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Follow-up Experiments to Remove Confounding Between Location and

Dispersion Effects in Unreplicated Two-Level Factorial Designs

André L. S. de Pinho*+

Harold J. Steudel*

Søren Bisgaard#

*Department of Industrial Engineering - University of Wisconsin-Madison+Department of Statistics - Federal University of Rio Grande do Norte (UFRN) - Brazil#Eugene M. Isenberg School of Management - University of Massachusetts, Amherst

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Outline

• Introduction– Motivation

• Montgomery’s (1990) Injection Molding Experiment

• Research Proposal

• Current Research Results

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Introduction• Motivation

– Ferocious competition in the market

– High pressure for lowering cost, shortening time-to-market and increase reliability

– Need to have faster, better and cheaper processes

• Current trend: Design for Six Sigma (DSS)• Approach: Robust product design

– Making products robust to process variability

– DOE provides the means to achieve this goal

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Montgomery’s (1990) Injection Molding Experiment

• fractional factorial design plus four center points with the objective of reducing the average parts shrinkage and also reducing the variability in shrinkage from run to run.

• The factors studied– mold temperature (A), screw speed (B), holding time

(C), gate size (D), cycle time (E), moisture content (F), and holding pressure (G).

• The generators of the design were E = ABC, F = BCD, and G = ACD

372 IV

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Injection Molding Experiment Datai ↓ j →

A

1

B

2

C

3

D

4

AB

5

AC

6

CG

7

AE

8

BD

9

AG

10

E

11

ABD

12

G

13

F

14

AF

15Y

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

-1

1

-1

1

-1

1

-1

1

-1

1

-1

1

-1

1

-1

1

-1

-1

1

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1

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1

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-1

1

1

1

1

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1

1

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1

1

1

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1

6

10

32

60

4

15

26

60

8

12

34

60

16

5

37

52

6

0 10 20 30

-1

0

1

Effect

Nor

mal

Sco

re

B-2

A-1

AB-5

G-13

CG-7

Probability Plot of Effects

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Two possible location models:

Montgomery’s Model

(M1) = 27.3125 + 6.9375A + 17.8125B + 5.9375AB

McGrath and Lin’s model

(M2) = 27.3125 + 6.9375A + 17.8125B + 5.9375AB –2.6875CG – 2.4375G

Plausible Location Models

y

y

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-1 0 1 2

Box & Meyer Statistics

Model

M1

M2

C

Dispersion Effect Analysis

2

2

i

iBMj s

sLogD

ContrastsModel A B C D E F G AB AC CG AE

M1 -0.3804 -0.18748 2.50254 0.51262 -0.03627 -0.30452 0.22873 0.10663 -0.41301 0.41897 -0.23544M2 0.39864 0.34435 -0.06419 -0.1887 0.36335 -1.29224 -1.1744 1.35584 -0.01814 -0.3511 0.35699

Box and Meyer Dispersion Effect Statistics

Dispersion effect

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Conclusion

Montgomery's Model(M1) = 27.3125 + 6.9375A + 17.8125B + 5.9375AB

(dispersion effect in C)

McGrath and Lin’s Model(M2) = 27.3125 + 6.9375A + 17.8125B + 5.9375AB

–2.6875CG – 2.4375G

(no dispersion effects [d.e.])

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Minimum Number of Trials

Montgomery’s (1990) injection molding

• Addressed by McGrath (2001), 4 extra runs

• The selection is done in such a way that A and B are fixed and each combination of the settings for columns 7 and 13 occurs

• There are four sets of rows, (1, 5, 9, 13), (2, 6, 10, 14), (3, 7, 11, 15), and (4, 8, 12, 16). He selected (1, 5, 9, 13)

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C

Minimum Number of Trials Graphical Representation

32, 34 R = 2

4, 16 R = 12

60, 60 R = 0

6, 8 R = 2 10, 12 R = 2

15, 5 R = 10

60, 52 R = 826, 27 R = 11

A

BRecommended runs for replication

A B C G

- -- -- -- -

- -+ -- ++ +

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Research Proposal• Expanding Meyer, Steinberg and Box (1996) to

accommodate the presence of d.e. in the models

3 - Sequential design method for discrimination among concurrent models

[Box and Hill (1967)]

1 - Bayesian method of finding active factors

in fractionated screening experiments

[Box and Meyer (1993)]

2 - Apply a suitable transformation to ensure constant variance

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1 - Bayesian Method of Finding Active Factors

Scenario• Fractionated Factorial Designs • Sparsity Principle Underlines the Process

Being Studied• Allow the Inclusion of Non-Structured

Models

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Bayesian Method of Finding Active Factors Cont.

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0

'

21'

21

0'0

ˆ

ˆˆˆ

1

n

iiii

iii

tf

iS

S

XX

XXCYMp i

i

Interpretation of the Posterior Probability

The first one can be regarded as a penalty for increasing the number of variables in the model Mi.

The second component is nothing less than a measure of fit

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0.00

0.20

0.40

0.60

0.80

1.00

FactorPj

Pj 0.000 0.998 0.999 0.299 0.012 0.002 0.001 0.305

NONE A B C D E F G

Finding Active Factors – Injection Molding Experiment

0.00

0.20

0.40

0.60

0.80

1.00

Factor

Pj

Pj 0.000 0.997 0.999 0.003 0.018 0.002 0.002 0.010

NONE A B C D E F G

Marginal Posterior Probabilities – Pj

Considering non-structured modelsConsidering structured models

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Model Discrimination (MD) Criterion Overview

Two Possible Models(M1) and (M2) to describe a Response

X

Response

M2

M1

Two Rival Models

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MD Criterion Cont.

, 1 0

**1****1**

mji

ijijjijiji SYYVYYnVVtrnYMPYMPMD

MD in the context of DOE:

iiiiiiii XYXYSwhere

''

Remark: Must have constant variance for all models considered!

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• Use WLS of the expanded location model is in the sense of the Bergman and Hynén (1997) method of identifying dispersion effects

• Once we have available the residuals from the expanded location model we can then calculate the ratio,

dddddd ˆ*ˆˆˆˆˆ 2222

Outlines of the Transformation Procedure

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d (+)

Rearranged Covariance Matrix of Y

1616

2

ˆ0ˆ00ˆ000ˆ0000ˆ00000ˆ000000ˆ0000000ˆ000000001

0000000001

00000000001

000000000001

0000000000001

00000000000001

000000000000001

0000000000000001

d

d

d

d

d

d

d

dd

Symmetric

}d (-)

}

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Transformation – Injection Molding Experiment

Montgomery’s (1990) Injection Molding Experiment

(M1) = 27.3125 + 6.9375A + 17.8125B + 5.9375AB. (d.e. in C)

(M2) = 27.3125 + 6.9375A + 17.8125B + 5.9375AB –2.6875CG – 2.4375G. (no d.e.)

• The minimum number of trials to resolve the confounding problem is four• The possible sets of four runs that can be used for the follow-up experiment are (1,

5, 9, 13), (2, 6, 10, 14), (3, 7, 11, 15), and (4, 8, 12, 16)• McGrath then suggested (1, 5, 9, 13) for replication because it is near the optimum

condition.

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Finding the Expanded Model• The set of active location effects is L = {I, A,

B, AB}• The set of dispersion effect is D = {C}• M1-expanded model is represented by the set

= {I, A, B, C, AB, AC, BC, ABC}• (M1-expanded)

= 27.3125 + 6.9375A + 17.8125B – 0.4375C + 5.9375AB – 0.8125AC –

0.9375BC + 0.1875ABC • The estimated weight is = 0.167

y

d1

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MD Criterion – Injection Molding

MD criterion and the design points

MD Design Points

14.0189 4 8 12 16

12.2975 1 5 9 13

10.6127 2 6 10 14

9.1064 3 7 11 15

32, 34 R = 2

4, 16 R = 12

60, 60 R = 0

6, 8 R = 2 10, 12 R = 2

15, 5 R = 10

60, 52 R = 826, 27 R = 11

A

C

B

Recommendedruns for replication

A B C G

+ ++ ++ ++ +

- -+ -- ++ +

Remark: McGrath’s suggestion, (1, 5, 9, 13), was the second-bestdiscriminated follow-up design!

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References• Bergman, B. and Hynén, A. (1997). “Dispersion Effects from

Unreplicated Designs in the 2k-p Series”, Technometrics, 39, 2, 191-198.• Box, G. E. P. and Hill, W. J. (1967). “Discrimination Among

Mechanistic Models”, Technometrics, 9, 1, 57-71.• Box, G. E. P. and Meyer, R. D. (1993). “Finding the Active Factors in

Fractionated Screening Experiments”, Journal of Quality Technology, 25, 2, 94-105.

• McGrath, R. N. (2001). “Unreplicated Fractional Factorials: Two Location Effects or One Dispersion Effect?”, Joint Statistical Meetings (JSM) in Atlanta.

• Meyer, R. D., Steinberg, D. M., and Box, G. E. P. (1996). “Follow-up Designs to Resolve Confounding in Multifactor Experiments”, Technometrics, 38, 4, 303-313.

• Montgomery, D. C. (1990). “Using Fractional Factorial Designs for Robust Process Development”, Quality Engineering, 3, 2, 193-205.

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