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An Overview of

Designed Experiments

By: Larry A. ScottPrincipal: Process Technologies

Welcome to:

DOE Simplified:

Agenda

Intro to DOE:

Full Two-Level Factorials

Design: Eye-Hand Exercise

Analysis:

Interactions The Hidden Gold

Example: Engine PerformanceOverall Strategy Of DOE

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ProcessProcess

Controllable Factors (X)Controllable Factors (X)

Responses (Y)Responses (Y)

Uncontrollable Variables (Z)Uncontrollable Variables (Z)

DOE Works onDOE Works on AnyAny ProcessProcess

DOE is:

A series of tests,

in which purposeful changes

are made to input factors,

so that you may identify causes

for significant changes

in the output responses.

Design Terminology

OFAT vs. DOE (factorials)

Factor Effects

Main Effects

Interactions (effects)

Balanced Test Matrix

Array Codes

Prediction Equations (Y = mX + b)Y = b

0+ b

1X

1+ b

2X

2+ b

12X

1X

2

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Full TwoLevel Factorial Design

Run high/low combos of two or more factors

Use statistics to identify the critical few

Main effects

Interactions (the hidden gold!)

What could be simpler?

Exercise: Eye-Hand Coordination

Your mission: In a 10 second span, markas many dots in two circles as you can.You must alternate between circles anduse only one hand.

Response: Number of dots in both circles.

1 inch

2 inch

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Eye-Hand Coordination

Factors:

A. Which hand holds the pen: Non-Dominant hand (ND)+ Dominant (D)

B. Size of circle: Small+ Large

Eye-Hand CoordinationFactor Space and Test Matrix

AHand

BSize

AB

1 + ___

2 + ___

3 + ___

4 + + + ___

Y

3 4

1 2

A

B

+

-

- +ND D

Large

Small

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Eye-Hand CoordinationTemplates

1. Small, Dominant: ___

3. Small, Non-Dominant: ___

2. Large, Dominant: ___

4. Large, Non-Dominant: ___

Eye-Hand CoordinationProcedure for Doing Exercise

1. Write the numbers 1 through 4 on fourslips of paper. Put these in one handand blindly pull out at random.

2. Following procedure in your notes,perform tests in random run order.

Record data (Y) in blank at bottom oftemplates. Chart on following graph.

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Eye-Hand Exercise Outline

Reference:

DOE Simplified Practical Tools for Effective Experimentationby Mark J. Anderson and Patrick J. Whitcomb, Productivity, Inc., Portland, OR(2000).*

Eye-Hand CoordinationInteraction Graph

20

15

10

5

Non-Dominant

Hand

Dominant

The Y-axis showsthe number ofdots (cycles).Plot the data forD vs. ND for thesmall circles,connect with aline and label.

Then do same forthe large circles.

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Eye-Hand CoordinationInteraction Graph

Using statistical software, lets see howwell the Stat-Ease programming staff didon this exercise and compare results.

Eye-Hand CoordinationInteraction Graph (published*)

B: Circle Size

Bullseyes

A: Hand

Non-dominant Dominant

15

20

25

30

35

40

45

Small

Large

In this case, there

was a significant

interaction: the

effect of switching

hands became

more pronounced

with large circles.

"#$ $ %&' ( ) * ) + ) ,-.$/ 00$ *1$ *2$ ) ,

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Real-World Example: OFAT MissesBreakthrough Interaction!

Life

130

B

A+A

90

50

10

B+

Eye-Hand CoordinationFactor Space and Test Matrix

3 4

1 2

A

B

+

-

- +ND D

Large

Small

AHand

BSize

AB

1 + ___

2 + ___

3 + ___

4 + + + ___

Y

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(4 4 5 66666666666666

(4 5 66666666666666

(4 5 66666666666666

(4 5 66666666666666

(4 5 66666666666666

7 4 8 5 66666666666666

Relative Efficiency of Factorial vsOne Factor at a Time (OFAT)

Advantages of DOE vs. OFAT

9 .

9 ) : .8

9

9 2 ,

9 8 1 4 , )

9 .; 1 , )9

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Relative Efficiency of Factorial vs.One Factor at a Time (OFAT)

Relative efficiency =Relative efficiency = 1616//8 = 28 = 2

A- A+B-

B+

C-

C+

17

19

26

16

25

21

85

128

Case Study: 23 Factorial Design

Background: An engineer wants to study the effect

of three factors on mileage performance of an engine.

3010Spark Angle (deg)

528Engine Torque (lb/ft)

50002000Engine Speed (rpm)

High (+)Low ()Factor

! " #$$%&

' ( ! ) * ! ( +

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23 Full Factorial Array

101025.83032500083249.1303220007

21912.530850006

273.730820005

52829.0103250004

2019.8103220003

7715.510850002

134.510820001

Y2Y1CBAStdOrder

Responses:

Y1 = fuel flow (lb/hr)

Y2 = NOx (g/hr)

Fuel Flow:All Effects Calculated

13.73-2.13-0.03-1.184.02-1.929.38Effect25.8+++++++8

+

+

+

AB

+

+

+

AC

+

+

+

BC

+

+

+

ABC

9.1++7

29++4

4.51

12.5++6

3.7+5

9.8+3

15.5+2

Y1CBAStd

Fill in effect of A. Which, if any, effects are significant?

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Fuel Flow (Y1):Calculating Effect of A (Engine Speed)

B

A

C

EffectY

n

Y

n====

++++

++++

General formula:

Aeffect ====++++ ++++ ++++

++++ ++++ ++++====

15.5 29 12.5 25.8

4

4.5 9.8 3.7 9.1

413.4

9.1 25.8

299.8

4.5 15.5

3.7 12.5

Engine speed

+

Half-Normal Plot:

Identify the Big Effects

Effect

0.00 3.48 6.96 10.44 13.93

0

20

40

6070

808590

9597

99

A

B

AB

Effect

0.00 0.24 0.47 0.71 0.94

0

20

40

70

8085

90

97

99

A

B

C

AB

=

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Sparsity of Effects Principle

Trivial Many: the remainder that result from random variation.

These effects will be centered on zero.

Since they are based on averages,you can assume normality

by the Central Limit Theorem*.

Two types of effects:

Vital Few: the big ones we want to catch

20 % of ME's and 2FI's will be significant.

Half-Normal Probability Paper

Significant effects(the vital few) fallabnormally high (tothe right) on theabsolute effectscale. These arethe keepers.

What do you do withthe little ones?

7.14

21.4335.71

50.00

64.29

78.57

92.86

Pi

0

|Effect|

A

B

AB

3 6 9 12 15

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Interactions The Hidden Gold

The effect of speeddepends on the levelof torque. This is an

interaction.

Can you explainwhats happening tothe fuel flow?

B - B +

30

24

6

0

Fuel flow

torque

Lowsp

eed

High

Spe

ed

12

18

Analysis of Variance Report(ANOVA)

! "# $

* 000 03 !3= =0 + =3 !

2 !>. ) 0?. +@ 0

>? =

%&%%%'

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New!Minimum-Run Resolution IV Designs*

The minimum number of runs forresolution IV design is only two times thenumber of factors (runs = 2k). This can offerquite a savings when compared to a regularresolution IV 2k-p fraction.

!" #" $% &% '()*+

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Minimum-Run Resolution IV Designs

506425303215486424283214

466423263213

446422243212

426421223211

406420203210

38641918329

36641816*168

34641714167

32*321612166

Min2k-pkMin2k-pk

506425303215486424283214

466423263213

446422243212

426421223211

406420203210

38641918329

36641816*168

34641714167

32*321612166

Min2k-pkMin2k-pk

, - ./

RSM: When to Apply It

Region of Operability

Region of InterestUse factorial

design to get close

to the peak. Then

RSM to climb it.

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RSM vs OFATOFAT

-2 -1 0 1 2

30

45

60

75

90

Factor A

Response

-2 -1 0 1 2

60

65

70

75

80

85

90

Factor B

Response

Response

65

73

80

88

95

Response

-4-2

02

4

-4

-2

0

2

4

Factor A

Factor B

Response

Advantages of DOE vs. OFAT

9 .

9 ) : .8

9

9 2 ,

9 8 1 4 , )

9 .; 1 , )9