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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 *+ 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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2 Outline Introduction –Motivation Montgomerys (1990) Injection Molding Experiment Research Proposal Current Research Results

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3 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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4 Montgomerys (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

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5 Injection Molding Experiment Data i j A1A1 B2B2 C3C3 D4D4 AB 5 AC 6 CG 7 AE 8 BD 9 AG 10 E 11 ABD 12 G 13 F 14 AF 15 Y 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 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 6 10 32 60 4 15 26 60 8 12 34 60 16 5 37 52

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6 Probability Plot of Effects

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7 Two possible location models: Montgomerys Model (M1) = 27.3125 + 6.9375A + 17.8125B + 5.9375AB McGrath and Lins model (M2) = 27.3125 + 6.9375A + 17.8125B + 5.9375AB –2.6875CG – 2.4375G Plausible Location Models

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8 Dispersion Effect Analysis Box and Meyer Dispersion Effect Statistics Dispersion effect

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9 Conclusion Montgomery's Model (M1) = 27.3125 + 6.9375A + 17.8125B + 5.9375AB (dispersion effect in C) McGrath and Lins Model (M2) = 27.3125 + 6.9375A + 17.8125B + 5.9375AB –2.6875CG – 2.4375G (no dispersion effects [d.e.])

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10 Minimum Number of Trials Montgomerys (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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11 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 = 8 26, 27 R = 11 A B Recommended runs for replication A BC G - + - - + +

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12 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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13 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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14 Bayesian Method of Finding Active Factors Cont. Interpretation of the Posterior Probability The first one can be regarded as a penalty for increasing the number of variables in the model M i. The second component is nothing less than a measure of fit

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15 Finding Active Factors – Injection Molding Experiment Marginal Posterior Probabilities – P j Considering non-structured models Considering structured models

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16 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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17 MD Criterion Cont. MD in the context of DOE: Remark: Must have constant variance for all models considered!

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18 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, Outlines of the Transformation Procedure

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19 d (+) Rearranged Covariance Matrix of Y Symmetric } d (-) }

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20 Transformation – Injection Molding Experiment Montgomerys (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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21 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

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22 MD Criterion – Injection Molding MD criterion and the design points MDDesign Points 14.0189481216 12.297515913 10.6127261014 9.1064371115 32, 34 R = 2 4, 16 R = 12 60, 60 R = 0 6, 8 R = 210, 12 R = 2 15, 5 R = 10 60, 52 R = 8 26, 27 R = 11 A C B Recommended runs for replication A BC G + - + - - + + Remark: McGraths suggestion, (1, 5, 9, 13), was the second-best discriminated follow-up design!

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23 References Bergman, B. and Hynén, A. (1997). Dispersion Effects from Unreplicated Designs in the 2 k-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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24 Thank you for your time!

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