum) of time series components

bull Multiplicative modelndash Yi = Ti middot Si middot Ci middot Ri (if quarterly or mo data)

bull Additive modelndash Yi = Ti + Si + Ci + Ri (if quarterly or mo

data)

General Time Series Models

Naive Approach

bull Assumes demand in next period is the same as demand in most recent periodndash eg If May sales were 48

then June sales will be 48

bull Sometimes cost effective amp efficient

copy 1995 Corel Corp

bull MA is a series of arithmetic means

bull Used if little or no trend

bull Used often for smoothingndash Provides overall impression of data over

timebull Equation

MAMAnn

nn Demand inDemand in PreviousPrevious PeriodsPeriods

Moving Average Method

Yoursquore manager of a museum store that sells historical replicas You want to forecast sales (000) for 2003 using a 3-period moving average

1998 41999 62000 52001 32002 7

copy 1995 Corel Corp

Moving Average Example

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 6+5+3=14 143=4 23 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153=50 2002 7 6+5+3=14 143=47 2003 NA 5+3+7=15 153=50

95 96 97 98 99 00Year

Sales

2

4

6

8 Actual

Forecast

Moving Average Graph

bull Used when trend is present ndash Older data usually less important

bull Weights based on intuitionndash Often lay between 0 amp 1 amp sum to 10

bull Equation

WMA =WMA =ΣΣ(Weight for period (Weight for period nn) (Demand in period ) (Demand in period nn))

ΣΣWeightsWeights

Weighted Moving Average Method

Actual Demand Moving Average Weighted Moving

Average

0

5

10

15

20

25

30

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Sal

es D

eman

d

Actual sales

Moving average

Weighted moving average

bull Increasing n makes forecast less sensitive to changes

bull Do not forecast trend wellbull Require much historical data

copy 1984-1994 TMaker Co

Disadvantages of Moving Average Methods

bull Form of weighted moving averagendash Weights decline exponentiallyndash Most recent data weighted most

bull Requires smoothing constant ()ndash Ranges from 0 to 1ndash Subjectively chosen

bull Involves little record keeping of past data

Exponential Smoothing Method

bull Ft = At - 1 + (1-)At - 2 + (1- )2middotAt - 3

+ (1- )3At - 4 + + (1- )t-1middotA0

ndash Ft = Forecast value

ndash At = Actual value = Smoothing constant

bull Ft = Ft-1 + (At-1 - Ft-1)ndash Use for computing forecast

Exponential Smoothing Equations

During the past 8 quarters the Port of Baltimore has unloaded large quantities of grain ( = 10) The first quarter forecast was 175 QuarterActual

1 180 2 168

3 1594 1755 190

6 2057 1808 1829

Exponential Smoothing Example

Find the forecast for the 9th quarter

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

11 180 17500 (Given)

22 168168

33 159159

44 175175

55 190190

66 205205

17500 +17500 +

Exponential Smoothing Solution

QuarterQuarter ActuaActualForecast F t

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010((

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - -

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Ft

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - 17500 - 17500))

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing SolutionFt = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα= = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 +17500 + 1010(180 (180 - 17500- 17500)) = 17550 = 17550

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

1 180 17500 (Given)

22 168168 17500 + 10(180 - 17500) = 1755017500 + 10(180 - 17500) = 17550

33 159159 1755017550 ++ 1010(168 -(168 - 1755017550)) = 17475= 17475

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1995 180 17500 (Given)

1996 168 17500 + 10(180 - 17500) = 17550

1997 159 17550 + 10(168 - 17550) = 17475

1998 175

1999 190

2000 205

17475 + 10(159 - 17475)= 17318

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9 17822 + 10(182 - 17822) = 17858 182 17802 + 10(180 - 17802) = 17822

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Forecasting Approaches

bull Used when situation is lsquostablersquo amp historical data existndash Existing productsndash Current technology

bull Involves mathematical techniquesndash eg forecasting sales of

color televisions

Quantitative Methodsbull Used when situation is

vague amp little data existndash New productsndash New technology

bull Involves intuition experiencendash eg forecasting sales on

Internet

Qualitative Methods

Overview of Qualitative Methods

bull Jury of executive opinionndash Pool opinions of high-level executives

sometimes augment by statistical models

bull Delphi methodndash Panel of experts queried iteratively

bull Sales force compositendash Estimates from individual salespersons are

reviewed for reasonableness then aggregated

bull Consumer Market Surveyndash Ask the customer

bull Involves small group of high-level managers

ndash Group estimates demand by working together

bull Combines managerial experience with statistical models

bull Relatively quick

bull lsquoGroup-thinkrsquodisadvantage

copy 1995 Corel Corp

Jury of Executive Opinion

Sales Force Composite

bull Each salesperson projects his or her sales

bull Combined at district amp national levels

bull Sales reps know customersrsquo wants

bull Tends to be overly optimistic

SalesSales

copy 1995 Corel Corp

Delphi Method

bull Iterative group process

bull 3 types of peoplendash Decision makers

ndash Staff

ndash Respondents

bull Reduces lsquogroup-thinkrsquo

Respondents Respondents

Staff Staff

Decision MakersDecision Makers(Sales)

(What will sales be survey)

(Sales will be 45 50 55)

(Sales will be 50)

Consumer Market Survey

bull Ask customers about purchasing plans

bull What consumers say and what they actually do are often different

bull Sometimes difficult to answer

How many hours will you use the Internet

next week

How many hours will you use the Internet

next week

copy 1995 Corel Corp

Overview of Quantitative Approaches

bull Naiumlve approach

bull Moving averages

bull Exponential smoothing

bull Trend projection

bull Linear regression

Time-series Models

Associative models

Quantitative Forecasting Methods (Non-Naive)

QuantitativeForecasting

LinearRegression

AssociativeModels

ExponentialSmoothing

MovingAverage

Time SeriesModels

TrendProjection

bull Set of evenly spaced numerical datandash Obtained by observing response variable at

regular time periods

bull Forecast based only on past valuesndash Assumes that factors influencing past and

present will continue influence in future

bull ExampleYear 1998 1999 2000 2001 2002Sales 787 635 897 932 921

What is a Time Series

TrendTrend

SeasonalSeasonal

CyclicalCyclical

RandomRandom

Time Series Components

bull Persistent overall upward or downward pattern

bull Due to population technology etcbull Several years duration

Mo Qtr Yr

Response

copy 1984-1994 TMaker Co

Trend Component

bull Regular pattern of up amp down fluctuations

bull Due to weather customs etcbull Occurs within 1 year

Mo Qtr

Response

Summer

copy 1984-1994 TMaker Co

Seasonal Component

Common Seasonal PatternsPeriod of Pattern

ldquoSeasonrdquo Length

Number of

ldquoSeasonsrdquo in

Pattern

Week Day 7

Month Week 4 ndash 4 frac12

Month Day 28 ndash 31

Year Quarter 4

Year Month 12

Year Week 52

bull Repeating up amp down movementsbull Due to interactions of factors

influencing economybull Usually 2-10 years duration

Mo Qtr YrMo Qtr Yr

ResponseResponseCycle

Cyclical Component

bull Erratic unsystematic lsquoresidualrsquo fluctuations

bull Due to random variation or unforeseen

events

ndash Union strike

ndash Tornado

bull Short duration amp

nonrepeating

copy 1984-1994 TMaker Co

Random Component

bull Any observed value in a time series is the product (or sum) of time series components

bull Multiplicative modelndash Yi = Ti middot Si middot Ci middot Ri (if quarterly or mo data)

bull Additive modelndash Yi = Ti + Si + Ci + Ri (if quarterly or mo

data)

General Time Series Models

Naive Approach

bull Assumes demand in next period is the same as demand in most recent periodndash eg If May sales were 48

then June sales will be 48

bull Sometimes cost effective amp efficient

copy 1995 Corel Corp

bull MA is a series of arithmetic means

bull Used if little or no trend

bull Used often for smoothingndash Provides overall impression of data over

timebull Equation

MAMAnn

nn Demand inDemand in PreviousPrevious PeriodsPeriods

Moving Average Method

Yoursquore manager of a museum store that sells historical replicas You want to forecast sales (000) for 2003 using a 3-period moving average

1998 41999 62000 52001 32002 7

copy 1995 Corel Corp

Moving Average Example

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 6+5+3=14 143=4 23 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153=50 2002 7 6+5+3=14 143=47 2003 NA 5+3+7=15 153=50

95 96 97 98 99 00Year

Sales

2

4

6

8 Actual

Forecast

Moving Average Graph

bull Used when trend is present ndash Older data usually less important

bull Weights based on intuitionndash Often lay between 0 amp 1 amp sum to 10

bull Equation

WMA =WMA =ΣΣ(Weight for period (Weight for period nn) (Demand in period ) (Demand in period nn))

ΣΣWeightsWeights

Weighted Moving Average Method

Actual Demand Moving Average Weighted Moving

Average

0

5

10

15

20

25

30

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Sal

es D

eman

d

Actual sales

Moving average

Weighted moving average

bull Increasing n makes forecast less sensitive to changes

bull Do not forecast trend wellbull Require much historical data

copy 1984-1994 TMaker Co

Disadvantages of Moving Average Methods

bull Form of weighted moving averagendash Weights decline exponentiallyndash Most recent data weighted most

bull Requires smoothing constant ()ndash Ranges from 0 to 1ndash Subjectively chosen

bull Involves little record keeping of past data

Exponential Smoothing Method

bull Ft = At - 1 + (1-)At - 2 + (1- )2middotAt - 3

+ (1- )3At - 4 + + (1- )t-1middotA0

ndash Ft = Forecast value

ndash At = Actual value = Smoothing constant

bull Ft = Ft-1 + (At-1 - Ft-1)ndash Use for computing forecast

Exponential Smoothing Equations

During the past 8 quarters the Port of Baltimore has unloaded large quantities of grain ( = 10) The first quarter forecast was 175 QuarterActual

1 180 2 168

3 1594 1755 190

6 2057 1808 1829

Exponential Smoothing Example

Find the forecast for the 9th quarter

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

11 180 17500 (Given)

22 168168

33 159159

44 175175

55 190190

66 205205

17500 +17500 +

Exponential Smoothing Solution

QuarterQuarter ActuaActualForecast F t

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010((

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - -

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Ft

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - 17500 - 17500))

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing SolutionFt = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα= = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 +17500 + 1010(180 (180 - 17500- 17500)) = 17550 = 17550

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

1 180 17500 (Given)

22 168168 17500 + 10(180 - 17500) = 1755017500 + 10(180 - 17500) = 17550

33 159159 1755017550 ++ 1010(168 -(168 - 1755017550)) = 17475= 17475

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1995 180 17500 (Given)

1996 168 17500 + 10(180 - 17500) = 17550

1997 159 17550 + 10(168 - 17550) = 17475

1998 175

1999 190

2000 205

17475 + 10(159 - 17475)= 17318

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9 17822 + 10(182 - 17822) = 17858 182 17802 + 10(180 - 17802) = 17822

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Overview of Qualitative Methods

bull Jury of executive opinionndash Pool opinions of high-level executives

sometimes augment by statistical models

bull Delphi methodndash Panel of experts queried iteratively

bull Sales force compositendash Estimates from individual salespersons are

reviewed for reasonableness then aggregated

bull Consumer Market Surveyndash Ask the customer

bull Involves small group of high-level managers

ndash Group estimates demand by working together

bull Combines managerial experience with statistical models

bull Relatively quick

bull lsquoGroup-thinkrsquodisadvantage

copy 1995 Corel Corp

Jury of Executive Opinion

Sales Force Composite

bull Each salesperson projects his or her sales

bull Combined at district amp national levels

bull Sales reps know customersrsquo wants

bull Tends to be overly optimistic

SalesSales

copy 1995 Corel Corp

Delphi Method

bull Iterative group process

bull 3 types of peoplendash Decision makers

ndash Staff

ndash Respondents

bull Reduces lsquogroup-thinkrsquo

Respondents Respondents

Staff Staff

Decision MakersDecision Makers(Sales)

(What will sales be survey)

(Sales will be 45 50 55)

(Sales will be 50)

Consumer Market Survey

bull Ask customers about purchasing plans

bull What consumers say and what they actually do are often different

bull Sometimes difficult to answer

How many hours will you use the Internet

next week

How many hours will you use the Internet

next week

copy 1995 Corel Corp

Overview of Quantitative Approaches

bull Naiumlve approach

bull Moving averages

bull Exponential smoothing

bull Trend projection

bull Linear regression

Time-series Models

Associative models

Quantitative Forecasting Methods (Non-Naive)

QuantitativeForecasting

LinearRegression

AssociativeModels

ExponentialSmoothing

MovingAverage

Time SeriesModels

TrendProjection

bull Set of evenly spaced numerical datandash Obtained by observing response variable at

regular time periods

bull Forecast based only on past valuesndash Assumes that factors influencing past and

present will continue influence in future

bull ExampleYear 1998 1999 2000 2001 2002Sales 787 635 897 932 921

What is a Time Series

TrendTrend

SeasonalSeasonal

CyclicalCyclical

RandomRandom

Time Series Components

bull Persistent overall upward or downward pattern

bull Due to population technology etcbull Several years duration

Mo Qtr Yr

Response

copy 1984-1994 TMaker Co

Trend Component

bull Regular pattern of up amp down fluctuations

bull Due to weather customs etcbull Occurs within 1 year

Mo Qtr

Response

Summer

copy 1984-1994 TMaker Co

Seasonal Component

Common Seasonal PatternsPeriod of Pattern

ldquoSeasonrdquo Length

Number of

ldquoSeasonsrdquo in

Pattern

Week Day 7

Month Week 4 ndash 4 frac12

Month Day 28 ndash 31

Year Quarter 4

Year Month 12

Year Week 52

bull Repeating up amp down movementsbull Due to interactions of factors

influencing economybull Usually 2-10 years duration

Mo Qtr YrMo Qtr Yr

ResponseResponseCycle

Cyclical Component

bull Erratic unsystematic lsquoresidualrsquo fluctuations

bull Due to random variation or unforeseen

events

ndash Union strike

ndash Tornado

bull Short duration amp

nonrepeating

copy 1984-1994 TMaker Co

Random Component

bull Any observed value in a time series is the product (or sum) of time series components

bull Multiplicative modelndash Yi = Ti middot Si middot Ci middot Ri (if quarterly or mo data)

bull Additive modelndash Yi = Ti + Si + Ci + Ri (if quarterly or mo

data)

General Time Series Models

Naive Approach

bull Assumes demand in next period is the same as demand in most recent periodndash eg If May sales were 48

then June sales will be 48

bull Sometimes cost effective amp efficient

copy 1995 Corel Corp

bull MA is a series of arithmetic means

bull Used if little or no trend

bull Used often for smoothingndash Provides overall impression of data over

timebull Equation

MAMAnn

nn Demand inDemand in PreviousPrevious PeriodsPeriods

Moving Average Method

Yoursquore manager of a museum store that sells historical replicas You want to forecast sales (000) for 2003 using a 3-period moving average

1998 41999 62000 52001 32002 7

copy 1995 Corel Corp

Moving Average Example

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 6+5+3=14 143=4 23 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153=50 2002 7 6+5+3=14 143=47 2003 NA 5+3+7=15 153=50

95 96 97 98 99 00Year

Sales

2

4

6

8 Actual

Forecast

Moving Average Graph

bull Used when trend is present ndash Older data usually less important

bull Weights based on intuitionndash Often lay between 0 amp 1 amp sum to 10

bull Equation

WMA =WMA =ΣΣ(Weight for period (Weight for period nn) (Demand in period ) (Demand in period nn))

ΣΣWeightsWeights

Weighted Moving Average Method

Actual Demand Moving Average Weighted Moving

Average

0

5

10

15

20

25

30

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Sal

es D

eman

d

Actual sales

Moving average

Weighted moving average

bull Increasing n makes forecast less sensitive to changes

bull Do not forecast trend wellbull Require much historical data

copy 1984-1994 TMaker Co

Disadvantages of Moving Average Methods

bull Form of weighted moving averagendash Weights decline exponentiallyndash Most recent data weighted most

bull Requires smoothing constant ()ndash Ranges from 0 to 1ndash Subjectively chosen

bull Involves little record keeping of past data

Exponential Smoothing Method

bull Ft = At - 1 + (1-)At - 2 + (1- )2middotAt - 3

+ (1- )3At - 4 + + (1- )t-1middotA0

ndash Ft = Forecast value

ndash At = Actual value = Smoothing constant

bull Ft = Ft-1 + (At-1 - Ft-1)ndash Use for computing forecast

Exponential Smoothing Equations

During the past 8 quarters the Port of Baltimore has unloaded large quantities of grain ( = 10) The first quarter forecast was 175 QuarterActual

1 180 2 168

3 1594 1755 190

6 2057 1808 1829

Exponential Smoothing Example

Find the forecast for the 9th quarter

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

11 180 17500 (Given)

22 168168

33 159159

44 175175

55 190190

66 205205

17500 +17500 +

Exponential Smoothing Solution

QuarterQuarter ActuaActualForecast F t

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010((

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - -

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Ft

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - 17500 - 17500))

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing SolutionFt = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα= = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 +17500 + 1010(180 (180 - 17500- 17500)) = 17550 = 17550

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

1 180 17500 (Given)

22 168168 17500 + 10(180 - 17500) = 1755017500 + 10(180 - 17500) = 17550

33 159159 1755017550 ++ 1010(168 -(168 - 1755017550)) = 17475= 17475

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1995 180 17500 (Given)

1996 168 17500 + 10(180 - 17500) = 17550

1997 159 17550 + 10(168 - 17550) = 17475

1998 175

1999 190

2000 205

17475 + 10(159 - 17475)= 17318

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9 17822 + 10(182 - 17822) = 17858 182 17802 + 10(180 - 17802) = 17822

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

bull Involves small group of high-level managers

ndash Group estimates demand by working together

bull Combines managerial experience with statistical models

bull Relatively quick

bull lsquoGroup-thinkrsquodisadvantage

copy 1995 Corel Corp

Jury of Executive Opinion

Sales Force Composite

bull Each salesperson projects his or her sales

bull Combined at district amp national levels

bull Sales reps know customersrsquo wants

bull Tends to be overly optimistic

SalesSales

copy 1995 Corel Corp

Delphi Method

bull Iterative group process

bull 3 types of peoplendash Decision makers

ndash Staff

ndash Respondents

bull Reduces lsquogroup-thinkrsquo

Respondents Respondents

Staff Staff

Decision MakersDecision Makers(Sales)

(What will sales be survey)

(Sales will be 45 50 55)

(Sales will be 50)

Consumer Market Survey

bull Ask customers about purchasing plans

bull What consumers say and what they actually do are often different

bull Sometimes difficult to answer

How many hours will you use the Internet

next week

How many hours will you use the Internet

next week

copy 1995 Corel Corp

Overview of Quantitative Approaches

bull Naiumlve approach

bull Moving averages

bull Exponential smoothing

bull Trend projection

bull Linear regression

Time-series Models

Associative models

Quantitative Forecasting Methods (Non-Naive)

QuantitativeForecasting

LinearRegression

AssociativeModels

ExponentialSmoothing

MovingAverage

Time SeriesModels

TrendProjection

bull Set of evenly spaced numerical datandash Obtained by observing response variable at

regular time periods

bull Forecast based only on past valuesndash Assumes that factors influencing past and

present will continue influence in future

bull ExampleYear 1998 1999 2000 2001 2002Sales 787 635 897 932 921

What is a Time Series

TrendTrend

SeasonalSeasonal

CyclicalCyclical

RandomRandom

Time Series Components

bull Persistent overall upward or downward pattern

bull Due to population technology etcbull Several years duration

Mo Qtr Yr

Response

copy 1984-1994 TMaker Co

Trend Component

bull Regular pattern of up amp down fluctuations

bull Due to weather customs etcbull Occurs within 1 year

Mo Qtr

Response

Summer

copy 1984-1994 TMaker Co

Seasonal Component

Common Seasonal PatternsPeriod of Pattern

ldquoSeasonrdquo Length

Number of

ldquoSeasonsrdquo in

Pattern

Week Day 7

Month Week 4 ndash 4 frac12

Month Day 28 ndash 31

Year Quarter 4

Year Month 12

Year Week 52

bull Repeating up amp down movementsbull Due to interactions of factors

influencing economybull Usually 2-10 years duration

Mo Qtr YrMo Qtr Yr

ResponseResponseCycle

Cyclical Component

bull Erratic unsystematic lsquoresidualrsquo fluctuations

bull Due to random variation or unforeseen

events

ndash Union strike

ndash Tornado

bull Short duration amp

nonrepeating

copy 1984-1994 TMaker Co

Random Component

bull Any observed value in a time series is the product (or sum) of time series components

bull Multiplicative modelndash Yi = Ti middot Si middot Ci middot Ri (if quarterly or mo data)

bull Additive modelndash Yi = Ti + Si + Ci + Ri (if quarterly or mo

data)

General Time Series Models

Naive Approach

bull Assumes demand in next period is the same as demand in most recent periodndash eg If May sales were 48

then June sales will be 48

bull Sometimes cost effective amp efficient

copy 1995 Corel Corp

bull MA is a series of arithmetic means

bull Used if little or no trend

bull Used often for smoothingndash Provides overall impression of data over

timebull Equation

MAMAnn

nn Demand inDemand in PreviousPrevious PeriodsPeriods

Moving Average Method

Yoursquore manager of a museum store that sells historical replicas You want to forecast sales (000) for 2003 using a 3-period moving average

1998 41999 62000 52001 32002 7

copy 1995 Corel Corp

Moving Average Example

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 6+5+3=14 143=4 23 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153=50 2002 7 6+5+3=14 143=47 2003 NA 5+3+7=15 153=50

95 96 97 98 99 00Year

Sales

2

4

6

8 Actual

Forecast

Moving Average Graph

bull Used when trend is present ndash Older data usually less important

bull Weights based on intuitionndash Often lay between 0 amp 1 amp sum to 10

bull Equation

WMA =WMA =ΣΣ(Weight for period (Weight for period nn) (Demand in period ) (Demand in period nn))

ΣΣWeightsWeights

Weighted Moving Average Method

Actual Demand Moving Average Weighted Moving

Average

0

5

10

15

20

25

30

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Sal

es D

eman

d

Actual sales

Moving average

Weighted moving average

bull Increasing n makes forecast less sensitive to changes

bull Do not forecast trend wellbull Require much historical data

copy 1984-1994 TMaker Co

Disadvantages of Moving Average Methods

bull Form of weighted moving averagendash Weights decline exponentiallyndash Most recent data weighted most

bull Requires smoothing constant ()ndash Ranges from 0 to 1ndash Subjectively chosen

bull Involves little record keeping of past data

Exponential Smoothing Method

bull Ft = At - 1 + (1-)At - 2 + (1- )2middotAt - 3

+ (1- )3At - 4 + + (1- )t-1middotA0

ndash Ft = Forecast value

ndash At = Actual value = Smoothing constant

bull Ft = Ft-1 + (At-1 - Ft-1)ndash Use for computing forecast

Exponential Smoothing Equations

During the past 8 quarters the Port of Baltimore has unloaded large quantities of grain ( = 10) The first quarter forecast was 175 QuarterActual

1 180 2 168

3 1594 1755 190

6 2057 1808 1829

Exponential Smoothing Example

Find the forecast for the 9th quarter

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

11 180 17500 (Given)

22 168168

33 159159

44 175175

55 190190

66 205205

17500 +17500 +

Exponential Smoothing Solution

QuarterQuarter ActuaActualForecast F t

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010((

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - -

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Ft

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - 17500 - 17500))

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing SolutionFt = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα= = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 +17500 + 1010(180 (180 - 17500- 17500)) = 17550 = 17550

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

1 180 17500 (Given)

22 168168 17500 + 10(180 - 17500) = 1755017500 + 10(180 - 17500) = 17550

33 159159 1755017550 ++ 1010(168 -(168 - 1755017550)) = 17475= 17475

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1995 180 17500 (Given)

1996 168 17500 + 10(180 - 17500) = 17550

1997 159 17550 + 10(168 - 17550) = 17475

1998 175

1999 190

2000 205

17475 + 10(159 - 17475)= 17318

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9 17822 + 10(182 - 17822) = 17858 182 17802 + 10(180 - 17802) = 17822

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Sales Force Composite

bull Each salesperson projects his or her sales

bull Combined at district amp national levels

bull Sales reps know customersrsquo wants

bull Tends to be overly optimistic

SalesSales

copy 1995 Corel Corp

Delphi Method

bull Iterative group process

bull 3 types of peoplendash Decision makers

ndash Staff

ndash Respondents

bull Reduces lsquogroup-thinkrsquo

Respondents Respondents

Staff Staff

Decision MakersDecision Makers(Sales)

(What will sales be survey)

(Sales will be 45 50 55)

(Sales will be 50)

Consumer Market Survey

bull Ask customers about purchasing plans

bull What consumers say and what they actually do are often different

bull Sometimes difficult to answer

How many hours will you use the Internet

next week

How many hours will you use the Internet

next week

copy 1995 Corel Corp

Overview of Quantitative Approaches

bull Naiumlve approach

bull Moving averages

bull Exponential smoothing

bull Trend projection

bull Linear regression

Time-series Models

Associative models

Quantitative Forecasting Methods (Non-Naive)

QuantitativeForecasting

LinearRegression

AssociativeModels

ExponentialSmoothing

MovingAverage

Time SeriesModels

TrendProjection

bull Set of evenly spaced numerical datandash Obtained by observing response variable at

regular time periods

bull Forecast based only on past valuesndash Assumes that factors influencing past and

present will continue influence in future

bull ExampleYear 1998 1999 2000 2001 2002Sales 787 635 897 932 921

What is a Time Series

TrendTrend

SeasonalSeasonal

CyclicalCyclical

RandomRandom

Time Series Components

bull Persistent overall upward or downward pattern

bull Due to population technology etcbull Several years duration

Mo Qtr Yr

Response

copy 1984-1994 TMaker Co

Trend Component

bull Regular pattern of up amp down fluctuations

bull Due to weather customs etcbull Occurs within 1 year

Mo Qtr

Response

Summer

copy 1984-1994 TMaker Co

Seasonal Component

Common Seasonal PatternsPeriod of Pattern

ldquoSeasonrdquo Length

Number of

ldquoSeasonsrdquo in

Pattern

Week Day 7

Month Week 4 ndash 4 frac12

Month Day 28 ndash 31

Year Quarter 4

Year Month 12

Year Week 52

bull Repeating up amp down movementsbull Due to interactions of factors

influencing economybull Usually 2-10 years duration

Mo Qtr YrMo Qtr Yr

ResponseResponseCycle

Cyclical Component

bull Erratic unsystematic lsquoresidualrsquo fluctuations

bull Due to random variation or unforeseen

events

ndash Union strike

ndash Tornado

bull Short duration amp

nonrepeating

copy 1984-1994 TMaker Co

Random Component

bull Any observed value in a time series is the product (or sum) of time series components

bull Multiplicative modelndash Yi = Ti middot Si middot Ci middot Ri (if quarterly or mo data)

bull Additive modelndash Yi = Ti + Si + Ci + Ri (if quarterly or mo

data)

General Time Series Models

Naive Approach

bull Assumes demand in next period is the same as demand in most recent periodndash eg If May sales were 48

then June sales will be 48

bull Sometimes cost effective amp efficient

copy 1995 Corel Corp

bull MA is a series of arithmetic means

bull Used if little or no trend

bull Used often for smoothingndash Provides overall impression of data over

timebull Equation

MAMAnn

nn Demand inDemand in PreviousPrevious PeriodsPeriods

Moving Average Method

Yoursquore manager of a museum store that sells historical replicas You want to forecast sales (000) for 2003 using a 3-period moving average

1998 41999 62000 52001 32002 7

copy 1995 Corel Corp

Moving Average Example

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 6+5+3=14 143=4 23 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153=50 2002 7 6+5+3=14 143=47 2003 NA 5+3+7=15 153=50

95 96 97 98 99 00Year

Sales

2

4

6

8 Actual

Forecast

Moving Average Graph

bull Used when trend is present ndash Older data usually less important

bull Weights based on intuitionndash Often lay between 0 amp 1 amp sum to 10

bull Equation

WMA =WMA =ΣΣ(Weight for period (Weight for period nn) (Demand in period ) (Demand in period nn))

ΣΣWeightsWeights

Weighted Moving Average Method

Actual Demand Moving Average Weighted Moving

Average

0

5

10

15

20

25

30

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Sal

es D

eman

d

Actual sales

Moving average

Weighted moving average

bull Increasing n makes forecast less sensitive to changes

bull Do not forecast trend wellbull Require much historical data

copy 1984-1994 TMaker Co

Disadvantages of Moving Average Methods

bull Form of weighted moving averagendash Weights decline exponentiallyndash Most recent data weighted most

bull Requires smoothing constant ()ndash Ranges from 0 to 1ndash Subjectively chosen

bull Involves little record keeping of past data

Exponential Smoothing Method

bull Ft = At - 1 + (1-)At - 2 + (1- )2middotAt - 3

+ (1- )3At - 4 + + (1- )t-1middotA0

ndash Ft = Forecast value

ndash At = Actual value = Smoothing constant

bull Ft = Ft-1 + (At-1 - Ft-1)ndash Use for computing forecast

Exponential Smoothing Equations

During the past 8 quarters the Port of Baltimore has unloaded large quantities of grain ( = 10) The first quarter forecast was 175 QuarterActual

1 180 2 168

3 1594 1755 190

6 2057 1808 1829

Exponential Smoothing Example

Find the forecast for the 9th quarter

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

11 180 17500 (Given)

22 168168

33 159159

44 175175

55 190190

66 205205

17500 +17500 +

Exponential Smoothing Solution

QuarterQuarter ActuaActualForecast F t

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010((

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - -

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Ft

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - 17500 - 17500))

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing SolutionFt = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα= = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 +17500 + 1010(180 (180 - 17500- 17500)) = 17550 = 17550

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

1 180 17500 (Given)

22 168168 17500 + 10(180 - 17500) = 1755017500 + 10(180 - 17500) = 17550

33 159159 1755017550 ++ 1010(168 -(168 - 1755017550)) = 17475= 17475

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1995 180 17500 (Given)

1996 168 17500 + 10(180 - 17500) = 17550

1997 159 17550 + 10(168 - 17550) = 17475

1998 175

1999 190

2000 205

17475 + 10(159 - 17475)= 17318

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9 17822 + 10(182 - 17822) = 17858 182 17802 + 10(180 - 17802) = 17822

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Delphi Method

bull Iterative group process

bull 3 types of peoplendash Decision makers

ndash Staff

ndash Respondents

bull Reduces lsquogroup-thinkrsquo

Respondents Respondents

Staff Staff

Decision MakersDecision Makers(Sales)

(What will sales be survey)

(Sales will be 45 50 55)

(Sales will be 50)

Consumer Market Survey

bull Ask customers about purchasing plans

bull What consumers say and what they actually do are often different

bull Sometimes difficult to answer

How many hours will you use the Internet

next week

How many hours will you use the Internet

next week

copy 1995 Corel Corp

Overview of Quantitative Approaches

bull Naiumlve approach

bull Moving averages

bull Exponential smoothing

bull Trend projection

bull Linear regression

Time-series Models

Associative models

Quantitative Forecasting Methods (Non-Naive)

QuantitativeForecasting

LinearRegression

AssociativeModels

ExponentialSmoothing

MovingAverage

Time SeriesModels

TrendProjection

bull Set of evenly spaced numerical datandash Obtained by observing response variable at

regular time periods

bull Forecast based only on past valuesndash Assumes that factors influencing past and

present will continue influence in future

bull ExampleYear 1998 1999 2000 2001 2002Sales 787 635 897 932 921

What is a Time Series

TrendTrend

SeasonalSeasonal

CyclicalCyclical

RandomRandom

Time Series Components

bull Persistent overall upward or downward pattern

bull Due to population technology etcbull Several years duration

Mo Qtr Yr

Response

copy 1984-1994 TMaker Co

Trend Component

bull Regular pattern of up amp down fluctuations

bull Due to weather customs etcbull Occurs within 1 year

Mo Qtr

Response

Summer

copy 1984-1994 TMaker Co

Seasonal Component

Common Seasonal PatternsPeriod of Pattern

ldquoSeasonrdquo Length

Number of

ldquoSeasonsrdquo in

Pattern

Week Day 7

Month Week 4 ndash 4 frac12

Month Day 28 ndash 31

Year Quarter 4

Year Month 12

Year Week 52

bull Repeating up amp down movementsbull Due to interactions of factors

influencing economybull Usually 2-10 years duration

Mo Qtr YrMo Qtr Yr

ResponseResponseCycle

Cyclical Component

bull Erratic unsystematic lsquoresidualrsquo fluctuations

bull Due to random variation or unforeseen

events

ndash Union strike

ndash Tornado

bull Short duration amp

nonrepeating

copy 1984-1994 TMaker Co

Random Component

bull Any observed value in a time series is the product (or sum) of time series components

bull Multiplicative modelndash Yi = Ti middot Si middot Ci middot Ri (if quarterly or mo data)

bull Additive modelndash Yi = Ti + Si + Ci + Ri (if quarterly or mo

data)

General Time Series Models

Naive Approach

bull Assumes demand in next period is the same as demand in most recent periodndash eg If May sales were 48

then June sales will be 48

bull Sometimes cost effective amp efficient

copy 1995 Corel Corp

bull MA is a series of arithmetic means

bull Used if little or no trend

bull Used often for smoothingndash Provides overall impression of data over

timebull Equation

MAMAnn

nn Demand inDemand in PreviousPrevious PeriodsPeriods

Moving Average Method

Yoursquore manager of a museum store that sells historical replicas You want to forecast sales (000) for 2003 using a 3-period moving average

1998 41999 62000 52001 32002 7

copy 1995 Corel Corp

Moving Average Example

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 6+5+3=14 143=4 23 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153=50 2002 7 6+5+3=14 143=47 2003 NA 5+3+7=15 153=50

95 96 97 98 99 00Year

Sales

2

4

6

8 Actual

Forecast

Moving Average Graph

bull Used when trend is present ndash Older data usually less important

bull Weights based on intuitionndash Often lay between 0 amp 1 amp sum to 10

bull Equation

WMA =WMA =ΣΣ(Weight for period (Weight for period nn) (Demand in period ) (Demand in period nn))

ΣΣWeightsWeights

Weighted Moving Average Method

Actual Demand Moving Average Weighted Moving

Average

0

5

10

15

20

25

30

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Sal

es D

eman

d

Actual sales

Moving average

Weighted moving average

bull Increasing n makes forecast less sensitive to changes

bull Do not forecast trend wellbull Require much historical data

copy 1984-1994 TMaker Co

Disadvantages of Moving Average Methods

bull Form of weighted moving averagendash Weights decline exponentiallyndash Most recent data weighted most

bull Requires smoothing constant ()ndash Ranges from 0 to 1ndash Subjectively chosen

bull Involves little record keeping of past data

Exponential Smoothing Method

bull Ft = At - 1 + (1-)At - 2 + (1- )2middotAt - 3

+ (1- )3At - 4 + + (1- )t-1middotA0

ndash Ft = Forecast value

ndash At = Actual value = Smoothing constant

bull Ft = Ft-1 + (At-1 - Ft-1)ndash Use for computing forecast

Exponential Smoothing Equations

During the past 8 quarters the Port of Baltimore has unloaded large quantities of grain ( = 10) The first quarter forecast was 175 QuarterActual

1 180 2 168

3 1594 1755 190

6 2057 1808 1829

Exponential Smoothing Example

Find the forecast for the 9th quarter

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

11 180 17500 (Given)

22 168168

33 159159

44 175175

55 190190

66 205205

17500 +17500 +

Exponential Smoothing Solution

QuarterQuarter ActuaActualForecast F t

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010((

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - -

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Ft

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - 17500 - 17500))

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing SolutionFt = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα= = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 +17500 + 1010(180 (180 - 17500- 17500)) = 17550 = 17550

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

1 180 17500 (Given)

22 168168 17500 + 10(180 - 17500) = 1755017500 + 10(180 - 17500) = 17550

33 159159 1755017550 ++ 1010(168 -(168 - 1755017550)) = 17475= 17475

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1995 180 17500 (Given)

1996 168 17500 + 10(180 - 17500) = 17550

1997 159 17550 + 10(168 - 17550) = 17475

1998 175

1999 190

2000 205

17475 + 10(159 - 17475)= 17318

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9 17822 + 10(182 - 17822) = 17858 182 17802 + 10(180 - 17802) = 17822

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Consumer Market Survey

bull Ask customers about purchasing plans

bull What consumers say and what they actually do are often different

bull Sometimes difficult to answer

How many hours will you use the Internet

next week

How many hours will you use the Internet

next week

copy 1995 Corel Corp

Overview of Quantitative Approaches

bull Naiumlve approach

bull Moving averages

bull Exponential smoothing

bull Trend projection

bull Linear regression

Time-series Models

Associative models

Quantitative Forecasting Methods (Non-Naive)

QuantitativeForecasting

LinearRegression

AssociativeModels

ExponentialSmoothing

MovingAverage

Time SeriesModels

TrendProjection

bull Set of evenly spaced numerical datandash Obtained by observing response variable at

regular time periods

bull Forecast based only on past valuesndash Assumes that factors influencing past and

present will continue influence in future

bull ExampleYear 1998 1999 2000 2001 2002Sales 787 635 897 932 921

What is a Time Series

TrendTrend

SeasonalSeasonal

CyclicalCyclical

RandomRandom

Time Series Components

bull Persistent overall upward or downward pattern

bull Due to population technology etcbull Several years duration

Mo Qtr Yr

Response

copy 1984-1994 TMaker Co

Trend Component

bull Regular pattern of up amp down fluctuations

bull Due to weather customs etcbull Occurs within 1 year

Mo Qtr

Response

Summer

copy 1984-1994 TMaker Co

Seasonal Component

Common Seasonal PatternsPeriod of Pattern

ldquoSeasonrdquo Length

Number of

ldquoSeasonsrdquo in

Pattern

Week Day 7

Month Week 4 ndash 4 frac12

Month Day 28 ndash 31

Year Quarter 4

Year Month 12

Year Week 52

bull Repeating up amp down movementsbull Due to interactions of factors

influencing economybull Usually 2-10 years duration

Mo Qtr YrMo Qtr Yr

ResponseResponseCycle

Cyclical Component

bull Erratic unsystematic lsquoresidualrsquo fluctuations

bull Due to random variation or unforeseen

events

ndash Union strike

ndash Tornado

bull Short duration amp

nonrepeating

copy 1984-1994 TMaker Co

Random Component

bull Any observed value in a time series is the product (or sum) of time series components

bull Multiplicative modelndash Yi = Ti middot Si middot Ci middot Ri (if quarterly or mo data)

bull Additive modelndash Yi = Ti + Si + Ci + Ri (if quarterly or mo

data)

General Time Series Models

Naive Approach

bull Assumes demand in next period is the same as demand in most recent periodndash eg If May sales were 48

then June sales will be 48

bull Sometimes cost effective amp efficient

copy 1995 Corel Corp

bull MA is a series of arithmetic means

bull Used if little or no trend

bull Used often for smoothingndash Provides overall impression of data over

timebull Equation

MAMAnn

nn Demand inDemand in PreviousPrevious PeriodsPeriods

Moving Average Method

Yoursquore manager of a museum store that sells historical replicas You want to forecast sales (000) for 2003 using a 3-period moving average

1998 41999 62000 52001 32002 7

copy 1995 Corel Corp

Moving Average Example

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 6+5+3=14 143=4 23 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153=50 2002 7 6+5+3=14 143=47 2003 NA 5+3+7=15 153=50

95 96 97 98 99 00Year

Sales

2

4

6

8 Actual

Forecast

Moving Average Graph

bull Used when trend is present ndash Older data usually less important

bull Weights based on intuitionndash Often lay between 0 amp 1 amp sum to 10

bull Equation

WMA =WMA =ΣΣ(Weight for period (Weight for period nn) (Demand in period ) (Demand in period nn))

ΣΣWeightsWeights

Weighted Moving Average Method

Actual Demand Moving Average Weighted Moving

Average

0

5

10

15

20

25

30

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Sal

es D

eman

d

Actual sales

Moving average

Weighted moving average

bull Increasing n makes forecast less sensitive to changes

bull Do not forecast trend wellbull Require much historical data

copy 1984-1994 TMaker Co

Disadvantages of Moving Average Methods

bull Form of weighted moving averagendash Weights decline exponentiallyndash Most recent data weighted most

bull Requires smoothing constant ()ndash Ranges from 0 to 1ndash Subjectively chosen

bull Involves little record keeping of past data

Exponential Smoothing Method

bull Ft = At - 1 + (1-)At - 2 + (1- )2middotAt - 3

+ (1- )3At - 4 + + (1- )t-1middotA0

ndash Ft = Forecast value

ndash At = Actual value = Smoothing constant

bull Ft = Ft-1 + (At-1 - Ft-1)ndash Use for computing forecast

Exponential Smoothing Equations

During the past 8 quarters the Port of Baltimore has unloaded large quantities of grain ( = 10) The first quarter forecast was 175 QuarterActual

1 180 2 168

3 1594 1755 190

6 2057 1808 1829

Exponential Smoothing Example

Find the forecast for the 9th quarter

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

11 180 17500 (Given)

22 168168

33 159159

44 175175

55 190190

66 205205

17500 +17500 +

Exponential Smoothing Solution

QuarterQuarter ActuaActualForecast F t

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010((

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - -

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Ft

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - 17500 - 17500))

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing SolutionFt = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα= = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 +17500 + 1010(180 (180 - 17500- 17500)) = 17550 = 17550

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

1 180 17500 (Given)

22 168168 17500 + 10(180 - 17500) = 1755017500 + 10(180 - 17500) = 17550

33 159159 1755017550 ++ 1010(168 -(168 - 1755017550)) = 17475= 17475

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1995 180 17500 (Given)

1996 168 17500 + 10(180 - 17500) = 17550

1997 159 17550 + 10(168 - 17550) = 17475

1998 175

1999 190

2000 205

17475 + 10(159 - 17475)= 17318

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9 17822 + 10(182 - 17822) = 17858 182 17802 + 10(180 - 17802) = 17822

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Overview of Quantitative Approaches

bull Naiumlve approach

bull Moving averages

bull Exponential smoothing

bull Trend projection

bull Linear regression

Time-series Models

Associative models

Quantitative Forecasting Methods (Non-Naive)

QuantitativeForecasting

LinearRegression

AssociativeModels

ExponentialSmoothing

MovingAverage

Time SeriesModels

TrendProjection

bull Set of evenly spaced numerical datandash Obtained by observing response variable at

regular time periods

bull Forecast based only on past valuesndash Assumes that factors influencing past and

present will continue influence in future

bull ExampleYear 1998 1999 2000 2001 2002Sales 787 635 897 932 921

What is a Time Series

TrendTrend

SeasonalSeasonal

CyclicalCyclical

RandomRandom

Time Series Components

bull Persistent overall upward or downward pattern

bull Due to population technology etcbull Several years duration

Mo Qtr Yr

Response

copy 1984-1994 TMaker Co

Trend Component

bull Regular pattern of up amp down fluctuations

bull Due to weather customs etcbull Occurs within 1 year

Mo Qtr

Response

Summer

copy 1984-1994 TMaker Co

Seasonal Component

Common Seasonal PatternsPeriod of Pattern

ldquoSeasonrdquo Length

Number of

ldquoSeasonsrdquo in

Pattern

Week Day 7

Month Week 4 ndash 4 frac12

Month Day 28 ndash 31

Year Quarter 4

Year Month 12

Year Week 52

bull Repeating up amp down movementsbull Due to interactions of factors

influencing economybull Usually 2-10 years duration

Mo Qtr YrMo Qtr Yr

ResponseResponseCycle

Cyclical Component

bull Erratic unsystematic lsquoresidualrsquo fluctuations

bull Due to random variation or unforeseen

events

ndash Union strike

ndash Tornado

bull Short duration amp

nonrepeating

copy 1984-1994 TMaker Co

Random Component

bull Any observed value in a time series is the product (or sum) of time series components

bull Multiplicative modelndash Yi = Ti middot Si middot Ci middot Ri (if quarterly or mo data)

bull Additive modelndash Yi = Ti + Si + Ci + Ri (if quarterly or mo

data)

General Time Series Models

Naive Approach

bull Assumes demand in next period is the same as demand in most recent periodndash eg If May sales were 48

then June sales will be 48

bull Sometimes cost effective amp efficient

copy 1995 Corel Corp

bull MA is a series of arithmetic means

bull Used if little or no trend

bull Used often for smoothingndash Provides overall impression of data over

timebull Equation

MAMAnn

nn Demand inDemand in PreviousPrevious PeriodsPeriods

Moving Average Method

Yoursquore manager of a museum store that sells historical replicas You want to forecast sales (000) for 2003 using a 3-period moving average

1998 41999 62000 52001 32002 7

copy 1995 Corel Corp

Moving Average Example

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 6+5+3=14 143=4 23 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153=50 2002 7 6+5+3=14 143=47 2003 NA 5+3+7=15 153=50

95 96 97 98 99 00Year

Sales

2

4

6

8 Actual

Forecast

Moving Average Graph

bull Used when trend is present ndash Older data usually less important

bull Weights based on intuitionndash Often lay between 0 amp 1 amp sum to 10

bull Equation

WMA =WMA =ΣΣ(Weight for period (Weight for period nn) (Demand in period ) (Demand in period nn))

ΣΣWeightsWeights

Weighted Moving Average Method

Actual Demand Moving Average Weighted Moving

Average

0

5

10

15

20

25

30

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Sal

es D

eman

d

Actual sales

Moving average

Weighted moving average

bull Increasing n makes forecast less sensitive to changes

bull Do not forecast trend wellbull Require much historical data

copy 1984-1994 TMaker Co

Disadvantages of Moving Average Methods

bull Form of weighted moving averagendash Weights decline exponentiallyndash Most recent data weighted most

bull Requires smoothing constant ()ndash Ranges from 0 to 1ndash Subjectively chosen

bull Involves little record keeping of past data

Exponential Smoothing Method

bull Ft = At - 1 + (1-)At - 2 + (1- )2middotAt - 3

+ (1- )3At - 4 + + (1- )t-1middotA0

ndash Ft = Forecast value

ndash At = Actual value = Smoothing constant

bull Ft = Ft-1 + (At-1 - Ft-1)ndash Use for computing forecast

Exponential Smoothing Equations

During the past 8 quarters the Port of Baltimore has unloaded large quantities of grain ( = 10) The first quarter forecast was 175 QuarterActual

1 180 2 168

3 1594 1755 190

6 2057 1808 1829

Exponential Smoothing Example

Find the forecast for the 9th quarter

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

11 180 17500 (Given)

22 168168

33 159159

44 175175

55 190190

66 205205

17500 +17500 +

Exponential Smoothing Solution

QuarterQuarter ActuaActualForecast F t

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010((

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - -

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Ft

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - 17500 - 17500))

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing SolutionFt = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα= = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 +17500 + 1010(180 (180 - 17500- 17500)) = 17550 = 17550

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

1 180 17500 (Given)

22 168168 17500 + 10(180 - 17500) = 1755017500 + 10(180 - 17500) = 17550

33 159159 1755017550 ++ 1010(168 -(168 - 1755017550)) = 17475= 17475

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1995 180 17500 (Given)

1996 168 17500 + 10(180 - 17500) = 17550

1997 159 17550 + 10(168 - 17550) = 17475

1998 175

1999 190

2000 205

17475 + 10(159 - 17475)= 17318

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9 17822 + 10(182 - 17822) = 17858 182 17802 + 10(180 - 17802) = 17822

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Quantitative Forecasting Methods (Non-Naive)

QuantitativeForecasting

LinearRegression

AssociativeModels

ExponentialSmoothing

MovingAverage

Time SeriesModels

TrendProjection

bull Set of evenly spaced numerical datandash Obtained by observing response variable at

regular time periods

bull Forecast based only on past valuesndash Assumes that factors influencing past and

present will continue influence in future

bull ExampleYear 1998 1999 2000 2001 2002Sales 787 635 897 932 921

What is a Time Series

TrendTrend

SeasonalSeasonal

CyclicalCyclical

RandomRandom

Time Series Components

bull Persistent overall upward or downward pattern

bull Due to population technology etcbull Several years duration

Mo Qtr Yr

Response

copy 1984-1994 TMaker Co

Trend Component

bull Regular pattern of up amp down fluctuations

bull Due to weather customs etcbull Occurs within 1 year

Mo Qtr

Response

Summer

copy 1984-1994 TMaker Co

Seasonal Component

Common Seasonal PatternsPeriod of Pattern

ldquoSeasonrdquo Length

Number of

ldquoSeasonsrdquo in

Pattern

Week Day 7

Month Week 4 ndash 4 frac12

Month Day 28 ndash 31

Year Quarter 4

Year Month 12

Year Week 52

bull Repeating up amp down movementsbull Due to interactions of factors

influencing economybull Usually 2-10 years duration

Mo Qtr YrMo Qtr Yr

ResponseResponseCycle

Cyclical Component

bull Erratic unsystematic lsquoresidualrsquo fluctuations

bull Due to random variation or unforeseen

events

ndash Union strike

ndash Tornado

bull Short duration amp

nonrepeating

copy 1984-1994 TMaker Co

Random Component

bull Any observed value in a time series is the product (or sum) of time series components

bull Multiplicative modelndash Yi = Ti middot Si middot Ci middot Ri (if quarterly or mo data)

bull Additive modelndash Yi = Ti + Si + Ci + Ri (if quarterly or mo

data)

General Time Series Models

Naive Approach

bull Assumes demand in next period is the same as demand in most recent periodndash eg If May sales were 48

then June sales will be 48

bull Sometimes cost effective amp efficient

copy 1995 Corel Corp

bull MA is a series of arithmetic means

bull Used if little or no trend

bull Used often for smoothingndash Provides overall impression of data over

timebull Equation

MAMAnn

nn Demand inDemand in PreviousPrevious PeriodsPeriods

Moving Average Method

Yoursquore manager of a museum store that sells historical replicas You want to forecast sales (000) for 2003 using a 3-period moving average

1998 41999 62000 52001 32002 7

copy 1995 Corel Corp

Moving Average Example

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 6+5+3=14 143=4 23 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153=50 2002 7 6+5+3=14 143=47 2003 NA 5+3+7=15 153=50

95 96 97 98 99 00Year

Sales

2

4

6

8 Actual

Forecast

Moving Average Graph

bull Used when trend is present ndash Older data usually less important

bull Weights based on intuitionndash Often lay between 0 amp 1 amp sum to 10

bull Equation

WMA =WMA =ΣΣ(Weight for period (Weight for period nn) (Demand in period ) (Demand in period nn))

ΣΣWeightsWeights

Weighted Moving Average Method

Actual Demand Moving Average Weighted Moving

Average

0

5

10

15

20

25

30

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Sal

es D

eman

d

Actual sales

Moving average

Weighted moving average

bull Increasing n makes forecast less sensitive to changes

bull Do not forecast trend wellbull Require much historical data

copy 1984-1994 TMaker Co

Disadvantages of Moving Average Methods

bull Form of weighted moving averagendash Weights decline exponentiallyndash Most recent data weighted most

bull Requires smoothing constant ()ndash Ranges from 0 to 1ndash Subjectively chosen

bull Involves little record keeping of past data

Exponential Smoothing Method

bull Ft = At - 1 + (1-)At - 2 + (1- )2middotAt - 3

+ (1- )3At - 4 + + (1- )t-1middotA0

ndash Ft = Forecast value

ndash At = Actual value = Smoothing constant

bull Ft = Ft-1 + (At-1 - Ft-1)ndash Use for computing forecast

Exponential Smoothing Equations

During the past 8 quarters the Port of Baltimore has unloaded large quantities of grain ( = 10) The first quarter forecast was 175 QuarterActual

1 180 2 168

3 1594 1755 190

6 2057 1808 1829

Exponential Smoothing Example

Find the forecast for the 9th quarter

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

11 180 17500 (Given)

22 168168

33 159159

44 175175

55 190190

66 205205

17500 +17500 +

Exponential Smoothing Solution

QuarterQuarter ActuaActualForecast F t

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010((

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - -

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Ft

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - 17500 - 17500))

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing SolutionFt = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα= = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 +17500 + 1010(180 (180 - 17500- 17500)) = 17550 = 17550

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

1 180 17500 (Given)

22 168168 17500 + 10(180 - 17500) = 1755017500 + 10(180 - 17500) = 17550

33 159159 1755017550 ++ 1010(168 -(168 - 1755017550)) = 17475= 17475

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1995 180 17500 (Given)

1996 168 17500 + 10(180 - 17500) = 17550

1997 159 17550 + 10(168 - 17550) = 17475

1998 175

1999 190

2000 205

17475 + 10(159 - 17475)= 17318

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9 17822 + 10(182 - 17822) = 17858 182 17802 + 10(180 - 17802) = 17822

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

bull Set of evenly spaced numerical datandash Obtained by observing response variable at

regular time periods

bull Forecast based only on past valuesndash Assumes that factors influencing past and

present will continue influence in future

bull ExampleYear 1998 1999 2000 2001 2002Sales 787 635 897 932 921

What is a Time Series

TrendTrend

SeasonalSeasonal

CyclicalCyclical

RandomRandom

Time Series Components

bull Persistent overall upward or downward pattern

bull Due to population technology etcbull Several years duration

Mo Qtr Yr

Response

copy 1984-1994 TMaker Co

Trend Component

bull Regular pattern of up amp down fluctuations

bull Due to weather customs etcbull Occurs within 1 year

Mo Qtr

Response

Summer

copy 1984-1994 TMaker Co

Seasonal Component

Common Seasonal PatternsPeriod of Pattern

ldquoSeasonrdquo Length

Number of

ldquoSeasonsrdquo in

Pattern

Week Day 7

Month Week 4 ndash 4 frac12

Month Day 28 ndash 31

Year Quarter 4

Year Month 12

Year Week 52

bull Repeating up amp down movementsbull Due to interactions of factors

influencing economybull Usually 2-10 years duration

Mo Qtr YrMo Qtr Yr

ResponseResponseCycle

Cyclical Component

bull Erratic unsystematic lsquoresidualrsquo fluctuations

bull Due to random variation or unforeseen

events

ndash Union strike

ndash Tornado

bull Short duration amp

nonrepeating

copy 1984-1994 TMaker Co

Random Component

bull Any observed value in a time series is the product (or sum) of time series components

bull Multiplicative modelndash Yi = Ti middot Si middot Ci middot Ri (if quarterly or mo data)

bull Additive modelndash Yi = Ti + Si + Ci + Ri (if quarterly or mo

data)

General Time Series Models

Naive Approach

bull Assumes demand in next period is the same as demand in most recent periodndash eg If May sales were 48

then June sales will be 48

bull Sometimes cost effective amp efficient

copy 1995 Corel Corp

bull MA is a series of arithmetic means

bull Used if little or no trend

bull Used often for smoothingndash Provides overall impression of data over

timebull Equation

MAMAnn

nn Demand inDemand in PreviousPrevious PeriodsPeriods

Moving Average Method

Yoursquore manager of a museum store that sells historical replicas You want to forecast sales (000) for 2003 using a 3-period moving average

1998 41999 62000 52001 32002 7

copy 1995 Corel Corp

Moving Average Example

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 6+5+3=14 143=4 23 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153=50 2002 7 6+5+3=14 143=47 2003 NA 5+3+7=15 153=50

95 96 97 98 99 00Year

Sales

2

4

6

8 Actual

Forecast

Moving Average Graph

bull Used when trend is present ndash Older data usually less important

bull Weights based on intuitionndash Often lay between 0 amp 1 amp sum to 10

bull Equation

WMA =WMA =ΣΣ(Weight for period (Weight for period nn) (Demand in period ) (Demand in period nn))

ΣΣWeightsWeights

Weighted Moving Average Method

Actual Demand Moving Average Weighted Moving

Average

0

5

10

15

20

25

30

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Sal

es D

eman

d

Actual sales

Moving average

Weighted moving average

bull Increasing n makes forecast less sensitive to changes

bull Do not forecast trend wellbull Require much historical data

copy 1984-1994 TMaker Co

Disadvantages of Moving Average Methods

bull Form of weighted moving averagendash Weights decline exponentiallyndash Most recent data weighted most

bull Requires smoothing constant ()ndash Ranges from 0 to 1ndash Subjectively chosen

bull Involves little record keeping of past data

Exponential Smoothing Method

bull Ft = At - 1 + (1-)At - 2 + (1- )2middotAt - 3

+ (1- )3At - 4 + + (1- )t-1middotA0

ndash Ft = Forecast value

ndash At = Actual value = Smoothing constant

bull Ft = Ft-1 + (At-1 - Ft-1)ndash Use for computing forecast

Exponential Smoothing Equations

During the past 8 quarters the Port of Baltimore has unloaded large quantities of grain ( = 10) The first quarter forecast was 175 QuarterActual

1 180 2 168

3 1594 1755 190

6 2057 1808 1829

Exponential Smoothing Example

Find the forecast for the 9th quarter

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

11 180 17500 (Given)

22 168168

33 159159

44 175175

55 190190

66 205205

17500 +17500 +

Exponential Smoothing Solution

QuarterQuarter ActuaActualForecast F t

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010((

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - -

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Ft

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - 17500 - 17500))

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing SolutionFt = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα= = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 +17500 + 1010(180 (180 - 17500- 17500)) = 17550 = 17550

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

1 180 17500 (Given)

22 168168 17500 + 10(180 - 17500) = 1755017500 + 10(180 - 17500) = 17550

33 159159 1755017550 ++ 1010(168 -(168 - 1755017550)) = 17475= 17475

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1995 180 17500 (Given)

1996 168 17500 + 10(180 - 17500) = 17550

1997 159 17550 + 10(168 - 17550) = 17475

1998 175

1999 190

2000 205

17475 + 10(159 - 17475)= 17318

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9 17822 + 10(182 - 17822) = 17858 182 17802 + 10(180 - 17802) = 17822

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

TrendTrend

SeasonalSeasonal

CyclicalCyclical

RandomRandom

Time Series Components

bull Persistent overall upward or downward pattern

bull Due to population technology etcbull Several years duration

Mo Qtr Yr

Response

copy 1984-1994 TMaker Co

Trend Component

bull Regular pattern of up amp down fluctuations

bull Due to weather customs etcbull Occurs within 1 year

Mo Qtr

Response

Summer

copy 1984-1994 TMaker Co

Seasonal Component

Common Seasonal PatternsPeriod of Pattern

ldquoSeasonrdquo Length

Number of

ldquoSeasonsrdquo in

Pattern

Week Day 7

Month Week 4 ndash 4 frac12

Month Day 28 ndash 31

Year Quarter 4

Year Month 12

Year Week 52

bull Repeating up amp down movementsbull Due to interactions of factors

influencing economybull Usually 2-10 years duration

Mo Qtr YrMo Qtr Yr

ResponseResponseCycle

Cyclical Component

bull Erratic unsystematic lsquoresidualrsquo fluctuations

bull Due to random variation or unforeseen

events

ndash Union strike

ndash Tornado

bull Short duration amp

nonrepeating

copy 1984-1994 TMaker Co

Random Component

bull Any observed value in a time series is the product (or sum) of time series components

bull Multiplicative modelndash Yi = Ti middot Si middot Ci middot Ri (if quarterly or mo data)

bull Additive modelndash Yi = Ti + Si + Ci + Ri (if quarterly or mo

data)

General Time Series Models

Naive Approach

bull Assumes demand in next period is the same as demand in most recent periodndash eg If May sales were 48

then June sales will be 48

bull Sometimes cost effective amp efficient

copy 1995 Corel Corp

bull MA is a series of arithmetic means

bull Used if little or no trend

bull Used often for smoothingndash Provides overall impression of data over

timebull Equation

MAMAnn

nn Demand inDemand in PreviousPrevious PeriodsPeriods

Moving Average Method

Yoursquore manager of a museum store that sells historical replicas You want to forecast sales (000) for 2003 using a 3-period moving average

1998 41999 62000 52001 32002 7

copy 1995 Corel Corp

Moving Average Example

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 6+5+3=14 143=4 23 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153=50 2002 7 6+5+3=14 143=47 2003 NA 5+3+7=15 153=50

95 96 97 98 99 00Year

Sales

2

4

6

8 Actual

Forecast

Moving Average Graph

bull Used when trend is present ndash Older data usually less important

bull Weights based on intuitionndash Often lay between 0 amp 1 amp sum to 10

bull Equation

WMA =WMA =ΣΣ(Weight for period (Weight for period nn) (Demand in period ) (Demand in period nn))

ΣΣWeightsWeights

Weighted Moving Average Method

Actual Demand Moving Average Weighted Moving

Average

0

5

10

15

20

25

30

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Sal

es D

eman

d

Actual sales

Moving average

Weighted moving average

bull Increasing n makes forecast less sensitive to changes

bull Do not forecast trend wellbull Require much historical data

copy 1984-1994 TMaker Co

Disadvantages of Moving Average Methods

bull Form of weighted moving averagendash Weights decline exponentiallyndash Most recent data weighted most

bull Requires smoothing constant ()ndash Ranges from 0 to 1ndash Subjectively chosen

bull Involves little record keeping of past data

Exponential Smoothing Method

bull Ft = At - 1 + (1-)At - 2 + (1- )2middotAt - 3

+ (1- )3At - 4 + + (1- )t-1middotA0

ndash Ft = Forecast value

ndash At = Actual value = Smoothing constant

bull Ft = Ft-1 + (At-1 - Ft-1)ndash Use for computing forecast

Exponential Smoothing Equations

During the past 8 quarters the Port of Baltimore has unloaded large quantities of grain ( = 10) The first quarter forecast was 175 QuarterActual

1 180 2 168

3 1594 1755 190

6 2057 1808 1829

Exponential Smoothing Example

Find the forecast for the 9th quarter

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

11 180 17500 (Given)

22 168168

33 159159

44 175175

55 190190

66 205205

17500 +17500 +

Exponential Smoothing Solution

QuarterQuarter ActuaActualForecast F t

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010((

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - -

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Ft

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - 17500 - 17500))

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing SolutionFt = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα= = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 +17500 + 1010(180 (180 - 17500- 17500)) = 17550 = 17550

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

1 180 17500 (Given)

22 168168 17500 + 10(180 - 17500) = 1755017500 + 10(180 - 17500) = 17550

33 159159 1755017550 ++ 1010(168 -(168 - 1755017550)) = 17475= 17475

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1995 180 17500 (Given)

1996 168 17500 + 10(180 - 17500) = 17550

1997 159 17550 + 10(168 - 17550) = 17475

1998 175

1999 190

2000 205

17475 + 10(159 - 17475)= 17318

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9 17822 + 10(182 - 17822) = 17858 182 17802 + 10(180 - 17802) = 17822

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

bull Persistent overall upward or downward pattern

bull Due to population technology etcbull Several years duration

Mo Qtr Yr

Response

copy 1984-1994 TMaker Co

Trend Component

bull Regular pattern of up amp down fluctuations

bull Due to weather customs etcbull Occurs within 1 year

Mo Qtr

Response

Summer

copy 1984-1994 TMaker Co

Seasonal Component

Common Seasonal PatternsPeriod of Pattern

ldquoSeasonrdquo Length

Number of

ldquoSeasonsrdquo in

Pattern

Week Day 7

Month Week 4 ndash 4 frac12

Month Day 28 ndash 31

Year Quarter 4

Year Month 12

Year Week 52

bull Repeating up amp down movementsbull Due to interactions of factors

influencing economybull Usually 2-10 years duration

Mo Qtr YrMo Qtr Yr

ResponseResponseCycle

Cyclical Component

bull Erratic unsystematic lsquoresidualrsquo fluctuations

bull Due to random variation or unforeseen

events

ndash Union strike

ndash Tornado

bull Short duration amp

nonrepeating

copy 1984-1994 TMaker Co

Random Component

bull Any observed value in a time series is the product (or sum) of time series components

bull Multiplicative modelndash Yi = Ti middot Si middot Ci middot Ri (if quarterly or mo data)

bull Additive modelndash Yi = Ti + Si + Ci + Ri (if quarterly or mo

data)

General Time Series Models

Naive Approach

bull Assumes demand in next period is the same as demand in most recent periodndash eg If May sales were 48

then June sales will be 48

bull Sometimes cost effective amp efficient

copy 1995 Corel Corp

bull MA is a series of arithmetic means

bull Used if little or no trend

bull Used often for smoothingndash Provides overall impression of data over

timebull Equation

MAMAnn

nn Demand inDemand in PreviousPrevious PeriodsPeriods

Moving Average Method

Yoursquore manager of a museum store that sells historical replicas You want to forecast sales (000) for 2003 using a 3-period moving average

1998 41999 62000 52001 32002 7

copy 1995 Corel Corp

Moving Average Example

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 6+5+3=14 143=4 23 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153=50 2002 7 6+5+3=14 143=47 2003 NA 5+3+7=15 153=50

95 96 97 98 99 00Year

Sales

2

4

6

8 Actual

Forecast

Moving Average Graph

bull Used when trend is present ndash Older data usually less important

bull Weights based on intuitionndash Often lay between 0 amp 1 amp sum to 10

bull Equation

WMA =WMA =ΣΣ(Weight for period (Weight for period nn) (Demand in period ) (Demand in period nn))

ΣΣWeightsWeights

Weighted Moving Average Method

Actual Demand Moving Average Weighted Moving

Average

0

5

10

15

20

25

30

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Sal

es D

eman

d

Actual sales

Moving average

Weighted moving average

bull Increasing n makes forecast less sensitive to changes

bull Do not forecast trend wellbull Require much historical data

copy 1984-1994 TMaker Co

Disadvantages of Moving Average Methods

bull Form of weighted moving averagendash Weights decline exponentiallyndash Most recent data weighted most

bull Requires smoothing constant ()ndash Ranges from 0 to 1ndash Subjectively chosen

bull Involves little record keeping of past data

Exponential Smoothing Method

bull Ft = At - 1 + (1-)At - 2 + (1- )2middotAt - 3

+ (1- )3At - 4 + + (1- )t-1middotA0

ndash Ft = Forecast value

ndash At = Actual value = Smoothing constant

bull Ft = Ft-1 + (At-1 - Ft-1)ndash Use for computing forecast

Exponential Smoothing Equations

During the past 8 quarters the Port of Baltimore has unloaded large quantities of grain ( = 10) The first quarter forecast was 175 QuarterActual

1 180 2 168

3 1594 1755 190

6 2057 1808 1829

Exponential Smoothing Example

Find the forecast for the 9th quarter

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

11 180 17500 (Given)

22 168168

33 159159

44 175175

55 190190

66 205205

17500 +17500 +

Exponential Smoothing Solution

QuarterQuarter ActuaActualForecast F t

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010((

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - -

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Ft

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - 17500 - 17500))

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing SolutionFt = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα= = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 +17500 + 1010(180 (180 - 17500- 17500)) = 17550 = 17550

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

1 180 17500 (Given)

22 168168 17500 + 10(180 - 17500) = 1755017500 + 10(180 - 17500) = 17550

33 159159 1755017550 ++ 1010(168 -(168 - 1755017550)) = 17475= 17475

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1995 180 17500 (Given)

1996 168 17500 + 10(180 - 17500) = 17550

1997 159 17550 + 10(168 - 17550) = 17475

1998 175

1999 190

2000 205

17475 + 10(159 - 17475)= 17318

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9 17822 + 10(182 - 17822) = 17858 182 17802 + 10(180 - 17802) = 17822

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

bull Regular pattern of up amp down fluctuations

bull Due to weather customs etcbull Occurs within 1 year

Mo Qtr

Response

Summer

copy 1984-1994 TMaker Co

Seasonal Component

Common Seasonal PatternsPeriod of Pattern

ldquoSeasonrdquo Length

Number of

ldquoSeasonsrdquo in

Pattern

Week Day 7

Month Week 4 ndash 4 frac12

Month Day 28 ndash 31

Year Quarter 4

Year Month 12

Year Week 52

bull Repeating up amp down movementsbull Due to interactions of factors

influencing economybull Usually 2-10 years duration

Mo Qtr YrMo Qtr Yr

ResponseResponseCycle

Cyclical Component

bull Erratic unsystematic lsquoresidualrsquo fluctuations

bull Due to random variation or unforeseen

events

ndash Union strike

ndash Tornado

bull Short duration amp

nonrepeating

copy 1984-1994 TMaker Co

Random Component

bull Any observed value in a time series is the product (or sum) of time series components

bull Multiplicative modelndash Yi = Ti middot Si middot Ci middot Ri (if quarterly or mo data)

bull Additive modelndash Yi = Ti + Si + Ci + Ri (if quarterly or mo

data)

General Time Series Models

Naive Approach

bull Assumes demand in next period is the same as demand in most recent periodndash eg If May sales were 48

then June sales will be 48

bull Sometimes cost effective amp efficient

copy 1995 Corel Corp

bull MA is a series of arithmetic means

bull Used if little or no trend

bull Used often for smoothingndash Provides overall impression of data over

timebull Equation

MAMAnn

nn Demand inDemand in PreviousPrevious PeriodsPeriods

Moving Average Method

Yoursquore manager of a museum store that sells historical replicas You want to forecast sales (000) for 2003 using a 3-period moving average

1998 41999 62000 52001 32002 7

copy 1995 Corel Corp

Moving Average Example

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 6+5+3=14 143=4 23 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153=50 2002 7 6+5+3=14 143=47 2003 NA 5+3+7=15 153=50

95 96 97 98 99 00Year

Sales

2

4

6

8 Actual

Forecast

Moving Average Graph

bull Used when trend is present ndash Older data usually less important

bull Weights based on intuitionndash Often lay between 0 amp 1 amp sum to 10

bull Equation

WMA =WMA =ΣΣ(Weight for period (Weight for period nn) (Demand in period ) (Demand in period nn))

ΣΣWeightsWeights

Weighted Moving Average Method

Actual Demand Moving Average Weighted Moving

Average

0

5

10

15

20

25

30

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Sal

es D

eman

d

Actual sales

Moving average

Weighted moving average

bull Increasing n makes forecast less sensitive to changes

bull Do not forecast trend wellbull Require much historical data

copy 1984-1994 TMaker Co

Disadvantages of Moving Average Methods

bull Form of weighted moving averagendash Weights decline exponentiallyndash Most recent data weighted most

bull Requires smoothing constant ()ndash Ranges from 0 to 1ndash Subjectively chosen

bull Involves little record keeping of past data

Exponential Smoothing Method

bull Ft = At - 1 + (1-)At - 2 + (1- )2middotAt - 3

+ (1- )3At - 4 + + (1- )t-1middotA0

ndash Ft = Forecast value

ndash At = Actual value = Smoothing constant

bull Ft = Ft-1 + (At-1 - Ft-1)ndash Use for computing forecast

Exponential Smoothing Equations

During the past 8 quarters the Port of Baltimore has unloaded large quantities of grain ( = 10) The first quarter forecast was 175 QuarterActual

1 180 2 168

3 1594 1755 190

6 2057 1808 1829

Exponential Smoothing Example

Find the forecast for the 9th quarter

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

11 180 17500 (Given)

22 168168

33 159159

44 175175

55 190190

66 205205

17500 +17500 +

Exponential Smoothing Solution

QuarterQuarter ActuaActualForecast F t

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010((

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - -

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Ft

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - 17500 - 17500))

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing SolutionFt = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα= = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 +17500 + 1010(180 (180 - 17500- 17500)) = 17550 = 17550

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

1 180 17500 (Given)

22 168168 17500 + 10(180 - 17500) = 1755017500 + 10(180 - 17500) = 17550

33 159159 1755017550 ++ 1010(168 -(168 - 1755017550)) = 17475= 17475

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1995 180 17500 (Given)

1996 168 17500 + 10(180 - 17500) = 17550

1997 159 17550 + 10(168 - 17550) = 17475

1998 175

1999 190

2000 205

17475 + 10(159 - 17475)= 17318

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9 17822 + 10(182 - 17822) = 17858 182 17802 + 10(180 - 17802) = 17822

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Common Seasonal PatternsPeriod of Pattern

ldquoSeasonrdquo Length

Number of

ldquoSeasonsrdquo in

Pattern

Week Day 7

Month Week 4 ndash 4 frac12

Month Day 28 ndash 31

Year Quarter 4

Year Month 12

Year Week 52

bull Repeating up amp down movementsbull Due to interactions of factors

influencing economybull Usually 2-10 years duration

Mo Qtr YrMo Qtr Yr

ResponseResponseCycle

Cyclical Component

bull Erratic unsystematic lsquoresidualrsquo fluctuations

bull Due to random variation or unforeseen

events

ndash Union strike

ndash Tornado

bull Short duration amp

nonrepeating

copy 1984-1994 TMaker Co

Random Component

bull Any observed value in a time series is the product (or sum) of time series components

bull Multiplicative modelndash Yi = Ti middot Si middot Ci middot Ri (if quarterly or mo data)

bull Additive modelndash Yi = Ti + Si + Ci + Ri (if quarterly or mo

data)

General Time Series Models

Naive Approach

bull Assumes demand in next period is the same as demand in most recent periodndash eg If May sales were 48

then June sales will be 48

bull Sometimes cost effective amp efficient

copy 1995 Corel Corp

bull MA is a series of arithmetic means

bull Used if little or no trend

bull Used often for smoothingndash Provides overall impression of data over

timebull Equation

MAMAnn

nn Demand inDemand in PreviousPrevious PeriodsPeriods

Moving Average Method

Yoursquore manager of a museum store that sells historical replicas You want to forecast sales (000) for 2003 using a 3-period moving average

1998 41999 62000 52001 32002 7

copy 1995 Corel Corp

Moving Average Example

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 6+5+3=14 143=4 23 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153=50 2002 7 6+5+3=14 143=47 2003 NA 5+3+7=15 153=50

95 96 97 98 99 00Year

Sales

2

4

6

8 Actual

Forecast

Moving Average Graph

bull Used when trend is present ndash Older data usually less important

bull Weights based on intuitionndash Often lay between 0 amp 1 amp sum to 10

bull Equation

WMA =WMA =ΣΣ(Weight for period (Weight for period nn) (Demand in period ) (Demand in period nn))

ΣΣWeightsWeights

Weighted Moving Average Method

Actual Demand Moving Average Weighted Moving

Average

0

5

10

15

20

25

30

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Sal

es D

eman

d

Actual sales

Moving average

Weighted moving average

bull Increasing n makes forecast less sensitive to changes

bull Do not forecast trend wellbull Require much historical data

copy 1984-1994 TMaker Co

Disadvantages of Moving Average Methods

bull Form of weighted moving averagendash Weights decline exponentiallyndash Most recent data weighted most

bull Requires smoothing constant ()ndash Ranges from 0 to 1ndash Subjectively chosen

bull Involves little record keeping of past data

Exponential Smoothing Method

bull Ft = At - 1 + (1-)At - 2 + (1- )2middotAt - 3

+ (1- )3At - 4 + + (1- )t-1middotA0

ndash Ft = Forecast value

ndash At = Actual value = Smoothing constant

bull Ft = Ft-1 + (At-1 - Ft-1)ndash Use for computing forecast

Exponential Smoothing Equations

During the past 8 quarters the Port of Baltimore has unloaded large quantities of grain ( = 10) The first quarter forecast was 175 QuarterActual

1 180 2 168

3 1594 1755 190

6 2057 1808 1829

Exponential Smoothing Example

Find the forecast for the 9th quarter

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

11 180 17500 (Given)

22 168168

33 159159

44 175175

55 190190

66 205205

17500 +17500 +

Exponential Smoothing Solution

QuarterQuarter ActuaActualForecast F t

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010((

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - -

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Ft

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - 17500 - 17500))

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing SolutionFt = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα= = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 +17500 + 1010(180 (180 - 17500- 17500)) = 17550 = 17550

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

1 180 17500 (Given)

22 168168 17500 + 10(180 - 17500) = 1755017500 + 10(180 - 17500) = 17550

33 159159 1755017550 ++ 1010(168 -(168 - 1755017550)) = 17475= 17475

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1995 180 17500 (Given)

1996 168 17500 + 10(180 - 17500) = 17550

1997 159 17550 + 10(168 - 17550) = 17475

1998 175

1999 190

2000 205

17475 + 10(159 - 17475)= 17318

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9 17822 + 10(182 - 17822) = 17858 182 17802 + 10(180 - 17802) = 17822

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

bull Repeating up amp down movementsbull Due to interactions of factors

influencing economybull Usually 2-10 years duration

Mo Qtr YrMo Qtr Yr

ResponseResponseCycle

Cyclical Component

bull Erratic unsystematic lsquoresidualrsquo fluctuations

bull Due to random variation or unforeseen

events

ndash Union strike

ndash Tornado

bull Short duration amp

nonrepeating

copy 1984-1994 TMaker Co

Random Component

bull Any observed value in a time series is the product (or sum) of time series components

bull Multiplicative modelndash Yi = Ti middot Si middot Ci middot Ri (if quarterly or mo data)

bull Additive modelndash Yi = Ti + Si + Ci + Ri (if quarterly or mo

data)

General Time Series Models

Naive Approach

bull Assumes demand in next period is the same as demand in most recent periodndash eg If May sales were 48

then June sales will be 48

bull Sometimes cost effective amp efficient

copy 1995 Corel Corp

bull MA is a series of arithmetic means

bull Used if little or no trend

bull Used often for smoothingndash Provides overall impression of data over

timebull Equation

MAMAnn

nn Demand inDemand in PreviousPrevious PeriodsPeriods

Moving Average Method

Yoursquore manager of a museum store that sells historical replicas You want to forecast sales (000) for 2003 using a 3-period moving average

1998 41999 62000 52001 32002 7

copy 1995 Corel Corp

Moving Average Example

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 6+5+3=14 143=4 23 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153=50 2002 7 6+5+3=14 143=47 2003 NA 5+3+7=15 153=50

95 96 97 98 99 00Year

Sales

2

4

6

8 Actual

Forecast

Moving Average Graph

bull Used when trend is present ndash Older data usually less important

bull Weights based on intuitionndash Often lay between 0 amp 1 amp sum to 10

bull Equation

WMA =WMA =ΣΣ(Weight for period (Weight for period nn) (Demand in period ) (Demand in period nn))

ΣΣWeightsWeights

Weighted Moving Average Method

Actual Demand Moving Average Weighted Moving

Average

0

5

10

15

20

25

30

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Sal

es D

eman

d

Actual sales

Moving average

Weighted moving average

bull Increasing n makes forecast less sensitive to changes

bull Do not forecast trend wellbull Require much historical data

copy 1984-1994 TMaker Co

Disadvantages of Moving Average Methods

bull Form of weighted moving averagendash Weights decline exponentiallyndash Most recent data weighted most

bull Requires smoothing constant ()ndash Ranges from 0 to 1ndash Subjectively chosen

bull Involves little record keeping of past data

Exponential Smoothing Method

bull Ft = At - 1 + (1-)At - 2 + (1- )2middotAt - 3

+ (1- )3At - 4 + + (1- )t-1middotA0

ndash Ft = Forecast value

ndash At = Actual value = Smoothing constant

bull Ft = Ft-1 + (At-1 - Ft-1)ndash Use for computing forecast

Exponential Smoothing Equations

During the past 8 quarters the Port of Baltimore has unloaded large quantities of grain ( = 10) The first quarter forecast was 175 QuarterActual

1 180 2 168

3 1594 1755 190

6 2057 1808 1829

Exponential Smoothing Example

Find the forecast for the 9th quarter

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

11 180 17500 (Given)

22 168168

33 159159

44 175175

55 190190

66 205205

17500 +17500 +

Exponential Smoothing Solution

QuarterQuarter ActuaActualForecast F t

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010((

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - -

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Ft

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - 17500 - 17500))

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing SolutionFt = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα= = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 +17500 + 1010(180 (180 - 17500- 17500)) = 17550 = 17550

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

1 180 17500 (Given)

22 168168 17500 + 10(180 - 17500) = 1755017500 + 10(180 - 17500) = 17550

33 159159 1755017550 ++ 1010(168 -(168 - 1755017550)) = 17475= 17475

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1995 180 17500 (Given)

1996 168 17500 + 10(180 - 17500) = 17550

1997 159 17550 + 10(168 - 17550) = 17475

1998 175

1999 190

2000 205

17475 + 10(159 - 17475)= 17318

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9 17822 + 10(182 - 17822) = 17858 182 17802 + 10(180 - 17802) = 17822

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

bull Erratic unsystematic lsquoresidualrsquo fluctuations

bull Due to random variation or unforeseen

events

ndash Union strike

ndash Tornado

bull Short duration amp

nonrepeating

copy 1984-1994 TMaker Co

Random Component

bull Any observed value in a time series is the product (or sum) of time series components

bull Multiplicative modelndash Yi = Ti middot Si middot Ci middot Ri (if quarterly or mo data)

bull Additive modelndash Yi = Ti + Si + Ci + Ri (if quarterly or mo

data)

General Time Series Models

Naive Approach

bull Assumes demand in next period is the same as demand in most recent periodndash eg If May sales were 48

then June sales will be 48

bull Sometimes cost effective amp efficient

copy 1995 Corel Corp

bull MA is a series of arithmetic means

bull Used if little or no trend

bull Used often for smoothingndash Provides overall impression of data over

timebull Equation

MAMAnn

nn Demand inDemand in PreviousPrevious PeriodsPeriods

Moving Average Method

Yoursquore manager of a museum store that sells historical replicas You want to forecast sales (000) for 2003 using a 3-period moving average

1998 41999 62000 52001 32002 7

copy 1995 Corel Corp

Moving Average Example

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 6+5+3=14 143=4 23 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153=50 2002 7 6+5+3=14 143=47 2003 NA 5+3+7=15 153=50

95 96 97 98 99 00Year

Sales

2

4

6

8 Actual

Forecast

Moving Average Graph

bull Used when trend is present ndash Older data usually less important

bull Weights based on intuitionndash Often lay between 0 amp 1 amp sum to 10

bull Equation

WMA =WMA =ΣΣ(Weight for period (Weight for period nn) (Demand in period ) (Demand in period nn))

ΣΣWeightsWeights

Weighted Moving Average Method

Actual Demand Moving Average Weighted Moving

Average

0

5

10

15

20

25

30

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Sal

es D

eman

d

Actual sales

Moving average

Weighted moving average

bull Increasing n makes forecast less sensitive to changes

bull Do not forecast trend wellbull Require much historical data

copy 1984-1994 TMaker Co

Disadvantages of Moving Average Methods

bull Form of weighted moving averagendash Weights decline exponentiallyndash Most recent data weighted most

bull Requires smoothing constant ()ndash Ranges from 0 to 1ndash Subjectively chosen

bull Involves little record keeping of past data

Exponential Smoothing Method

bull Ft = At - 1 + (1-)At - 2 + (1- )2middotAt - 3

+ (1- )3At - 4 + + (1- )t-1middotA0

ndash Ft = Forecast value

ndash At = Actual value = Smoothing constant

bull Ft = Ft-1 + (At-1 - Ft-1)ndash Use for computing forecast

Exponential Smoothing Equations

During the past 8 quarters the Port of Baltimore has unloaded large quantities of grain ( = 10) The first quarter forecast was 175 QuarterActual

1 180 2 168

3 1594 1755 190

6 2057 1808 1829

Exponential Smoothing Example

Find the forecast for the 9th quarter

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

11 180 17500 (Given)

22 168168

33 159159

44 175175

55 190190

66 205205

17500 +17500 +

Exponential Smoothing Solution

QuarterQuarter ActuaActualForecast F t

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010((

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - -

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Ft

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - 17500 - 17500))

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing SolutionFt = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα= = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 +17500 + 1010(180 (180 - 17500- 17500)) = 17550 = 17550

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

1 180 17500 (Given)

22 168168 17500 + 10(180 - 17500) = 1755017500 + 10(180 - 17500) = 17550

33 159159 1755017550 ++ 1010(168 -(168 - 1755017550)) = 17475= 17475

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1995 180 17500 (Given)

1996 168 17500 + 10(180 - 17500) = 17550

1997 159 17550 + 10(168 - 17550) = 17475

1998 175

1999 190

2000 205

17475 + 10(159 - 17475)= 17318

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9 17822 + 10(182 - 17822) = 17858 182 17802 + 10(180 - 17802) = 17822

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

bull Any observed value in a time series is the product (or sum) of time series components

bull Multiplicative modelndash Yi = Ti middot Si middot Ci middot Ri (if quarterly or mo data)

bull Additive modelndash Yi = Ti + Si + Ci + Ri (if quarterly or mo

data)

General Time Series Models

Naive Approach

bull Assumes demand in next period is the same as demand in most recent periodndash eg If May sales were 48

then June sales will be 48

bull Sometimes cost effective amp efficient

copy 1995 Corel Corp

bull MA is a series of arithmetic means

bull Used if little or no trend

bull Used often for smoothingndash Provides overall impression of data over

timebull Equation

MAMAnn

nn Demand inDemand in PreviousPrevious PeriodsPeriods

Moving Average Method

Yoursquore manager of a museum store that sells historical replicas You want to forecast sales (000) for 2003 using a 3-period moving average

1998 41999 62000 52001 32002 7

copy 1995 Corel Corp

Moving Average Example

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 6+5+3=14 143=4 23 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153=50 2002 7 6+5+3=14 143=47 2003 NA 5+3+7=15 153=50

95 96 97 98 99 00Year

Sales

2

4

6

8 Actual

Forecast

Moving Average Graph

bull Used when trend is present ndash Older data usually less important

bull Weights based on intuitionndash Often lay between 0 amp 1 amp sum to 10

bull Equation

WMA =WMA =ΣΣ(Weight for period (Weight for period nn) (Demand in period ) (Demand in period nn))

ΣΣWeightsWeights

Weighted Moving Average Method

Actual Demand Moving Average Weighted Moving

Average

0

5

10

15

20

25

30

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Sal

es D

eman

d

Actual sales

Moving average

Weighted moving average

bull Increasing n makes forecast less sensitive to changes

bull Do not forecast trend wellbull Require much historical data

copy 1984-1994 TMaker Co

Disadvantages of Moving Average Methods

bull Form of weighted moving averagendash Weights decline exponentiallyndash Most recent data weighted most

bull Requires smoothing constant ()ndash Ranges from 0 to 1ndash Subjectively chosen

bull Involves little record keeping of past data

Exponential Smoothing Method

bull Ft = At - 1 + (1-)At - 2 + (1- )2middotAt - 3

+ (1- )3At - 4 + + (1- )t-1middotA0

ndash Ft = Forecast value

ndash At = Actual value = Smoothing constant

bull Ft = Ft-1 + (At-1 - Ft-1)ndash Use for computing forecast

Exponential Smoothing Equations

During the past 8 quarters the Port of Baltimore has unloaded large quantities of grain ( = 10) The first quarter forecast was 175 QuarterActual

1 180 2 168

3 1594 1755 190

6 2057 1808 1829

Exponential Smoothing Example

Find the forecast for the 9th quarter

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

11 180 17500 (Given)

22 168168

33 159159

44 175175

55 190190

66 205205

17500 +17500 +

Exponential Smoothing Solution

QuarterQuarter ActuaActualForecast F t

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010((

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - -

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Ft

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - 17500 - 17500))

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing SolutionFt = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα= = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 +17500 + 1010(180 (180 - 17500- 17500)) = 17550 = 17550

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

1 180 17500 (Given)

22 168168 17500 + 10(180 - 17500) = 1755017500 + 10(180 - 17500) = 17550

33 159159 1755017550 ++ 1010(168 -(168 - 1755017550)) = 17475= 17475

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1995 180 17500 (Given)

1996 168 17500 + 10(180 - 17500) = 17550

1997 159 17550 + 10(168 - 17550) = 17475

1998 175

1999 190

2000 205

17475 + 10(159 - 17475)= 17318

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9 17822 + 10(182 - 17822) = 17858 182 17802 + 10(180 - 17802) = 17822

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Naive Approach

bull Assumes demand in next period is the same as demand in most recent periodndash eg If May sales were 48

then June sales will be 48

bull Sometimes cost effective amp efficient

copy 1995 Corel Corp

bull MA is a series of arithmetic means

bull Used if little or no trend

bull Used often for smoothingndash Provides overall impression of data over

timebull Equation

MAMAnn

nn Demand inDemand in PreviousPrevious PeriodsPeriods

Moving Average Method

Yoursquore manager of a museum store that sells historical replicas You want to forecast sales (000) for 2003 using a 3-period moving average

1998 41999 62000 52001 32002 7

copy 1995 Corel Corp

Moving Average Example

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 6+5+3=14 143=4 23 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153=50 2002 7 6+5+3=14 143=47 2003 NA 5+3+7=15 153=50

95 96 97 98 99 00Year

Sales

2

4

6

8 Actual

Forecast

Moving Average Graph

bull Used when trend is present ndash Older data usually less important

bull Weights based on intuitionndash Often lay between 0 amp 1 amp sum to 10

bull Equation

WMA =WMA =ΣΣ(Weight for period (Weight for period nn) (Demand in period ) (Demand in period nn))

ΣΣWeightsWeights

Weighted Moving Average Method

Actual Demand Moving Average Weighted Moving

Average

0

5

10

15

20

25

30

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Sal

es D

eman

d

Actual sales

Moving average

Weighted moving average

bull Increasing n makes forecast less sensitive to changes

bull Do not forecast trend wellbull Require much historical data

copy 1984-1994 TMaker Co

Disadvantages of Moving Average Methods

bull Form of weighted moving averagendash Weights decline exponentiallyndash Most recent data weighted most

bull Requires smoothing constant ()ndash Ranges from 0 to 1ndash Subjectively chosen

bull Involves little record keeping of past data

Exponential Smoothing Method

bull Ft = At - 1 + (1-)At - 2 + (1- )2middotAt - 3

+ (1- )3At - 4 + + (1- )t-1middotA0

ndash Ft = Forecast value

ndash At = Actual value = Smoothing constant

bull Ft = Ft-1 + (At-1 - Ft-1)ndash Use for computing forecast

Exponential Smoothing Equations

During the past 8 quarters the Port of Baltimore has unloaded large quantities of grain ( = 10) The first quarter forecast was 175 QuarterActual

1 180 2 168

3 1594 1755 190

6 2057 1808 1829

Exponential Smoothing Example

Find the forecast for the 9th quarter

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

11 180 17500 (Given)

22 168168

33 159159

44 175175

55 190190

66 205205

17500 +17500 +

Exponential Smoothing Solution

QuarterQuarter ActuaActualForecast F t

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010((

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - -

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Ft

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - 17500 - 17500))

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing SolutionFt = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα= = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 +17500 + 1010(180 (180 - 17500- 17500)) = 17550 = 17550

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

1 180 17500 (Given)

22 168168 17500 + 10(180 - 17500) = 1755017500 + 10(180 - 17500) = 17550

33 159159 1755017550 ++ 1010(168 -(168 - 1755017550)) = 17475= 17475

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1995 180 17500 (Given)

1996 168 17500 + 10(180 - 17500) = 17550

1997 159 17550 + 10(168 - 17550) = 17475

1998 175

1999 190

2000 205

17475 + 10(159 - 17475)= 17318

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9 17822 + 10(182 - 17822) = 17858 182 17802 + 10(180 - 17802) = 17822

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

bull MA is a series of arithmetic means

bull Used if little or no trend

bull Used often for smoothingndash Provides overall impression of data over

timebull Equation

MAMAnn

nn Demand inDemand in PreviousPrevious PeriodsPeriods

Moving Average Method

Yoursquore manager of a museum store that sells historical replicas You want to forecast sales (000) for 2003 using a 3-period moving average

1998 41999 62000 52001 32002 7

copy 1995 Corel Corp

Moving Average Example

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 6+5+3=14 143=4 23 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153=50 2002 7 6+5+3=14 143=47 2003 NA 5+3+7=15 153=50

95 96 97 98 99 00Year

Sales

2

4

6

8 Actual

Forecast

Moving Average Graph

bull Used when trend is present ndash Older data usually less important

bull Weights based on intuitionndash Often lay between 0 amp 1 amp sum to 10

bull Equation

WMA =WMA =ΣΣ(Weight for period (Weight for period nn) (Demand in period ) (Demand in period nn))

ΣΣWeightsWeights

Weighted Moving Average Method

Actual Demand Moving Average Weighted Moving

Average

0

5

10

15

20

25

30

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Sal

es D

eman

d

Actual sales

Moving average

Weighted moving average

bull Increasing n makes forecast less sensitive to changes

bull Do not forecast trend wellbull Require much historical data

copy 1984-1994 TMaker Co

Disadvantages of Moving Average Methods

bull Form of weighted moving averagendash Weights decline exponentiallyndash Most recent data weighted most

bull Requires smoothing constant ()ndash Ranges from 0 to 1ndash Subjectively chosen

bull Involves little record keeping of past data

Exponential Smoothing Method

bull Ft = At - 1 + (1-)At - 2 + (1- )2middotAt - 3

+ (1- )3At - 4 + + (1- )t-1middotA0

ndash Ft = Forecast value

ndash At = Actual value = Smoothing constant

bull Ft = Ft-1 + (At-1 - Ft-1)ndash Use for computing forecast

Exponential Smoothing Equations

During the past 8 quarters the Port of Baltimore has unloaded large quantities of grain ( = 10) The first quarter forecast was 175 QuarterActual

1 180 2 168

3 1594 1755 190

6 2057 1808 1829

Exponential Smoothing Example

Find the forecast for the 9th quarter

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

11 180 17500 (Given)

22 168168

33 159159

44 175175

55 190190

66 205205

17500 +17500 +

Exponential Smoothing Solution

QuarterQuarter ActuaActualForecast F t

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010((

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - -

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Ft

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - 17500 - 17500))

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing SolutionFt = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα= = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 +17500 + 1010(180 (180 - 17500- 17500)) = 17550 = 17550

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

1 180 17500 (Given)

22 168168 17500 + 10(180 - 17500) = 1755017500 + 10(180 - 17500) = 17550

33 159159 1755017550 ++ 1010(168 -(168 - 1755017550)) = 17475= 17475

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1995 180 17500 (Given)

1996 168 17500 + 10(180 - 17500) = 17550

1997 159 17550 + 10(168 - 17550) = 17475

1998 175

1999 190

2000 205

17475 + 10(159 - 17475)= 17318

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9 17822 + 10(182 - 17822) = 17858 182 17802 + 10(180 - 17802) = 17822

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Yoursquore manager of a museum store that sells historical replicas You want to forecast sales (000) for 2003 using a 3-period moving average

1998 41999 62000 52001 32002 7

copy 1995 Corel Corp

Moving Average Example

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 6+5+3=14 143=4 23 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153=50 2002 7 6+5+3=14 143=47 2003 NA 5+3+7=15 153=50

95 96 97 98 99 00Year

Sales

2

4

6

8 Actual

Forecast

Moving Average Graph

bull Used when trend is present ndash Older data usually less important

bull Weights based on intuitionndash Often lay between 0 amp 1 amp sum to 10

bull Equation

WMA =WMA =ΣΣ(Weight for period (Weight for period nn) (Demand in period ) (Demand in period nn))

ΣΣWeightsWeights

Weighted Moving Average Method

Actual Demand Moving Average Weighted Moving

Average

0

5

10

15

20

25

30

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Sal

es D

eman

d

Actual sales

Moving average

Weighted moving average

bull Increasing n makes forecast less sensitive to changes

bull Do not forecast trend wellbull Require much historical data

copy 1984-1994 TMaker Co

Disadvantages of Moving Average Methods

bull Form of weighted moving averagendash Weights decline exponentiallyndash Most recent data weighted most

bull Requires smoothing constant ()ndash Ranges from 0 to 1ndash Subjectively chosen

bull Involves little record keeping of past data

Exponential Smoothing Method

bull Ft = At - 1 + (1-)At - 2 + (1- )2middotAt - 3

+ (1- )3At - 4 + + (1- )t-1middotA0

ndash Ft = Forecast value

ndash At = Actual value = Smoothing constant

bull Ft = Ft-1 + (At-1 - Ft-1)ndash Use for computing forecast

Exponential Smoothing Equations

During the past 8 quarters the Port of Baltimore has unloaded large quantities of grain ( = 10) The first quarter forecast was 175 QuarterActual

1 180 2 168

3 1594 1755 190

6 2057 1808 1829

Exponential Smoothing Example

Find the forecast for the 9th quarter

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

11 180 17500 (Given)

22 168168

33 159159

44 175175

55 190190

66 205205

17500 +17500 +

Exponential Smoothing Solution

QuarterQuarter ActuaActualForecast F t

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010((

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - -

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Ft

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - 17500 - 17500))

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing SolutionFt = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα= = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 +17500 + 1010(180 (180 - 17500- 17500)) = 17550 = 17550

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

1 180 17500 (Given)

22 168168 17500 + 10(180 - 17500) = 1755017500 + 10(180 - 17500) = 17550

33 159159 1755017550 ++ 1010(168 -(168 - 1755017550)) = 17475= 17475

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1995 180 17500 (Given)

1996 168 17500 + 10(180 - 17500) = 17550

1997 159 17550 + 10(168 - 17550) = 17475

1998 175

1999 190

2000 205

17475 + 10(159 - 17475)= 17318

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9 17822 + 10(182 - 17822) = 17858 182 17802 + 10(180 - 17802) = 17822

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 6+5+3=14 143=4 23 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153=50 2002 7 6+5+3=14 143=47 2003 NA 5+3+7=15 153=50

95 96 97 98 99 00Year

Sales

2

4

6

8 Actual

Forecast

Moving Average Graph

bull Used when trend is present ndash Older data usually less important

bull Weights based on intuitionndash Often lay between 0 amp 1 amp sum to 10

bull Equation

WMA =WMA =ΣΣ(Weight for period (Weight for period nn) (Demand in period ) (Demand in period nn))

ΣΣWeightsWeights

Weighted Moving Average Method

Actual Demand Moving Average Weighted Moving

Average

0

5

10

15

20

25

30

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Sal

es D

eman

d

Actual sales

Moving average

Weighted moving average

bull Increasing n makes forecast less sensitive to changes

bull Do not forecast trend wellbull Require much historical data

copy 1984-1994 TMaker Co

Disadvantages of Moving Average Methods

bull Form of weighted moving averagendash Weights decline exponentiallyndash Most recent data weighted most

bull Requires smoothing constant ()ndash Ranges from 0 to 1ndash Subjectively chosen

bull Involves little record keeping of past data

Exponential Smoothing Method

bull Ft = At - 1 + (1-)At - 2 + (1- )2middotAt - 3

+ (1- )3At - 4 + + (1- )t-1middotA0

ndash Ft = Forecast value

ndash At = Actual value = Smoothing constant

bull Ft = Ft-1 + (At-1 - Ft-1)ndash Use for computing forecast

Exponential Smoothing Equations

During the past 8 quarters the Port of Baltimore has unloaded large quantities of grain ( = 10) The first quarter forecast was 175 QuarterActual

1 180 2 168

3 1594 1755 190

6 2057 1808 1829

Exponential Smoothing Example

Find the forecast for the 9th quarter

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

11 180 17500 (Given)

22 168168

33 159159

44 175175

55 190190

66 205205

17500 +17500 +

Exponential Smoothing Solution

QuarterQuarter ActuaActualForecast F t

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010((

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - -

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Ft

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - 17500 - 17500))

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing SolutionFt = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα= = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 +17500 + 1010(180 (180 - 17500- 17500)) = 17550 = 17550

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

1 180 17500 (Given)

22 168168 17500 + 10(180 - 17500) = 1755017500 + 10(180 - 17500) = 17550

33 159159 1755017550 ++ 1010(168 -(168 - 1755017550)) = 17475= 17475

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1995 180 17500 (Given)

1996 168 17500 + 10(180 - 17500) = 17550

1997 159 17550 + 10(168 - 17550) = 17475

1998 175

1999 190

2000 205

17475 + 10(159 - 17475)= 17318

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9 17822 + 10(182 - 17822) = 17858 182 17802 + 10(180 - 17802) = 17822

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153 = 5 2002 7 6+5+3=14 143=4 23 2003 NA

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153=50 2002 7 6+5+3=14 143=47 2003 NA 5+3+7=15 153=50

95 96 97 98 99 00Year

Sales

2

4

6

8 Actual

Forecast

Moving Average Graph

bull Used when trend is present ndash Older data usually less important

bull Weights based on intuitionndash Often lay between 0 amp 1 amp sum to 10

bull Equation

WMA =WMA =ΣΣ(Weight for period (Weight for period nn) (Demand in period ) (Demand in period nn))

ΣΣWeightsWeights

Weighted Moving Average Method

Actual Demand Moving Average Weighted Moving

Average

0

5

10

15

20

25

30

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Sal

es D

eman

d

Actual sales

Moving average

Weighted moving average

bull Increasing n makes forecast less sensitive to changes

bull Do not forecast trend wellbull Require much historical data

copy 1984-1994 TMaker Co

Disadvantages of Moving Average Methods

bull Form of weighted moving averagendash Weights decline exponentiallyndash Most recent data weighted most

bull Requires smoothing constant ()ndash Ranges from 0 to 1ndash Subjectively chosen

bull Involves little record keeping of past data

Exponential Smoothing Method

bull Ft = At - 1 + (1-)At - 2 + (1- )2middotAt - 3

+ (1- )3At - 4 + + (1- )t-1middotA0

ndash Ft = Forecast value

ndash At = Actual value = Smoothing constant

bull Ft = Ft-1 + (At-1 - Ft-1)ndash Use for computing forecast

Exponential Smoothing Equations

During the past 8 quarters the Port of Baltimore has unloaded large quantities of grain ( = 10) The first quarter forecast was 175 QuarterActual

1 180 2 168

3 1594 1755 190

6 2057 1808 1829

Exponential Smoothing Example

Find the forecast for the 9th quarter

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

11 180 17500 (Given)

22 168168

33 159159

44 175175

55 190190

66 205205

17500 +17500 +

Exponential Smoothing Solution

QuarterQuarter ActuaActualForecast F t

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010((

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - -

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Ft

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - 17500 - 17500))

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing SolutionFt = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα= = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 +17500 + 1010(180 (180 - 17500- 17500)) = 17550 = 17550

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

1 180 17500 (Given)

22 168168 17500 + 10(180 - 17500) = 1755017500 + 10(180 - 17500) = 17550

33 159159 1755017550 ++ 1010(168 -(168 - 1755017550)) = 17475= 17475

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1995 180 17500 (Given)

1996 168 17500 + 10(180 - 17500) = 17550

1997 159 17550 + 10(168 - 17550) = 17475

1998 175

1999 190

2000 205

17475 + 10(159 - 17475)= 17318

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9 17822 + 10(182 - 17822) = 17858 182 17802 + 10(180 - 17802) = 17822

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Moving Average SolutionTime Response

Yi Moving Total (n=3)

Moving Average

(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 153=50 2002 7 6+5+3=14 143=47 2003 NA 5+3+7=15 153=50

95 96 97 98 99 00Year

Sales

2

4

6

8 Actual

Forecast

Moving Average Graph

bull Used when trend is present ndash Older data usually less important

bull Weights based on intuitionndash Often lay between 0 amp 1 amp sum to 10

bull Equation

WMA =WMA =ΣΣ(Weight for period (Weight for period nn) (Demand in period ) (Demand in period nn))

ΣΣWeightsWeights

Weighted Moving Average Method

Actual Demand Moving Average Weighted Moving

Average

0

5

10

15

20

25

30

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Sal

es D

eman

d

Actual sales

Moving average

Weighted moving average

bull Increasing n makes forecast less sensitive to changes

bull Do not forecast trend wellbull Require much historical data

copy 1984-1994 TMaker Co

Disadvantages of Moving Average Methods

bull Form of weighted moving averagendash Weights decline exponentiallyndash Most recent data weighted most

bull Requires smoothing constant ()ndash Ranges from 0 to 1ndash Subjectively chosen

bull Involves little record keeping of past data

Exponential Smoothing Method

bull Ft = At - 1 + (1-)At - 2 + (1- )2middotAt - 3

+ (1- )3At - 4 + + (1- )t-1middotA0

ndash Ft = Forecast value

ndash At = Actual value = Smoothing constant

bull Ft = Ft-1 + (At-1 - Ft-1)ndash Use for computing forecast

Exponential Smoothing Equations

During the past 8 quarters the Port of Baltimore has unloaded large quantities of grain ( = 10) The first quarter forecast was 175 QuarterActual

1 180 2 168

3 1594 1755 190

6 2057 1808 1829

Exponential Smoothing Example

Find the forecast for the 9th quarter

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

11 180 17500 (Given)

22 168168

33 159159

44 175175

55 190190

66 205205

17500 +17500 +

Exponential Smoothing Solution

QuarterQuarter ActuaActualForecast F t

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010((

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - -

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Ft

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - 17500 - 17500))

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing SolutionFt = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα= = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 +17500 + 1010(180 (180 - 17500- 17500)) = 17550 = 17550

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

1 180 17500 (Given)

22 168168 17500 + 10(180 - 17500) = 1755017500 + 10(180 - 17500) = 17550

33 159159 1755017550 ++ 1010(168 -(168 - 1755017550)) = 17475= 17475

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1995 180 17500 (Given)

1996 168 17500 + 10(180 - 17500) = 17550

1997 159 17550 + 10(168 - 17550) = 17475

1998 175

1999 190

2000 205

17475 + 10(159 - 17475)= 17318

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9 17822 + 10(182 - 17822) = 17858 182 17802 + 10(180 - 17802) = 17822

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

95 96 97 98 99 00Year

Sales

2

4

6

8 Actual

Forecast

Moving Average Graph

bull Used when trend is present ndash Older data usually less important

bull Weights based on intuitionndash Often lay between 0 amp 1 amp sum to 10

bull Equation

WMA =WMA =ΣΣ(Weight for period (Weight for period nn) (Demand in period ) (Demand in period nn))

ΣΣWeightsWeights

Weighted Moving Average Method

Actual Demand Moving Average Weighted Moving

Average

0

5

10

15

20

25

30

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Sal

es D

eman

d

Actual sales

Moving average

Weighted moving average

bull Increasing n makes forecast less sensitive to changes

bull Do not forecast trend wellbull Require much historical data

copy 1984-1994 TMaker Co

Disadvantages of Moving Average Methods

bull Form of weighted moving averagendash Weights decline exponentiallyndash Most recent data weighted most

bull Requires smoothing constant ()ndash Ranges from 0 to 1ndash Subjectively chosen

bull Involves little record keeping of past data

Exponential Smoothing Method

bull Ft = At - 1 + (1-)At - 2 + (1- )2middotAt - 3

+ (1- )3At - 4 + + (1- )t-1middotA0

ndash Ft = Forecast value

ndash At = Actual value = Smoothing constant

bull Ft = Ft-1 + (At-1 - Ft-1)ndash Use for computing forecast

Exponential Smoothing Equations

During the past 8 quarters the Port of Baltimore has unloaded large quantities of grain ( = 10) The first quarter forecast was 175 QuarterActual

1 180 2 168

3 1594 1755 190

6 2057 1808 1829

Exponential Smoothing Example

Find the forecast for the 9th quarter

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

11 180 17500 (Given)

22 168168

33 159159

44 175175

55 190190

66 205205

17500 +17500 +

Exponential Smoothing Solution

QuarterQuarter ActuaActualForecast F t

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010((

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - -

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Ft

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - 17500 - 17500))

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing SolutionFt = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα= = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 +17500 + 1010(180 (180 - 17500- 17500)) = 17550 = 17550

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

1 180 17500 (Given)

22 168168 17500 + 10(180 - 17500) = 1755017500 + 10(180 - 17500) = 17550

33 159159 1755017550 ++ 1010(168 -(168 - 1755017550)) = 17475= 17475

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1995 180 17500 (Given)

1996 168 17500 + 10(180 - 17500) = 17550

1997 159 17550 + 10(168 - 17550) = 17475

1998 175

1999 190

2000 205

17475 + 10(159 - 17475)= 17318

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9 17822 + 10(182 - 17822) = 17858 182 17802 + 10(180 - 17802) = 17822

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

bull Used when trend is present ndash Older data usually less important

bull Weights based on intuitionndash Often lay between 0 amp 1 amp sum to 10

bull Equation

WMA =WMA =ΣΣ(Weight for period (Weight for period nn) (Demand in period ) (Demand in period nn))

ΣΣWeightsWeights

Weighted Moving Average Method

Actual Demand Moving Average Weighted Moving

Average

0

5

10

15

20

25

30

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Sal

es D

eman

d

Actual sales

Moving average

Weighted moving average

bull Increasing n makes forecast less sensitive to changes

bull Do not forecast trend wellbull Require much historical data

copy 1984-1994 TMaker Co

Disadvantages of Moving Average Methods

bull Form of weighted moving averagendash Weights decline exponentiallyndash Most recent data weighted most

bull Requires smoothing constant ()ndash Ranges from 0 to 1ndash Subjectively chosen

bull Involves little record keeping of past data

Exponential Smoothing Method

bull Ft = At - 1 + (1-)At - 2 + (1- )2middotAt - 3

+ (1- )3At - 4 + + (1- )t-1middotA0

ndash Ft = Forecast value

ndash At = Actual value = Smoothing constant

bull Ft = Ft-1 + (At-1 - Ft-1)ndash Use for computing forecast

Exponential Smoothing Equations

During the past 8 quarters the Port of Baltimore has unloaded large quantities of grain ( = 10) The first quarter forecast was 175 QuarterActual

1 180 2 168

3 1594 1755 190

6 2057 1808 1829

Exponential Smoothing Example

Find the forecast for the 9th quarter

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

11 180 17500 (Given)

22 168168

33 159159

44 175175

55 190190

66 205205

17500 +17500 +

Exponential Smoothing Solution

QuarterQuarter ActuaActualForecast F t

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010((

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - -

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Ft

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - 17500 - 17500))

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing SolutionFt = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα= = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 +17500 + 1010(180 (180 - 17500- 17500)) = 17550 = 17550

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

1 180 17500 (Given)

22 168168 17500 + 10(180 - 17500) = 1755017500 + 10(180 - 17500) = 17550

33 159159 1755017550 ++ 1010(168 -(168 - 1755017550)) = 17475= 17475

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1995 180 17500 (Given)

1996 168 17500 + 10(180 - 17500) = 17550

1997 159 17550 + 10(168 - 17550) = 17475

1998 175

1999 190

2000 205

17475 + 10(159 - 17475)= 17318

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9 17822 + 10(182 - 17822) = 17858 182 17802 + 10(180 - 17802) = 17822

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Actual Demand Moving Average Weighted Moving

Average

0

5

10

15

20

25

30

35

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

Sal

es D

eman

d

Actual sales

Moving average

Weighted moving average

bull Increasing n makes forecast less sensitive to changes

bull Do not forecast trend wellbull Require much historical data

copy 1984-1994 TMaker Co

Disadvantages of Moving Average Methods

bull Form of weighted moving averagendash Weights decline exponentiallyndash Most recent data weighted most

bull Requires smoothing constant ()ndash Ranges from 0 to 1ndash Subjectively chosen

bull Involves little record keeping of past data

Exponential Smoothing Method

bull Ft = At - 1 + (1-)At - 2 + (1- )2middotAt - 3

+ (1- )3At - 4 + + (1- )t-1middotA0

ndash Ft = Forecast value

ndash At = Actual value = Smoothing constant

bull Ft = Ft-1 + (At-1 - Ft-1)ndash Use for computing forecast

Exponential Smoothing Equations

During the past 8 quarters the Port of Baltimore has unloaded large quantities of grain ( = 10) The first quarter forecast was 175 QuarterActual

1 180 2 168

3 1594 1755 190

6 2057 1808 1829

Exponential Smoothing Example

Find the forecast for the 9th quarter

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

11 180 17500 (Given)

22 168168

33 159159

44 175175

55 190190

66 205205

17500 +17500 +

Exponential Smoothing Solution

QuarterQuarter ActuaActualForecast F t

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010((

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - -

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Ft

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - 17500 - 17500))

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing SolutionFt = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα= = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 +17500 + 1010(180 (180 - 17500- 17500)) = 17550 = 17550

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

1 180 17500 (Given)

22 168168 17500 + 10(180 - 17500) = 1755017500 + 10(180 - 17500) = 17550

33 159159 1755017550 ++ 1010(168 -(168 - 1755017550)) = 17475= 17475

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1995 180 17500 (Given)

1996 168 17500 + 10(180 - 17500) = 17550

1997 159 17550 + 10(168 - 17550) = 17475

1998 175

1999 190

2000 205

17475 + 10(159 - 17475)= 17318

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9 17822 + 10(182 - 17822) = 17858 182 17802 + 10(180 - 17802) = 17822

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

bull Increasing n makes forecast less sensitive to changes

bull Do not forecast trend wellbull Require much historical data

copy 1984-1994 TMaker Co

Disadvantages of Moving Average Methods

bull Form of weighted moving averagendash Weights decline exponentiallyndash Most recent data weighted most

bull Requires smoothing constant ()ndash Ranges from 0 to 1ndash Subjectively chosen

bull Involves little record keeping of past data

Exponential Smoothing Method

bull Ft = At - 1 + (1-)At - 2 + (1- )2middotAt - 3

+ (1- )3At - 4 + + (1- )t-1middotA0

ndash Ft = Forecast value

ndash At = Actual value = Smoothing constant

bull Ft = Ft-1 + (At-1 - Ft-1)ndash Use for computing forecast

Exponential Smoothing Equations

During the past 8 quarters the Port of Baltimore has unloaded large quantities of grain ( = 10) The first quarter forecast was 175 QuarterActual

1 180 2 168

3 1594 1755 190

6 2057 1808 1829

Exponential Smoothing Example

Find the forecast for the 9th quarter

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

11 180 17500 (Given)

22 168168

33 159159

44 175175

55 190190

66 205205

17500 +17500 +

Exponential Smoothing Solution

QuarterQuarter ActuaActualForecast F t

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010((

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - -

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Ft

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - 17500 - 17500))

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing SolutionFt = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα= = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 +17500 + 1010(180 (180 - 17500- 17500)) = 17550 = 17550

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

1 180 17500 (Given)

22 168168 17500 + 10(180 - 17500) = 1755017500 + 10(180 - 17500) = 17550

33 159159 1755017550 ++ 1010(168 -(168 - 1755017550)) = 17475= 17475

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1995 180 17500 (Given)

1996 168 17500 + 10(180 - 17500) = 17550

1997 159 17550 + 10(168 - 17550) = 17475

1998 175

1999 190

2000 205

17475 + 10(159 - 17475)= 17318

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9 17822 + 10(182 - 17822) = 17858 182 17802 + 10(180 - 17802) = 17822

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

bull Form of weighted moving averagendash Weights decline exponentiallyndash Most recent data weighted most

bull Requires smoothing constant ()ndash Ranges from 0 to 1ndash Subjectively chosen

bull Involves little record keeping of past data

Exponential Smoothing Method

bull Ft = At - 1 + (1-)At - 2 + (1- )2middotAt - 3

+ (1- )3At - 4 + + (1- )t-1middotA0

ndash Ft = Forecast value

ndash At = Actual value = Smoothing constant

bull Ft = Ft-1 + (At-1 - Ft-1)ndash Use for computing forecast

Exponential Smoothing Equations

During the past 8 quarters the Port of Baltimore has unloaded large quantities of grain ( = 10) The first quarter forecast was 175 QuarterActual

1 180 2 168

3 1594 1755 190

6 2057 1808 1829

Exponential Smoothing Example

Find the forecast for the 9th quarter

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

11 180 17500 (Given)

22 168168

33 159159

44 175175

55 190190

66 205205

17500 +17500 +

Exponential Smoothing Solution

QuarterQuarter ActuaActualForecast F t

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010((

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - -

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Ft

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - 17500 - 17500))

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing SolutionFt = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα= = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 +17500 + 1010(180 (180 - 17500- 17500)) = 17550 = 17550

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

1 180 17500 (Given)

22 168168 17500 + 10(180 - 17500) = 1755017500 + 10(180 - 17500) = 17550

33 159159 1755017550 ++ 1010(168 -(168 - 1755017550)) = 17475= 17475

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1995 180 17500 (Given)

1996 168 17500 + 10(180 - 17500) = 17550

1997 159 17550 + 10(168 - 17550) = 17475

1998 175

1999 190

2000 205

17475 + 10(159 - 17475)= 17318

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9 17822 + 10(182 - 17822) = 17858 182 17802 + 10(180 - 17802) = 17822

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

bull Ft = At - 1 + (1-)At - 2 + (1- )2middotAt - 3

+ (1- )3At - 4 + + (1- )t-1middotA0

ndash Ft = Forecast value

ndash At = Actual value = Smoothing constant

bull Ft = Ft-1 + (At-1 - Ft-1)ndash Use for computing forecast

Exponential Smoothing Equations

During the past 8 quarters the Port of Baltimore has unloaded large quantities of grain ( = 10) The first quarter forecast was 175 QuarterActual

1 180 2 168

3 1594 1755 190

6 2057 1808 1829

Exponential Smoothing Example

Find the forecast for the 9th quarter

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

11 180 17500 (Given)

22 168168

33 159159

44 175175

55 190190

66 205205

17500 +17500 +

Exponential Smoothing Solution

QuarterQuarter ActuaActualForecast F t

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010((

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - -

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Ft

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - 17500 - 17500))

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing SolutionFt = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα= = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 +17500 + 1010(180 (180 - 17500- 17500)) = 17550 = 17550

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

1 180 17500 (Given)

22 168168 17500 + 10(180 - 17500) = 1755017500 + 10(180 - 17500) = 17550

33 159159 1755017550 ++ 1010(168 -(168 - 1755017550)) = 17475= 17475

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1995 180 17500 (Given)

1996 168 17500 + 10(180 - 17500) = 17550

1997 159 17550 + 10(168 - 17550) = 17475

1998 175

1999 190

2000 205

17475 + 10(159 - 17475)= 17318

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9 17822 + 10(182 - 17822) = 17858 182 17802 + 10(180 - 17802) = 17822

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

During the past 8 quarters the Port of Baltimore has unloaded large quantities of grain ( = 10) The first quarter forecast was 175 QuarterActual

1 180 2 168

3 1594 1755 190

6 2057 1808 1829

Exponential Smoothing Example

Find the forecast for the 9th quarter

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

11 180 17500 (Given)

22 168168

33 159159

44 175175

55 190190

66 205205

17500 +17500 +

Exponential Smoothing Solution

QuarterQuarter ActuaActualForecast F t

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010((

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - -

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Ft

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - 17500 - 17500))

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing SolutionFt = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα= = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 +17500 + 1010(180 (180 - 17500- 17500)) = 17550 = 17550

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

1 180 17500 (Given)

22 168168 17500 + 10(180 - 17500) = 1755017500 + 10(180 - 17500) = 17550

33 159159 1755017550 ++ 1010(168 -(168 - 1755017550)) = 17475= 17475

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1995 180 17500 (Given)

1996 168 17500 + 10(180 - 17500) = 17550

1997 159 17550 + 10(168 - 17550) = 17475

1998 175

1999 190

2000 205

17475 + 10(159 - 17475)= 17318

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9 17822 + 10(182 - 17822) = 17858 182 17802 + 10(180 - 17802) = 17822

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

11 180 17500 (Given)

22 168168

33 159159

44 175175

55 190190

66 205205

17500 +17500 +

Exponential Smoothing Solution

QuarterQuarter ActuaActualForecast F t

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010((

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - -

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Ft

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - 17500 - 17500))

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing SolutionFt = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα= = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 +17500 + 1010(180 (180 - 17500- 17500)) = 17550 = 17550

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

1 180 17500 (Given)

22 168168 17500 + 10(180 - 17500) = 1755017500 + 10(180 - 17500) = 17550

33 159159 1755017550 ++ 1010(168 -(168 - 1755017550)) = 17475= 17475

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1995 180 17500 (Given)

1996 168 17500 + 10(180 - 17500) = 17550

1997 159 17550 + 10(168 - 17550) = 17475

1998 175

1999 190

2000 205

17475 + 10(159 - 17475)= 17318

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9 17822 + 10(182 - 17822) = 17858 182 17802 + 10(180 - 17802) = 17822

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

QuarterQuarter ActuaActualForecast F t

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010((

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - -

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Ft

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - 17500 - 17500))

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing SolutionFt = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα= = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 +17500 + 1010(180 (180 - 17500- 17500)) = 17550 = 17550

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

1 180 17500 (Given)

22 168168 17500 + 10(180 - 17500) = 1755017500 + 10(180 - 17500) = 17550

33 159159 1755017550 ++ 1010(168 -(168 - 1755017550)) = 17475= 17475

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1995 180 17500 (Given)

1996 168 17500 + 10(180 - 17500) = 17550

1997 159 17550 + 10(168 - 17550) = 17475

1998 175

1999 190

2000 205

17475 + 10(159 - 17475)= 17318

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9 17822 + 10(182 - 17822) = 17858 182 17802 + 10(180 - 17802) = 17822

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

QuarterQuarter ActualActualForecast Forecast FFtt

((αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - -

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Ft

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - 17500 - 17500))

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing SolutionFt = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα= = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 +17500 + 1010(180 (180 - 17500- 17500)) = 17550 = 17550

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

1 180 17500 (Given)

22 168168 17500 + 10(180 - 17500) = 1755017500 + 10(180 - 17500) = 17550

33 159159 1755017550 ++ 1010(168 -(168 - 1755017550)) = 17475= 17475

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1995 180 17500 (Given)

1996 168 17500 + 10(180 - 17500) = 17550

1997 159 17550 + 10(168 - 17550) = 17475

1998 175

1999 190

2000 205

17475 + 10(159 - 17475)= 17318

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9 17822 + 10(182 - 17822) = 17858 182 17802 + 10(180 - 17802) = 17822

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

QuarterQuarter ActualActualForecast Ft

(αα = = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 + 17500 + 1010(180(180 - 17500 - 17500))

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing SolutionFt = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast Forecast FFtt

((αα= = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 +17500 + 1010(180 (180 - 17500- 17500)) = 17550 = 17550

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

1 180 17500 (Given)

22 168168 17500 + 10(180 - 17500) = 1755017500 + 10(180 - 17500) = 17550

33 159159 1755017550 ++ 1010(168 -(168 - 1755017550)) = 17475= 17475

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1995 180 17500 (Given)

1996 168 17500 + 10(180 - 17500) = 17550

1997 159 17550 + 10(168 - 17550) = 17475

1998 175

1999 190

2000 205

17475 + 10(159 - 17475)= 17318

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9 17822 + 10(182 - 17822) = 17858 182 17802 + 10(180 - 17802) = 17822

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

QuarterQuarter ActualActualForecast Forecast FFtt

((αα= = 1010))

11 180180 17500 (Given)17500 (Given)

22 168168 17500 +17500 + 1010(180 (180 - 17500- 17500)) = 17550 = 17550

33 159159

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

1 180 17500 (Given)

22 168168 17500 + 10(180 - 17500) = 1755017500 + 10(180 - 17500) = 17550

33 159159 1755017550 ++ 1010(168 -(168 - 1755017550)) = 17475= 17475

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1995 180 17500 (Given)

1996 168 17500 + 10(180 - 17500) = 17550

1997 159 17550 + 10(168 - 17550) = 17475

1998 175

1999 190

2000 205

17475 + 10(159 - 17475)= 17318

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9 17822 + 10(182 - 17822) = 17858 182 17802 + 10(180 - 17802) = 17822

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Ft = Ft-1 + 01(At-1 - Ft-1)

QuarterQuarter ActualActualForecast F t

(αα = = 1010))

1 180 17500 (Given)

22 168168 17500 + 10(180 - 17500) = 1755017500 + 10(180 - 17500) = 17550

33 159159 1755017550 ++ 1010(168 -(168 - 1755017550)) = 17475= 17475

44 175175

55 190190

66 205205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1995 180 17500 (Given)

1996 168 17500 + 10(180 - 17500) = 17550

1997 159 17550 + 10(168 - 17550) = 17475

1998 175

1999 190

2000 205

17475 + 10(159 - 17475)= 17318

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9 17822 + 10(182 - 17822) = 17858 182 17802 + 10(180 - 17802) = 17822

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1995 180 17500 (Given)

1996 168 17500 + 10(180 - 17500) = 17550

1997 159 17550 + 10(168 - 17550) = 17475

1998 175

1999 190

2000 205

17475 + 10(159 - 17475)= 17318

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9 17822 + 10(182 - 17822) = 17858 182 17802 + 10(180 - 17802) = 17822

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9 17822 + 10(182 - 17822) = 17858 182 17802 + 10(180 - 17802) = 17822

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Ft = Ft-1 + 01(At-1 - Ft-1)

Quarter ActualForecast F t

(α = 10)

1 180 17500 (Given)

2 168 17500 + 10(180 - 17500) = 17550

3 159 17550 + 10(168 - 17550) = 17475

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9 17822 + 10(182 - 17822) = 17858 182 17802 + 10(180 - 17802) = 17822

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9 17822 + 10(182 - 17822) = 17858 182 17802 + 10(180 - 17802) = 17822

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Ft = Ft-1 + 01(At-1 - Ft-1)

Time ActualForecast F t

(α = 10)

4 175 17475 + 10(159 - 17475) = 17318

5 190 17318 + 10(175 - 17318) = 17336

6 205 17336 + 10(190 - 17336) = 17502

Exponential Smoothing Solution

7 180

8

17502 + 10(205 - 17502) = 17802

9 17822 + 10(182 - 17822) = 17858 182 17802 + 10(180 - 17802) = 17822

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Ft = At - 1 + (1- )At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Ft = At - 1 + (1- ) At - 2 + (1- )2At - 3 +

Forecast Effects of Smoothing Constant

Weights

Prior Period

2 periods ago

(1 - )

3 periods ago

(1 - )2

=

= 010

= 090

10 9 81

90 9 09

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Impact of

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9

Quarter

Actu

al To

nage

ActualForecast (01)

Forecast (05)

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Choosing

Seek to minimize the Mean Absolute Deviation (MAD)

If Forecast error = demand - forecast

Then n

errorsforecast MAD

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Exponential Smoothing with Trend Adjustment

Forecast including trend (FITt)

= exponentially smoothed forecast (Ft)

+ exponentially smoothed trend (Tt)

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Ft = Last periodrsquos forecast + (Last periodrsquos actual ndash Last periodrsquos forecast)

Ft = Ft-1 + (At-1 ndash Ft-1)or

Tt = (Forecast this period - Forecast last period) + (1-)(Trend estimate last period

Tt = (Ft - Ft-1) + (1- )Tt-1 or

Exponential Smoothing with Trend Adjustment -

continued

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

bull Ft = exponentially smoothed forecast of the data series in period t

bull Tt = exponentially smoothed trend in period t

bull At = actual demand in period t = smoothing constant for the

average = smoothing constant for the trend

Exponential Smoothing with Trend Adjustment -

continued

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Comparing Actual and Forecasts

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10

Month

Dem

and

ActualDemand

SmoothedForecast

Smoothed Trend

Forecast includingtrend

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Regression

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Least Squares

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Deviation

Time

Valu

es o

f Dep

ende

nt V

aria

ble

bxaY ˆ

Actual observation

Point on regression line

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Actual and the Least Squares Line

0

20

40

60

80

100

120

140

160

1996 1997 1998 1999 2000 2001 2002 2003 2004

Year

Regression Line

Actual Demand

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

bull Used for forecasting linear trend linebull Assumes relationship between

response variable Y and time X is a linear function

bull Estimated by least squares methodndash Minimizes sum of squared errors

iY a bX i

Linear Trend Projection

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Scatter DiagramSales versus Payroll

0

1

2

3

4

0 1 2 3 4 5 6 7 8

Area Payroll (in $ hundreds of millions)

Sales

(in $

hund

reds

of

thou

sand

s)

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Least Squares Equations

Equation ii bxaY

Slope

xnx

yxnyxb

i

n

i

ii

n

i

Y-Intercept xbya

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

X i Y i X i2 Y i

2 X iY i

X 1 Y 1 X 12 Y 1

2 X 1Y 1

X 2 Y 2 X 22 Y 2

2 X 2Y 2

X n Y n X n2 Y n

2 X nY n

ΣX i ΣY i ΣX i2 ΣY i

2 ΣX iY i

Computation Table

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Using a Trend Line

Year Demand1997 741998 791999 802000 902001 1052002 1422003 122

The demand for electrical power at NYEdison over the years 1997 ndash 2003 is given at the left Find the overall trend

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Finding a Trend LineYea

rTime Perio

d

Power Deman

d

x2 xy

1997

1 74 1 74

1998

2 79 4 158

1999

3 80 9 240

2000

4 90 16 360

2001

5 105 25 525

2002

6 142 36 852

2003

7 122 49 854

x=28

y=692

x2=140

xy=3063

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

The Trend Line Equation

megawatts 15156 1054(9) 5670 2005in Demand

megawatts 14102 1054(8) 5670 2004in Demand

5670 1054(4) - 9886 xb - y a

105428

295

(7)(4)140

86)(7)(4)(983063

xnΣx

yxn -Σxy b

98867

692

n

Σyy 4

7

28

n

Σxx

222

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Actual and Trend ForecastElectric Power Demand

60

70

80

90

100

110

120

130

140

150

160

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Monthly Sales of Laptop ComputersSales Demand Average

DemandMonth 2000 2001 2002 2000-

2002Monthl

ySeasonal

IndexJan 80 85 105 90 94 0957Feb 70 85 85 80 94 0851Mar 80 93 82 85 94 0904Apr 90 95 115 100 94 1064May 113 125 131 123 94 1309Jun 110 115 120 115 94 1223Jul 100 102 113 105 94 1117Aug 88 102 110 100 94 1064Sept 85 90 95 90 94 0957Oct 77 78 85 80 94 0851Nov 75 72 83 80 94 0851Dec 82 78 80 80 94 0851

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Demand for IBM Laptops

0

20

40

60

80

100

120

140

J an Feb Mar Apr May J un J ul Aug Sep Oct Nov Dec

Month

000

020

040

060

080

100

120

140

Trend

Seasonal Index

Forecast trend + seasonal index

Monthly Average

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

San Diego Hospital ndash Inpatient Days

8800

9000

9200

9400

9600

9800

10000

10200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

092

094

096

098

1

102

104

106

Seasonal Index

Trend

Combined Forecast

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Multiplicative Seasonal Modelbull Find average historical demand for each ldquoseasonrdquo by

summing the demand for that season in each year and dividing by the number of years for which you have data

bull Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons

bull Compute a seasonal index by dividing that seasonrsquos historical demand (from step 1) by the average demand over all seasons

bull Estimate next yearrsquos total demandbull Divide this estimate of total demand by the number of

seasons then multiply it by the seasonal index for that season This provides the seasonal forecast

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Y Xi i= a b

bull Shows linear relationship between dependent amp explanatory variablesndash Example Sales amp advertising (not time)

Dependent (response) variable

Independent (explanatory) variable

SlopeY-intercept

^

Linear Regression Model

+

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Linear Regression Equations

Equation ii bxaY

Slope22

i

n

1i

ii

n

1i

xnx

yxnyx b

Y-Intercept xby a

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

X i Y i X i2 Y i

2 X iY i

X1 Y 1 X12 Y 1

2 X1Y 1

X2 Y 2 X22 Y 2

2 X2Y 2

Xn Y n Xn2 Y n

2 XnY n

Σ X i Σ Y i Σ X i2 Σ Y i

2 Σ X iY i

Computation Table

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

bull Slope (b)ndash Estimated Y changes by b for each 1 unit

increase in Xbull If b = 2 then sales (Y) is expected to increase

by 2 for each 1 unit increase in advertising (X)

bull Y-intercept (a)ndash Average value of Y when X = 0

bull If a = 4 then average sales (Y) is expected to be 4 when advertising (X) is 0

Interpretation of Coefficients

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

bull Variation of actual Y from predicted Ybull Measured by standard error of

estimatendash Sample standard deviation of errors

ndash Denoted SYX

bull Affects several factorsndash Parameter significancendash Prediction accuracy

Random Error Variation

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Least Squares Assumptions

bull Relationship is assumed to be linear Plot the data first - if curve appears to be present use curvilinear analysis

bull Relationship is assumed to hold only within or slightly outside data range Do not attempt to predict time periods far beyond the range of the data base

bull Deviations around least squares line are assumed to be random

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Standard Error of the Estimate

2

2

1 11

2

1

2

n

yxbyay

n

yyS

n

i

n

iiii

n

ii

n

ici

xy

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

bull Answers lsquohow strong is the linear relationship between the variablesrsquo

bull Coefficient of correlation Sample correlation coefficient denoted rndash Values range from -1 to +1ndash Measures degree of association

bull Used mainly for understanding

Correlation

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Sample Coefficient of Correlation

n

i

n

iii

n

i

n

iii

n

i

n

i

n

iiiii

yynxxn

yxyxnr

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

r = 1 r = -1

r = 89 r = 0

Y

XYi = a + b X i^

Y

X

Y

X

Y

XYi = a + b X i^ Yi = a + b X i

^

Yi = a + b X i^

Coefficient of Correlation and Regression Model

r2 = square of correlation coefficient (r) is the percent of the variation in y that is explained by the regression equation

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

bull You want to achievendash No pattern or direction in forecast error

bull Error = (Yi - Yi) = (Actual - Forecast)

bull Seen in plots of errors over time

ndash Smallest forecast errorbull Mean square error (MSE)bull Mean absolute deviation (MAD)

Guidelines for Selecting Forecasting Model

^

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Time (Years)

ErrorError

00

Desired Pattern

Time (Years)

Error

0

Trend Not Fully Accounted for

Pattern of Forecast Error

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

bull Mean Square Error (MSE)

bull Mean Absolute Deviation (MAD)

bull Mean Absolute Percent Error (MAPE)

Forecast Error Equations2

n

1i

2ii

n

errorsforecast

n

)y(yMSE

nn

yyMAD

n

iii

|errorsforecast |

|ˆ|1

n

actual

forecastactual

100MAPE

n

1i i

ii

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Yoursquore a marketing analyst for Hasbro Toys Yoursquove forecast sales with a linear model amp exponential smoothing Which model do you use

Actual Linear Model Exponential Smoothing

Year Sales Forecast Forecast (9)

1998 1 06 101999 1 13 102000 2 20 192001 2 27 202002 4 34 38

Selecting Forecasting Model Example

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

MSE = Σ Error2 n = 110 5 = 0220MAD = Σ |Error| n = 20 5 = 0400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Linear Model EvaluationY i

11224

Y i^

0613202734

Year

19981999200020012002Total

04-03 00-07 0600

Error

016009000049036110

Error2

040300070620

|Error||Error|Actual

040030000035015120

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

MSE = Σ Error2 n = 005 5 = 001MAD = Σ |Error| n = 03 5 = 006MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

Exponential Smoothing Model Evaluation

Year

19981999200020012002Total

Y i11224

Y i10 0010 0019 0120 0038 02

03

^ Error

000000001000004005 03

Error2

0000010002

|Error||Error|Actual

000000005000005

010

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Exponential Smoothing Model Evaluation

Linear ModelMSE = Σ Error2 n = 110 5 = 220MAD = Σ |Error| n = 20 5 = 400MAPE = 100 Σ|absolute percent errors|n= 1205 = 0240

Exponential Smoothing ModelMSE = Σ Error2 n = 005 5 = 001

MAD = Σ |Error| n = 03 5 = 006

MAPE = 100 Σ |Absolute percent errors|n = 0105 = 002

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

bull Measures how well the forecast is predicting actual values

bull Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD)ndash Good tracking signal has low values

bull Should be within upper and lower control limits

Tracking Signal

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Tracking Signal Equation

MAD

errorforecast

MAD

yy

MADRSFE

TS

n

iii

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

MoMo FcstFcst ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10

Error = Actual - Forecast = 90 - 100 = -10

Error = Actual - Forecast = 90 - 100 = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

6 100 140

-10-10 -10-10

RSFE = Errors = NA + (-10) = -10

RSFE = Errors = NA + (-10) = -10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010

Abs Error = |Error| = |-10| = 10

Abs Error = |Error| = |-10| = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010

Cum |Error| = |Errors| = NA + 10 = 10

Cum |Error| = |Errors| = NA + 10 = 10

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum|Error||Error|

MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100

MAD = |Errors|n = 101 = 10

MAD = |Errors|n = 101 = 10

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

TS = RSFEMAD = -1010 = -1

TS = RSFEMAD = -1010 = -1

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5

Error = Actual - Forecast = 95 - 100 = -5

Error = Actual - Forecast = 95 - 100 = -5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

RSFE = Errors = (-10) + (-5) = -15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55

Abs Error = |Error| = |-5| = 5

Abs Error = |Error| = |-5| = 5

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515

Cum Error = |Errors| = 10 + 5 = 15

Cum Error = |Errors| = 10 + 5 = 15

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575

MAD = |Errors|n = 152 = 75

MAD = |Errors|n = 152 = 75

|Error||Error|

Tracking Signal Computation

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

MoMo ForcForc ActAct ErrorError RSFERSFE AbsAbsErrorError

CumCum MADMAD TSTS

11 100100 9090

22 100100 9595

33 100100 115115

44 100100 100100

55 100100 125125

66 100100 140140

-10-10 -10-10 1010 1010 100100 -1-1

-5-5 -15-15 55 1515 7575 -2-2

|Error||Error|

TS = RSFEMAD = -1575 = -2

TS = RSFEMAD = -1575 = -2

Tracking Signal Computation

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Plot of a Tracking Signal

Time

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

+

0

-

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Tracking Signals

020406080

100120140160

0 1 2 3 4 5 6 7

Time

Act

ual

Dem

and

-3

-2

-1

0

1

2

3

Tra

ckin

g S

inga

l

Tracking Signal

Forecast

Actual demand

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Forecasting in the Service Sector

bull Presents unusual challengesndash special need for short term recordsndash needs differ greatly as function of

industry and productndash issues of holidays and calendarndash unusual events

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant

Forecast of Sales by Hour for Fast Food Restaurant

0

5

10

15

20

+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

• Slide 1
• Outline
• Outline - Continued
• Slide 4
• Learning Objectives
• Learning Objectives-continued
• Forecasting at Tupperware
• Three Key Factors for Tupperware
• Tupperware - Forecast by Consensus
• What is Forecasting
• Types of Forecasts by Time Horizon
• Short-term vs Longer-term Forecasting
• Influence of Product Life Cycle
• Strategy and Issues During a Productrsquos Life
• Types of Forecasts
• Seven Steps in Forecasting (預測的七個基本步驟)
• Product Demand Charted over 4 Years with Trend and Seasonality
• Actual Demand Moving Average Weighted Moving Average
• Realities of Forecasting
• Forecasting Approaches
• Overview of Qualitative Methods
• Jury of Executive Opinion
• Sales Force Composite
• Delphi Method
• Consumer Market Survey
• Overview of Quantitative Approaches
• Quantitative Forecasting Methods (Non-Naive)
• What is a Time Series
• Time Series Components
• Trend Component
• Seasonal Component
• Common Seasonal Patterns
• Cyclical Component
• Random Component
• General Time Series Models
• Naive Approach
• Moving Average Method
• Moving Average Example
• Moving Average Solution
• Slide 40
• Slide 41
• Moving Average Graph
• Weighted Moving Average Method
• Slide 44
• Disadvantages of Moving Average Methods
• Exponential Smoothing Method
• Exponential Smoothing Equations
• Exponential Smoothing Example
• Exponential Smoothing Solution
• Slide 50
• Slide 51
• Slide 52
• Slide 53
• Slide 54
• Slide 55
• Slide 56
• Slide 57
• Slide 58
• Slide 59
• Forecast Effects of Smoothing Constant
• Slide 61
• Slide 62
• Slide 63
• Slide 64
• Slide 65
• Impact of
• Choosing
• Exponential Smoothing with Trend Adjustment
• Exponential Smoothing with Trend Adjustment - continued
• Slide 70
• Comparing Actual and Forecasts
• Regression
• Least Squares
• Actual and the Least Squares Line
• Linear Trend Projection
• Scatter Diagram
• Least Squares Equations
• Computation Table
• Using a Trend Line
• Finding a Trend Line
• The Trend Line Equation
• Actual and Trend Forecast
• Monthly Sales of Laptop Computers
• Demand for IBM Laptops
• San Diego Hospital ndash Inpatient Days
• Multiplicative Seasonal Model
• Linear Regression Model
• Linear Regression Equations
• Slide 91
• Interpretation of Coefficients
• Random Error Variation
• Least Squares Assumptions
• Standard Error of the Estimate
• Correlation
• Sample Coefficient of Correlation
• Coefficient of Correlation and Regression Model
• Guidelines for Selecting Forecasting Model
• Pattern of Forecast Error
• Forecast Error Equations
• Selecting Forecasting Model Example
• Linear Model Evaluation
• Exponential Smoothing Model Evaluation
• Slide 106
• Tracking Signal
• Tracking Signal Equation
• Tracking Signal Computation
• Slide 110
• Slide 111
• Slide 112
• Slide 113
• Slide 114
• Slide 115
• Slide 116
• Slide 117
• Slide 118
• Slide 119
• Slide 120
• Slide 121
• Plot of a Tracking Signal
• Tracking Signals
• Forecasting in the Service Sector
• Forecast of Sales by Hour for Fast Food Restaurant