Ch08 - Inventory

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Inventory ModelsInventory Models

Chapter 8

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8.1 Overview of Inventory Issues

• Proper control of inventory is crucial to the success of an enterprise.

• Typical inventory problems include:– Basic inventory – Planned shortage – Quantity discount – Periodic review– Production lot size – Single period

• Inventory models are often used to develop an optimal inventory policy, consisting of:– An order quantity, denoted Q.– A reorder point, denoted R.

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• Inventory analyses can be thought of as cost-control techniques.

• Categories of costs in inventory models:– Holding (carrying costs)– Order/ Setup costs– Customer satisfaction costs– Procurement/Manufacturing costs

Type of Costs in Inventory Models

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• Holding Costs (Carrying costs): These costs depend on the order size– Cost of capital – Storage space rental cost– Costs of utilities– Labor– Insurance– Security– Theft and breakage– Deterioration or Obsolescence

Ch = Annual holding cost per unit in inventoryH = Annual holding cost rateC = Unit cost of an item

Ch = H * C


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• Order/Setup CostsThese costs are independent of the order size.– Order costs are incurred when purchasing a good

from a supplier. They include costs such as• Telephone • Order checking• Labor • Transportation

– Setup costs are incurred when producing goods for sale to others. They can include costs of

• Cleaning machines• Calibrating equipment• Training staff


Co = Order cost or setup cost

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• Customer Satisfaction Costs– Measure the degree to which a

customer is satisfied.– Unsatisfied customers may:

• Switch to the competition (lost sales).• Wait until an order is supplied.

– When customers are willing to wait there are two types of costs incurred:


Cb= Fixed administrative costs of an out of stock item ($/stockout unit).

Cs = Annualized cost of a customer awaiting an out of stock item($/stockout unit per year).

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• Procurement/Manufacturing Cost– Represents the unit purchase cost

(including transportation) in case of a purchase.

– Unit production cost in case of in-house manufacturing.


C = Unit purchase or manufacturing cost.

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• Demand is a key component affecting an inventory policy.

• Projected demand patterns determine how an inventory problem is modeled.

• Typical demand patterns are:– Constant over time (deterministic inventory models)– Changing but known over time (dynamic models)– Variable (randomly) over time (probabilistic models)

Demand in Inventory Models

D = Demand rate (usually per year)

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Inventory can be classified in various ways:

By Process By Importance By Shelf Life Raw materials Perishable Work in progress A, B, C Nonperishable Finished goods

By Process By Importance By Shelf Life Raw materials Perishable Work in progress A, B, C Nonperishable Finished goods

Used typically by accountants at manufacturing firms.Enables management to track the production process.

Items are classified by their relative importancein terms of the firm’s capital needs.

Management of items with short shelf life and long shelf life is very different

Inventory Classifications

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• Two types of review systems are used:– Continuous review systems.

• The system is continuously monitored.• A new order is placed when the inventory reaches a critical

point.– Periodic review systems.

• The inventory position is investigated on a regular basis.• An order is placed only at these times.

Review Systems

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• The item has a sufficiently long shelf life.• The item is monitored using a continuous review

system.• All the cost parameters remain constant forever

(over an infinite time horizon).• A complete order is received in one batch.

8.2 Economic Order Quantity Model - Assumptions

• Demand occurs at a known and reasonably constant rate.

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• The constant environment described by the EOQ assumptions leads to the following observation:

The optimal EOQ policy consists of same-size orders.

Q QQ

The EOQ Model – Inventory profile

This observation results in the following inventory profile :

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Q QQ

Total Annual Inventory Costs

= Total Annual Holding Costs

Total Annual ordering Costs

Total Annual procurement Costs

++

TC(Q) = (Q/2)Ch + (D/Q)Co + DC

Ch

The optimal order Size

2DCoQ* =

Cost Equation for the EOQ Model

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Constructing the total annual variable cost curve

Total Holding Costs

Total ordering costs

Add the two curves to one anotherTotal annual holding and ordering costs

Q

TV(Q)

Q*

The optimal order size

o* * * * *

TV(Q) and Q*

Note: at the optimal order size

total holding costs and ordering costs

are equal

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The curve is reasonably flat around Q*.

Q*

Deviations from the optimal order size cause only small increase in the total cost.

Sensitivity Analysis in EOQ models

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• The cycle time, T, represents the time that

elapses between the placement of orders.

• Note, if the cycle time is greater than the shelf life, items will go bad, and the model must be modified.

T = Q/D

Cycle Time

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• To find the number of orders per years take the reciprocal of the cycle time

N = D/Q

• Example: The demand for a product is 1000 units per year. The order size is 250 units under an EOQ policy.• How many orders are placed per year? N = 1000/250 = 4 orders.• How often orders need to be placed (what is the cycle time)?

T = 250/1000 = ¼ years. {Note: the four orders are equally spaced}.

Number of Orders per Year

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• In reality lead time always exists, and must be accounted for when deciding when to place an order.

• The reorder point, R, is the inventory position when an order is placed.

• R is calculated by

L and D must be expressed in the same time unit.

R = L D

Lead Time and the Reorder Point

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Inventory position

LPlace the order now

Reorder Point

R = Inventory at hand at the beginning of lead time

Lead Time and the Reorder Point –Graphical demonstration: Short Lead Time

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Outstanding order

Place the order now

R = inventory at hand at the beginning of lead time + one outstanding order = demand during lead time = LD

Inventory at

hand

L

Lead Time and the Reorder Point –Graphical demonstration: Long Lead Time

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• Safety stocks act as buffers to handle:– Higher than average lead time demand.– Longer than expected lead time.

• With the inclusion of safety stock (SS), R is calculated by

• The size of the safety stock is based on having a desired service level.

R = LD + SS

Safety stock

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LPlace the order now

Reorder Point

R = LD

Safety stockPlanned situation

Actualsituation

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L

R = LD

Safety stock

Actualsituation

+ SS

Reorder Point

Place the order now

SS=Safety stock

?

The safety stockprevents excessiveshortages.

LD

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Inventory Costs Including safety stock

Total Annual Inventory Costs

= Total Annual Holding Costs

Total Annual ordering Costs

Total Annual procurement Costs

++

TC(Q) = (Q/2)Ch + (D/Q)Co + DC + ChSS

Safety stockholding cost

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ALLEN APPLIANCE COMPANY (AAC)

• AAC wholesales small appliances.

• AAC currently orders 600 units of the Citron brand juicer each time inventory drops to 205 units.

• Management wishes to determine an optimal ordering policy for the Citron brand juicer

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Sales of Juicers over the last 10 weeksWeek 1 2 3 4 5Sales 105 115 125 120 125Week 6 7 8 9 10Sales 120 135 115 110 130

• Data– Co = $12 ($8 for placing an order) + (20 min. to check)($12 per hr) – Ch = $1.40 [HC = (14%)($10)]– C = $10.– H = 14% (10% ann. interest rate) + (4% miscellaneous)– D = demand information of the last 10 weeks was collected:


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• Data– The constant demand rate seems to be a good

assumption.– Annual demand = (120/week)(52weeks) = 6240 juicers.


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• Current ordering policy calls for Q = 600 juicers.TV( 600) = (600 / 2)($1.40) + (6240 / 600)($12) = $544.80

• The EOQ policy calls for orders of size

AAC – Solution:EOQ and Total Variable Cost

Savings of 16%

2(6240)(12)1.40 = 327.065 327 =Q*

TV(327) = (327 / 2)($1.40) + (6240 / 327) ( $12) = $457.89

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TC(327) = 457.89 + 6240($10) + (13)($1.40) = $62,876.09

• Under the current ordering policy AAC holds 13 units safety stock (how come? Observe):

• AAC is open 5 day a week.– The average daily demand = 120/week)/5 = 24 juicers.– Lead time is 8 days. Lead time demand is (8)(24) = 192 juicers.– Reorder point without Safety stock = LD = 192.– Current policy: R = 205.– Safety stock = 205 – 192 = 13.

• For safety stock of 13 juicers the total cost is

TV(327) + Procurement + Safety stock cost holding cost

AAC – Solution:Reorder Point and Total Cost

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• Changing the order size

– Suppose juicers must be ordered in increments of 100 (order 300 or 400)– AAC will order Q = 300 juicers in each order.– There will be a total variable cost increase of $1.71.– This is less than 0.5% increase in variable costs.

• Changes in input parameters– Suppose there is a 20% increase in demand. D=7500 juicers.– The new optimal order quantity is Q* = 359.– The new variable total cost = TV(359) = $502 – If AAC still orders Q = 327, its total variable costs becomes

TV(327) = (3272)($1.40) + (7500327)($12) = $504.13

Only 0.4% increase

AAC – Solution:Sensitivity of the EOQ Results

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• For an order size of 327 juicers we have:– T = (3276240) = 0.0524 year.

= 0.0524(52)(5) = 14 days.

– This is useful information because:

• Shelf life may be a problem.• Coordinating orders with other items might be desirable.

AAC – Solution:Cycle Time

working days per week

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AAC – Excel Spreadsheet

=SQRT(2*$B$10*$B$14/$B$13)

=1/E11Copy to cell H12

=E10/B10Copy to cell

H11

=$B$10*$B$11+E14+$B$13*B16Copy to Cell H15

=(E10/2)*$B$13+($B$10/E10)*$B$14Copy to cell H14

=$B$15*$B$10+$B$16-INT(($B$15*$B$10+$B$16)/E10)*E10

Copy to cell H13

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Service Levels and Safety Stocks

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8.3 Determining Safety Stock Levels

• Businesses incorporate safety stock requirements when determining reorder points.

• A possible approach to determining safety stock levels is by specifying desired service level .

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• The unit service level – The percentage of demands

that are filled without incurring any delay.

– Applied when the percentage of unsatisfied demand should be under control.

Two Types of Service Level

• The cycle service level – The probability of not

incurring a stockout during an inventory cycle.

– Applied when the likelihood of a stockout, and not its magnitude, is important for the firm.

Service levels can be viewed in two ways.

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• In many cases short run demand is variable even though long run demand is assumed constant.

• Therefore, stockout events during lead time may occur unexpectedly in each cycle.

• Stockouts occur only if demand during lead time is greater than the reorder point.

The Cycle Service Level Approach

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• To determine the reorder point we need to know:– The lead time demand distribution.– The required service level.

• In many cases lead time demand is approximately normally distributed. For the normal distribution case the reorder point is calculated by


R = L + zL 1 –= service level

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=192

P(DL> R) = P(Z > (R – L)/L) = . SinceP(Z > Z) = , we have Z = (R – L)/L,

which gives…


P(DL>R) = Service level = P(DL<R) = 1 –

R

R = L + zL

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• Assume that lead time demand is normally distributed.

• Estimation of the normal distribution parameters:

– Estimation of the mean weekly demand = ten weeks average demand = 120 juicers per week.

– Estimation of the variance of the weekly demand = Sample variance = 83.33 juicers2.

AAC - Cycle Service Level Approach

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• To find Land L the parameters (per week)and(per week)must be adjusted since the lead time is longer than one week.

– Lead time is 8 days =(8/5) weeks = 1.6 weeks.

• Estimates for the lead time mean demand and variance of demand

L (1.6)(120) = 192; 2L (1.6)(83.33) = 133.33

AAC - Cycle Service Level Approach

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• Let us use the current reorder point of 205 juicers.

205 = 192 + z (11.55) z = 1.13

• From the normal distribution table we have that a reorder

point of 205 juicers results in an 87% cycle service level.

133 33.

AAC - Service Level for a given Reorder Point

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• Management wants to improve the cycle service level to 99%.

• The z value corresponding to 1% right hand tail is 2.33.

R = 192 + 2.33(11.55) = 219 juicers.

AAC – Reorder Point for a given Service Level

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• AAC is willing to run out of stock an average of at most one cycle per year with an order quantity of 327 juicers.

• What is the equivalent service level for this strategy?

AAC – Acceptable Number of Stockouts per Year

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AAC – Acceptable Number of Stockouts per Year

• There will be an average of

6240327 = 19.08 lead times per year.

• The likelihood of stockouts = 1/19 = 0.0524.

• This translates into a service level of 94.76%

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• When lead time demand follows a normal distributionservice level can be calculated as follows:– Determine the value of z that satisfy the equation

L(z) = Q* L

– Solve for R using the equation

R = L + zL

The Unit Service Level Approach

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=NORMDIST(B8,B5,B6,TRUE)

AAC – Cycle Service Level (Excel spreadsheet)

=NORMINV(B7,B5,B6)

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• Quantity Discounts are Common Practice in Business– By offering discounts buyers are encouraged to increase

their order sizes, thus reducing the seller’s holding costs.

– Quantity discounts reflect the savings inherent in large orders.

– With quantity discounts sellers can reward their biggest customers without violating the Robinson - Patman Act.

8.4 EOQ Models with Quantity Discounts

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• Quantity Discount Schedule– This is a list of per unit discounts and their corresponding

purchase volumes.– Normally, the price per unit declines as the order quantity

increases.– The order quantity at which the unit price changes is called a

break point.– There are two main discount plans:

• All unit schedules - the price paid for all the units purchased is based on the total purchase.

• Incremental schedules - The price discount is based only on the additional units ordered beyond each break point.

8.4 EOQ Models with Quantity Discounts

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• To determine the optimal order quantity, the total purchase cost must be included

TC(Q) = (Q2)Ch + (DQ)Co + DCi + ChSS

Ci represents the unit cost at the ith pricing level.

All Units Discount Schedule

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AAC - All Units Quantity Discounts

Quantity Discount Schedule

1-299 $10.00300-599 $9.75600-999 $9.40

1000-4999 $9.505000 $9.00

Quantity Discount Schedule

1-299 $10.00300-599 $9.75600-999 $9.40

1000-4999 $9.505000 $9.00

• AAC is offering all units quantity discounts to its customers.

• Data

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Should AAC increase its regular order of 327 juicers, to take advantage of the discount?

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AAC – All units discount procedure

– Step 1: Find the optimal order Qi* for each discount level “i”.

Use the formula– Step 2: For each discount level “i” modify Q i

* as follows• If Qi

* is lower than the smallest quantity that qualifies for the i th discount, increase Qi

* to that level.

• If Qi* is greater than the largest quantity that qualifies for the i th discount,

eliminate this level from further consideration.

– Step 3: Substitute the modified Q*i value in the total cost formula

TC(Q*i ).

– Step 4: Select the Q i

* that minimizes TC(Q i*)

Q DC Co h* ( ) / 2

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Step 1: Find the optimal order quantity Qi* for each

discount level “i” based on the EOQ formula

Lowest cost order size per discount levelDiscount Qualifying Price

level order per unit Q*0 1-299 10.00 3271 300-599 9.75 3312 600-999 9.50 3363 1000-4999 9.40 3374 5000 9.00 345

AAC – All units discount procedure

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– Step 2 : Modify Q i *

Modified Q* and total CostQualified Price Modified Total

Urder per Unit Q* Q* Cost1-299 10.00 300 **** ****

300-599 9.75 331 331 61,292.13600-999 9.50 336 600 59,803.80

1000-4999 9.40 337 1000 59,388.885000 9.00 345 5000 59,324.98



300-599 9.75 331 331 61,292.13600-999 9.50 336 600 59,803.80

1000-4999 9.40 337 1000 59,388.885000 9.00 345 5000 59,324.98

1 299Q1

*

300

$10/unit

599331Q2

*$9.75/unit

999999600Q3

*336

$9.50

AAC – All Units Discount Procedure

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– Step 2 : Modify Q i *



300-599 9.75 331 331 61,292.13600-999 9.50 336 600 59,803.80

1000-4999 9.40 337 1000 59,388.885000 9.00 345 5000 59,324.98



300-599 9.75 331 331 61,292.13600-999 9.50 336 600 59,803.80

1000-4999 9.40 337 1000 59,388.885000 9.00 345 5000 59,324.98

1 299Q1

*

300

$10/unit

331Q2

*

999999600Q3

*336

$9.50


Q3* Q3

* Q3* Q3

* Q3*Q3

*

Q3*

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– Step 3: Substitute Q I * in the total cost function

– Step 4



300-599 9.75 331 331 61,292.13600-999 9.50 336 600 59,803.80

1000-4999 9.40 337 1000 59,388.885000 9.00 345 5000 59,324.98



300-599 9.75 331 331 61,292.13600-999 9.50 336 600 59,803.80

1000-4999 9.40 337 1000 59,388.885000 9.00 345 5000 59,324.98

AAC should order 5000 juicers


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Calculation of Optimal Inventory Policy Under All-Units Quantity Discounts

OPTIMALINPUTS Values OUTPUTS Values

Annual Demand, D = 6240.00 Order quantity, Q* = 5000Per Unit Cost, C = 10.00 Cycle Time (in years), T = 0.801282051Annual Holding Cost Rate, H = 0.14 # of Cycles Per Year, N = 1.248Annual Holding Cost Per Unit, Ch = 1.40 Reorder Point, R = 205.0000Order Cost, Co = 12.00 Total Annual Cost, TC(Q*) = 59341.36Lead Time (in years), L = 0.03077 Safety Stock, SS = 13.00

DISCOUNTSLevel Breakpoint Discount Price Q* TC(Q*) Modified Q*

0 1 10.00 327 62876.09 3271 300 9.75 331 61309.88 3312 600 9.50 336 59821.09 6003 1000 9.40 337 59405.99 10004 5000 9.00 345 59341.36 50005678

AAC – All Units Discount Excel Worksheet

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• Demand rate is constant.

• Production rate is larger than demand rate.

• The production lot is not received instantaneously (at an

infinite rate), because production rate is finite.

• There is only one product to be scheduled.

• The rest of the EOQ assumptions stay in place.

8.4 Production Lot Size Model - Assumptions

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• The optimal production lot size policy orders the same amount each time.• This observation results in the inventory profile below:

Production Lot Size Model – Inventory profile

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ProductionLot Size = Q = PT1

The inventory increasesat a net rate of P - D

The production increases theinventory at a rate of P.

The demand decreases theinventory at a rate of D.

Production time

T1

Demand accumulationduring production run

Demand accumulationduring production run = DT1

Maximum inventory = (P – D)T1

= (P – D)(Q/P) = Q(1 – D/P)

Maximum inventory

Production Lot Size Model – Understanding the inventory profile

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• The parameters of the total variable costs function are similar to those used in the EOQ model.

• Instead of ordering cost, we have here a fixed setup cost per production run (Co).

• In addition, we need to incorporate the annual production rate (P) in the model.

Production Lot Size Model –Total Variable Cost

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TV(Q) = (Q2)(1 - DP)Ch + (DQ)Co

P is the annual production rate

Ch(1-D/P)

The Optimal Order Size

Q* = 2DCo

The average inventory

Production Lot Size Model –Total Variable Cost

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• Cycle time T = Q / D.

• Length of a production run T1 = Q / P.

• Time when machines are not busy producing the product T2 = T - T1 = Q(1/D - 1/P).

• Average inventory = (Q/2)(1-D/P).

Production Lot Size Model –Useful relationships

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FARAH COSMETICS COMPANY

• Farah needs to determine optimal production lot size for its most popular shade of lipstick.

• Data• The factory operates 7 days a week, 24 hours a day.• Production rate is 1000 tubes per hour.• It takes 30 minutes to prepare the machinery for production.• It costs $150 to setup the line.• Demand is 980 dozen tubes per week.• Unit production cost is $.50• Annual holding cost rate is 40%.

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• Input for the total variable cost function

D = 613,200 per year [(980 dozen/week (12) 7](365)

Ch = 0.4(0.5) = $0.20 per tube per year.

Co = $150

P = (1000)(24)(365) = 8,760,000 per year.

Dozens

FARAH COSMETICS COMPANY – Solution

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• Current Policy

Currently, Farah produces in lots of 84,000 tubes.

T = (84,000 tubes per run)(613,200 tubes per year)= 0.137 years (about 50 days).

T1 = (84,000 tubes per lot)(8,760,000 tubes per year)= 0.0096 years (about 3.5 days).

T2 = 0.137 - 0.0096 = 0.1274 years (about 46.5 days).

TV(Q = 84,000) = (84,0002) {1-(613,2008,760,000)}(0.2) + 613,20084,000)(150) = $8907.


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• The Optimal Policy

Using the input data we find

TV(Q* = 31,499) = (31,499/2) [1-(613,200/8,760,000)](0.2) + (613,200/31,499)(150) = $5,850.

The optimal order size

(0.2)(1-613,200/8760,000)Q* =

2(613,200)(150) = 31,499


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FARAH COSMETICS COMPANY – Production Lot Size Template (Excel)

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8.5 Planned Shortage Model

• When an item is out of stock, customers may: – Go somewhere else (lost sales).– Place their order and wait (backordering).

• In this model we consider the backordering case.

• All the other EOQ assumptions are in place.

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• The parameters of the total variable costs function are similar to those used in the EOQ model.

• In addition, we need to incorporate the shortage costs in the model.– Backorder cost per unit per year (loss of goodwill cost) - Cs.

• Reflects future reduction in profitability.• Can be estimated from market surveys and focus groups.

– Backorder administrative cost per unit - Cb• Reflects additional work needed to take care of the backorder.

Planned Shortage Model –the Total Variable Cost Equation

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Planned Shortage Model –the Total Variable Cost Equation

• The Annual holding cost = Ch[T1T](Average inventory) =

Ch[T1T] (Q-S)2

• The Annual shortage cost = Cb(number of backorders per year) + Cs(T2T)(Average number of backorders).

• To calculate the annual holding cost and shortage cost we need to find– The proportion of time inventory is carried, (T1/T)– The proportion of time demand is backordered, (T2/T).

T1 T2

T

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S

Q - S

Q

T1 T2

S T

Average inventory = (Q - S) / 2

Average shortage = S / 2

Proportion of time inventory exists = T1T

T1

T

Q - S

Q

Proportion of time shortage exists

= T2T

Finding T1/ T and T2/ T

= (Q - S) / Q

= S / Q

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• Annual holding cost:Ch[T1T](Q-S)2 = Ch[(Q-S) Q](Q-S)2

= Ch(Q-S)22Q

• Annual shortage cost:Cb(Units in short per year) + Cs[T2T](Average number of backorders) = Cb(S)(DQ) + CsS2/2Q

Planned Shortage Model –The Total Variable Cost Equation

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– The total annual variable cost equation

– The optimal solution to this problem is obtained under the following conditions

• Cs > 0 ;

• Cb < \/ 2CoCh / D

TV(Q,S) = (Q -S)2

2Q Ch + DQ

(Co + SCb S2

2QCS

Holding costs

Time dependent backorder costs

Time independent backorder costs

Ordering costs

Planned Shortage Model –The Total Variable Cost Equation

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The Optimal Backorder level

S*= Q* Ch - DCb

Ch + Cs Reorder Point R = L D - S*

Planned Shortage Model –The Optimal Inventory Policy

The Optimal Order Size

Ch

(DCb)2

ChCs

2DCo Q* = Ch + Cs

Cs

x

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SCANLON PLUMBING CORPORATION

• Scanlon distributes a portable sauna from Sweden. • Data

– A sauna costs Scanlon $2400.– Annual holding cost per unit $525.– Fixed ordering cost $1250 (fairly high, due to costly transportation).– Lead time is 4 weeks.– Demand is 15 saunas per week on the average.

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• Scanlon estimates a $20 goodwill cost for each week a

customer who orders a sauna has to wait for delivery.

• Administrative backordrer cost is $10.

• Management wishes to know:

– The optimal order quantity.

– The optimal number of backorders.

– Backorder costs

SCANLON PLUMBING CORPORATION

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SCANLON PLUMBING – Solution

• Input for the total variable cost function – D = 780 saunas [(15)(52)]

– Co = $1,250

– Ch = $525

– Cs = $1,040

– Cb = $10

79

x(780)(10)2

(525)(1040)5252(780)(1250) 525+1040

1040Q* = 74

• The optimal policy

R = (4 / 52)(780) 20 = 40

_S*= (74)(525) (780)(10)

525 + 104020

SCANLON PLUMBING – Solution

80

SCANLON PLUMBING – Spreadsheet Solution

Calculation of Optimal Inventory Policy for a Planned Shortage ModelOPTIMAL ASSIGNED

INPUTS Values OUTPUTS Values OUTPUTS ValuesAnnual Demand, D = 780.00 Order Quantity, Q* = 74.01 Q = 74.00Per Unit Cost, C = 2400.00 Backorder Level, S* = 19.84 S = 20.00Annual Holding Cost Rate, H = 0.22 Cycle Time (in years), T = 0.0949 T = 0.0949Annual Holding Cost Per Unit, Ch =525.00 # of Cycles Per Year, N = 10.5388 N = 10.5405Order Cost, Co = 1250.00 Reorder Point, R = 40.1531 R = 39.9976Annual Backorder Cost, Cs = 1040.00 Total Annual Variable Cost, TV(Q*) =28438.24 TV(Q) = 28438.51Fixed Admin. Backorder Cost, Cb =10.00 Total Annual Cost, TC(Q*) = 1900438.24 TC(Q) = 1900438.51Lead Time (in years), L = 0.07692 % of Customers Backordered = 26.81 % Back. = 27.03

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8.7 Review Systems – Continuous Review

• (R, Q) Policies – The EOQ, production lot size, and planned shortage

models assume that • inventory levels are continuously monitored

• Items are sold one at a time.

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• (R, Q) Policies – The above models call for order point (R) order

quantity (Q) inventory policies.

– Such policies can be implemented by• A point-of-sale computerized system.• The two-bin system.

8.7 Review Systems – Continuous Review

83

• (R, M) policies

– When items are not necessarily sold one at a time, the

reorder point might be missed, and out of stock

situations might occur more frequently.

– The order to level (R, M) policy may be implemented in

this situation.

Continuous Review Systems

84

• (R,M) policies

– The R, M policy replenishes inventory up to a pre-

determined level M.

Continuous Review Systems

– Order Q = Q* + (R – I) = (M – SS) + (R – I) each

time the inventory falls to the reorder point R or below.

(Order size may vary from one cycle to another).

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• It may be difficult or impossible to adopt a continuous review system, because of:

– The high price of a computerized system.– Lack of space to adopt the two-bin system.– Operations inefficiency when ordering different items from

the same vendor separately.• The periodic review system may be found more suitable

for these situations.

Periodic Review Systems

86

• Under this system the inventory position for each item is observed periodically.

• Orders for different items can be better coordinated periodically.


87

– (T,M) Policies• In a replenishment cycle policy (T, M), the

inventory position is reviewed every T time units.• An order is placed to bring the inventory level back up

to a maximum inventory level M.• M is determined by

– Forecasting the number of units demanded during the review period T.

– Adding the desired safety stock to the forecasted demand.


88

T =Review periodL = Lead timeSS= Safety stockQ = Inventory positionD = Annual demandI = Inventory position


• Calculation of the replenishment level and order size

Q = M + LD – I

M = TD + SS

89

• Every three weeks AAC receives deliveries of different products from Citron.

• Lead time is eight days for ordering Citron’s juicers.

• AAC is now reviewing its juicer inventory and finds 210 in stock.

• How many juicers should AAC order for a safety stock of 30 juicers?

AAC operates a (T, M) policy

90

• Data– Review period T = 3 weeks = 3/52 = .05769 years,– Lead time = L = 8 days = 8/260 = .03077 years,– Demand D = 6240 juicers per year,– Safety stock SS = 30 juicers,– Inventory position I = 210 juicers

AAC operates a (T, M) policy – Solution

AAC operates 260 days a year.(5)(52) = 260.

91

• Review period demand = TD = ( 3/52)(6240) = 360 juicers,

• M = TD + SS = 360 + 30 = 390 juicers,

• Q = M + LD – I = 390 + .03077(6240) - 210 = 372 juicers.


92

Reviewpoint

Reviewpoint


T

SS SS SS

Inventory position

Order Order

Replenishment level

Inventory position

L

Notice: I + Q is designed to satisfy the demand within an interval of T + L. To obtain the replenishment level add SS to I + Q.

M = maximum inventory

L

93

• Demand is stochastic with a known distribution.

• Shelf life of the item is limited.

• Inventory is saleable only within a single time period.

• Inventory is delivered only once during a time period.

8.8 Single Period Inventory Model -Assumptions

• At the end of each period, unsold inventory is disposed of for some salvage.

• The salvage value is less than the cost per item.

• Unsatisfied demand may result in shortage costs.

94

• To find an optimal order quantity we need to balance the expected cost of over-ordering and under ordering.

Expected Profit = (Profit when Demand=X)Prob(Demand=X) x

• The expected profit is a function of the order size, the random

demand, and the various costs.

The Expected Profit Function

95

– Developing an expression for EP(Q)• Notation

p = per unit selling price of the good.c = per unit cost of the good.s = per unit salvage value of unsold good.K = fixed purchasing costsQ = order quantity.EP(Q) = Expected Profit if Q units are ordered.

• Scenarios – Demand X is less than the order quantity (X < Q).– Demand X is greater than or equal to the order quantity (X Q.


96

• Scenario 1: Demand X is less than the units stocked, Q.

• Scenario 2: Demand X is greater than or equal to the units stocked.

Profit = pX + s(Q - X) - cQ - K

Profit = pQ - g(X - Q) - cQ - K

EP(Q) = [pX+s(Q - X) - cQ - K]P(X) + [pQ - g(X - Q) - cQ - K]P(X)X Q

X Q


97

– To maximize the expected profit order Q*

• For the discrete demand case take the smallest value of Q*

that satisfies the condition

P(D Q*) (p - c + g)(p - s + g)

• For the continuous demand case find the Q* that solves

F(Q*) = (p - c + g) (p - s + g)

The Optimal Solution

98

THE SENTINEL NEWSPAPER• Management at Sentinel wishes to know how

many newspapers to put in a new vending machine.

• Data– Unit selling price is $0.30– Unit production cost is $0.38.– Advertising revenue is $0.18 per newspaper.– Unsold newspaper can be recycled and net $0.01.– Unsatisfied demand costs $0.10 per newspaper.– Filling a vending machine costs $1.20.

Demand distribution isdiscrete uniform between 30 and 49 newspapers.

99

SENTINEL - Solution

• Input to the optimal order quantity formula

p = 0.30c = 0.20 [0.38-0.18]s = 0.01g = 0.10K = 1.20

The probability of the optimal service level = p+ g - cp+ g - s

0.30 + 0.10 - 0.200.30 + 0.10 - 0.01

= 0.513=

100

1.0

0.500.55

30 49

0.513

39 40

P(D 39) = 0.50P(D 40) = 0.55

Q* = 40

SENTINEL – SolutionFinding the optimal order quantity Q*

101

=(B5+B8-B6)/(B5+B8-B7)

=ROUNDUP(B10+E5*(B11-10),0)

=(E6-B10+1)/(B11-B10+1)

SENTINEL – Spreadsheet Solution

102

WENDELL’S BAKERY• Management in Wendell’s wishes to determine

the number of donuts to prepare for sale, on weekday evenings

• Data– Unit cost is $0.15.– Unit selling price is $0.35.– Unsold donuts are donated to charity for a tax credit

of $0.05 per donut.– Customer goodwill cost is $0.25.– Operating costs are $15 per evening.

Demand is normally distributedwith a mean of 120, and a standard deviation of 20 donuts.

103

WENDELL’S BAKERY - Solution

• Input to the optimal order quantity formulap = $0.35c = $0.15s = $0.05g = $0.25K = $15.00

The optimal service level = p+ g - cp+ g - s

0.35+ 0.25 - 0.150.35+0.25 - 0.05

= 0.8182=

104

.8182

=120 Q*

• From the relationship F(Q*) = 0.8182 we find the corresponding z value.

• From the standard normal table we have z = 0.3186.• The optimal order quantity is calculated by

Q* = + z

• For Wendell’s Q* = 120 + (0.3186)(20) 138

WENDELL’S BAKERY - SolutionFinding the optimal order quantity

105

EP(Q*) = (p - s) - (c - s)Q* - (p + g - s) ()L[(Q* - ) / - K

• For the normal distribution

L [(Q* - ) / is obtained from the partial expected value table. • For Wendell’s

EP(138) = (0.35 - 0.05)(120) - (0.15 - 0.05)(138) - (0.35 + 0.25 - 0.05)x(20)L[(138 - 120) / 20] - 15 = $6.10

L(0.9) = 0.1004

WENDELL’S BAKERY - SolutionCalculating the expected profit

106

=(B5-B7)*B10-(B6-B7)*E6-(B5+B8-B7)*B11*(EXP(-(((E6-B10)/B11)^2)/2)/((2*PI())^0.5)-((E6-B10)/B11)*(1-

NORMSDIST((E6-B10)/B11)))-B9

=NORMINV(E5,B10,B11) =(B5+B8-B6)/

(B5+B8-B7)

WENDELL’S BAKERY -

Spreadsheet Solution

107

WENDELL’S – The commission strategy

• When commission replaces fixed wages…– Compare the maximum expected profit of two strategies:

• $0.13 commission paid per donut sold,

• $15 fixed wage per evening (calculated before).

– Calculate first the optimal quantity for the alternative policy.

– Check the expected difference in pay for the operator.

108

WENDELL’S – The commission strategy - Solution

• The unit selling price changes to

c = 0.35 - 0.13 = $0.22

• The optimal order:F(Q*) = (0.22 + 0.25 - 0.15) / (0.22 + 0.25 - 0.05)= 0.7616.

• Z = .71

Q* = + z = 120 + (0.71)(20) 134 donuts.

109

• Will the bakery’s expected profit increase?EP(134) = (0.22 - 0.05)(20) - (0.15 - 0.05)(134) - (0.22 + 0.25 - 0.05)x(20)L[(134 - 120) / 20] = $5.80 < 6.10

• The bakery should not proceed with the alternative plan.


110

• Comments– The operator expected compensation will increase,

but not as much as the bakery’s expected loss.– An increase in the mean sales is probable when the

commission compensation plan is implemented. This may change the analysis results.


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Ch08 - Inventory