1 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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1 Follow-up Experiments to Remove Confounding Between Location and Dispersion Effects in Unreplicated Two- Level Factorial Designs André L. S. de Pinho.
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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
– 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
• 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
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.