Practical Design of Experiments (DOE)

Practical Design of Practical Experime Exper iments nts (DOE) ( DOE)

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Practical Design of Experiments (DOE) A Guide for Optimizing Designs and Processes

Mark Allen Durivage

ASQ Quality Press Milwaukee, Wisconsin

American Society for Quality, Quality Press, Milwaukee 53203 © 2016 by ASQ All rights reserved. Published 2016 Printed in the United States of America 22 21 20 19 18 17 16 5 4 3 2 1 Library of Congress Cataloging-in-Publication Data

Names: Durivage, Mark Allen. Title: Practical design of experiments (DOE) : a guide for optimizing designs and processes / Mark Allen Durivage. Description: Milwaukee, Wisconsin : ASQ Quality Press, 2016. | Includes bibliographical references and index. Identifiers: LCCN 2015049294 | ISBN 9780873899246 (hard cover : alk. paper) Subjects: LCSH: Engineering—Statistical methods. | Acceptance sampling. | Quality control—Statistical methods. | Distribution (Probability theory) | Science—Methodology. Classification: LCC TA340 .D868 2016 | DDC 001.4/34—dc23 LC record available at http://lccn.loc.gov/2015049294 ISBN: 978-0-87389-924-6 No part of this book may be reproduced in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written p ermission of the publisher. Publisher: Lynelle Korte Acquisitions Editor: Matt T. Meinholz Project Editor: Paul Daniel O’Mara Production Administrator: Randall Benson ASQ Mission: The American Society for Quality advances individual, organizational, and community excellence worldwide through learning, quality improvement, and knowledge exchange. Attention Bookstores, Wholesalers, Schools, and Corporations: ASQ Quality Press books, video, audio, and software are available at quantity discounts with bul k purchases for business, educational, or instructional use. For information, please contact ASQ Qual ity Press at 800-248-1946, or write to ASQ Quality Press, P.O. Box 3005, Milwaukee, WI 53201-3005. To place orders or to request ASQ membership information, call 800-248-1946. Visit our website at http://www.asq.org/quality-press. Printed on acid-free paper

Table of Contents

List of Figures and Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Limit of Liability /Disclaimer of Warranty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 1

ix xiii xv xvi

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Chapter 2 Statistical Tools and Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Dean and Dixon Outlier Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Hypothesis Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Type I and Type II Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alpha (`) and Beta ( a) Risks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Apportionment of Risk in Hypothesis Testing . . . . . . . . . . . . . . . . . . . . The Hypothesis Test for a One-Tail (Upper-Tailed) Test . . . . . . . . . . . . . The Hypothesis Test for a One-Tail (Lower-Tailed) Test . . . . . . . . . . . . . The Hypothesis Test for a Two-Tail Test . . . . . . . . . . . . . . . . . . . . . . . . . The Hypothesis Test Conclusion Statements . . . . . . . . . . . . . . . . . . . . . . Testing for a Difference between Two Observed Variances Using Sample Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Normal Probability Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Half-Normal Probability Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Interpreting Effect and Interaction Plots . . . . . . . . . . . . . . . . . . . . . . . . .

7 7 8 8 8 9 9 10 11 11

11 13 15 16

Chapter 3 ANOVA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 One-Way ANOVA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Two-Way ANOVA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19 19 22

Chapter 4 Experiments with Two Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Bond Strength Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nine Steps for Analysis of Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Nonlinear Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Corrosion Study Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nine Steps for Analysis of Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27 27 29 38 40 41

Chapter 5 Experiments with Three Factors . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Chemical Processing Yield Example . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47 47

v

vi

Table of Contents

Nine Steps for Analysis of Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Variation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Analysis with Unreplicated Experiments (Residual Analysis) . . . . . . . . .

47 55 60

Chapter 6 Experiments with Qualitative (Attribute Data) Responses . . . . 6.1 Plastic Welding Example (without Transformation) . . . . . . . . . . . . . . . . . Nine Steps for Analysis of Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Plastic Welding Example (with Transformation) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nine Steps for Analysis of Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Ordered Categorical Data Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nine Steps for Analysis of Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65 65 66

Chapter 7 Screening and Other Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Confounding, Aliases, and Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Screening Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Other Analytical Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Even Larger Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Other Types of Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87 88 91 92 94 94 94

Chapter 8 Taguchi Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Taguchi Orthogonal Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Signal-to-Noise (S/N) Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Taguchi L4 Orthogonal Array Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nine Steps for Analysis of Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Taguchi L8 Orthogonal Array Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nine Steps for Analysis of Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Taguchi L9 Orthogonal Array Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nine Steps for Analysis of Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97 97 101

71 72 78 81

103 104 108 110 116 118

Chapter 9 Mixture Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Three-Factor Simplex Design Example . . . . . . . . . . . . . . . . . . . . . . . . . .

123 125

Chapter 10 Procedural Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Common Problems and Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Review of the Basics in Managing a DOE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Obstacles to the Application of DOE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 DOE Spreadsheets and Software Considerations . . . . . . . . . . . . . . . . . .

129 129

Chapter 11

135

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

131 132 133

Table of Contents

vii

Appendix A

Critical Values of the Dean and Dixon Outlier Test . . . . . . . .

137

Appendix B

Percentages of the F-Distribution . . . . . . . . . . . . . . . . . . . . . . .

139

Appendix C

Percentage Points of the Student’s t-Distribution . . . . . . . . . .

151

Appendix D

Cumulative Percentage Points . . . . . . . . . . . . . . . . . . . . . . . . . .

153

Appendix E

z-Scores

of the Cumulative Percentage Points . . . . . . . . . . . . .

155

Appendix F

Normal Distribution Probability Points—Area below Z . . . .

157

Appendix G

Normal Distribution Probability Points—Area above Z . . . .

159

Appendix H

Selected Full and Fractional Factorial Designs . . . . . . . . . . . .

161

Appendix I

Selected Plackett-Burman Screening Designs . . . . . . . . . . . . .

165

Appendix J

Selected Taguchi Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

167

Appendix K

Selected Mixture Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

171

Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

173 181 183

(This page intentionally left blank)

List of Figures and Tables

Figure 1.1

Cause-and-effect diagram depicting inputs ( X ’s) and outputs (Y ’s). . . . . . .

2

Figure 1.2

Relationship between statistical control limits and product specifications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

Figure 1.3

Nine steps for analysis of effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

Table 2.1

Hypothesis truth table. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

Figure 2.1

Representation of a one-tail (upper-tailed) test. . . . . . . . . . . . . . . . . . . . . . .

10

Figure 2.2

Representation of a one-tail (lower-tailed) test. . . . . . . . . . . . . . . . . . . . . . .

10

Figure 2.3

Representation of a two-tail test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

Figure 2.4

Right-skewed distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

Figure 2.5

Left-skewed distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

Figure 2.6

Short-tailed distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

Figure 2.7

Long-tailed distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

Table 2.2

Calculation summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

Figure 2.8

Normal probability plot for strength. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

Table 2.3

Calculation summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

Figure 2.9

Half-normal probability plot for effects. . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

Figure 2.10 Factor effect plots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

Figure 2.11 Factor interaction plots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

Table 3.1

One-way ANOVA summary table. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

Table 3.2

One-way ANOVA summary data table. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

Figure 3.1

Decision limit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

Table 3. 3

Two-way ANOVA summary table. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

Table 3.4

Two-way ANOVA summary data table. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

Figure 3.2

Decision limit for rows (pressure). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

Figure 3.3

Decision limit for columns (dwell time). . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

Figure 3.4

Decision limit interaction (pressure and dwell time). . . . . . . . . . . . . . . . . .

26

Table 4.1

Bond strength example data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

Figure 4.1

Plot of effect A (temperature) on bond strength. . . . . . . . . . . . . . . . . . . . . .

30

Figure 4.2

Plot of effect B (vendor) on bond strength. . . . . . . . . . . . . . . . . . . . . . . . . .

31

ix

x


Figure 4.3

Plot of interaction AB (temperature-vendor). . . . . . . . . . . . . . . . . . . . . . . .

32

Figure 4.4

Pareto chart of the absolute values of the effects. . . . . . . . . . . . . . . . . . . . .

32

Figure 4.5

Decision limits for the effects and interactions. . . . . . . . . . . . . . . . . . . . . . .

34

Figure 4.6

Temperature scale range. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

Table 4.2

Corrosion study example data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

Figure 4.7


41

Figure 4.8

Decision limits for the effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

Figure 4.9

Plot of effect A (chrome) on weight loss. . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

Figure 4.10 Plot of effect B (nickel) on weight loss. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

Figure 4.11 Plot of interaction AB (chrome-nickel) on weight loss. . . . . . . . . . . . . . . . .

44

Figure 4.12 Decision limits for linearity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

Table 5.1

Chemical processing yield data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

Figure 5.1


50

Figure 5.2

Decision limit for the effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

Figure 5.3

Half-normal plot of the absolute effects. . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

Figure 5.4

Plot of effect A (temperature) on yield. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

Figure 5.5

Plot of effect B (catalyst) on yield. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

Figure 5.6

Plot of effect C (ramp time) on yield. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

Figure 5.7

Plot of interaction AB (temperature-catalyst) on yield. . . . . . . . . . . . . . . . .

53

Table 5.2

Chemical process yield example with variances. . . . . . . . . . . . . . . . . . . . . .

56

Figure 5.8

Decision limit for variances. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

Table 5.3

Chemical processing yield data (unreplicated). . . . . . . . . . . . . . . . . . . . . . .

61

Figure 5.9

Normal plot of residuals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63

Figure 5.10 Decision limit for variances. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

64

Table 6.1

Plastic welding without transformation example data. . . . . . . . . . . . . . . . .

67

Figure 6.1


68

Figure 6.2


68

Figure 6.3

Effects plot for effect A (time). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

Figure 6.4

Effects plot for effect B (temperature). . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

Figure 6.5

Effects plot for effect C (pressure). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

70

Table 6.2

Plastic welding with transformation example data. . . . . . . . . . . . . . . . . . . .

73

Figure 6.6


74

Figure 6.7


75

Figure 6.8

Plot of effect A (time) on transformed defects. . . . . . . . . . . . . . . . . . . . . . .

75

Figure 6.9

Plot of effect B (temperature) on transformed defects. . . . . . . . . . . . . . . . .

76

Figure 6.10 Plot of effect C (pressure) on transformed defects. . . . . . . . . . . . . . . . . . . .

76

Figure 6.11 Plot of interaction AB (time-temperature) on transformed defects. . . . . . .

77

Table 6.3

78

Quality characteristic scoring scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . .


xi

Table 6.4

Ordered categorical data example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

80

Table 6.5

Observations converted to probabilities. . . . . . . . . . . . . . . . . . . . . . . . . . . .

80

Table 6.6

Table of effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82

Figure 6.12 Pareto chart of the absolute values of the effects. . . . . . . . . . . . . . . . . . . . .

83

Figure 6.13 Half-normal plot of the absolute effects. . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

Figure 6.14 Plot of effect A (time). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

Figure 6.15 Plot of effect B (temperature). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

Figure 6.16 Plot of effect C (pressure). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85

Table 7.1

Comparison of the number of runs in factorial and screening designs. . . .

87

Table 7.2

Analysis table for three factors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

88

Table 7.3

Illustration of identical interactions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

88

Table 7.4

Aliases for a half-fractional factorial design with four factors. . . . . . . . . . .

89

Table 7.5

Experiment resolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

90

Table 7.6

Summary of effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93

Figure 8.1

Accuracy versus precision. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

98

Figure 8.2

Adjusting the process to the target value. . . . . . . . . . . . . . . . . . . . . . . . . . .

98

Figure 8.3

Taguchi’s view of a process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99

Figure 8.4

Taguchi L4 orthogonal array and interaction table. . . . . . . . . . . . . . . . . . . .

99

Figure 8.5

Determining the L4 interaction between factors 1 and 2. . . . . . . . . . . . . . .

100

Figure 8.6

Determining the L4 interaction between factors 2 and 3. . . . . . . . . . . . . . .

100

Figure 8.7

L4 array displaying the interaction column of factors 1 and 2. . . . . . . . . . .

100

Figure 8.8

Taguchi L4 orthogonal array using only two factors that does not require column 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

100

Table 8.1

Plastic sealing example data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

103

Figure 8.9


105

Figure 8.10 Plot of effect 1 (time) on opening force. . . . . . . . . . . . . . . . . . . . . . . . . . . .

105

Figure 8.11 Plot of effect 2 (temperature) on opening force. . . . . . . . . . . . . . . . . . . . . .

106

Figure 8.12 Plot of effect 3 (pressure) on opening force. . . . . . . . . . . . . . . . . . . . . . . . .

106

Table 8.2

Steel alloy heat-treating example data. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

109


111

Figure 8.14 Half-normal plot of the absolute effects. . . . . . . . . . . . . . . . . . . . . . . . . . . .

112

Figure 8.15 Plot of effect 1 (preheat) on hardness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

112

Figure 8.16 Plot of effect 2 (equalize) on hardness. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

113

Figure 8.17 Plot of effect 3 (austenize) on hardness. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

113

Figure 8.18 Plot of effect 4 (temper) on hardness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

114

Figure 8.19 Plot of effect 5 (quench) on hardness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

114

Table 8.3

Plastic processing example data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

117


119

xii


Figure 8.21 Plot of effect 1 (temperature) on the nominal specification. . . . . . . . . . . . .

119

Figure 8.22 Plot of effect 2 (time) on the nominal specification. . . . . . . . . . . . . . . . . . .

120

Figure 8.23 Plot of effect 3 (pressure) on the nominal specification. . . . . . . . . . . . . . . .

120

Figure 8.24 Plot of effect 4 (polymer) on the nominal specification. . . . . . . . . . . . . . . .

121

Figure 9.1

Three-component reduced cubic mixture design. . . . . . . . . . . . . . . . . . . . .

123

Figure 9.2

Linear, quadratic, reduced cubic, full cubic, and special quartic mixture designs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

124

Table 9.1

Blown film development example data. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

125

Figure 9.3

Three-component quadratic mixture design for the blown film example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

125

Blown film development example—additional data. . . . . . . . . . . . . . . . . . .

126

Table 9.2

Preface

T

his book—a result of 30 years of quality-related work experience—was written to aid quality technicians and engineers. To that end, the intent of this book is to provide the quality professional working in virtually any industry a quick, convenient, and comprehensive guide to properly conducting design of experiments (DOE) for the purpose of process optimization. This book is intended for people who have never been exposed to design of experiments, been intimidated in their attempts to learn about DOE, or have not appreciated the potential of this family of tools in their process improvement and optimization efforts. This is a practical introduction to the basics, and is not intended to provide complete coverage of DOE. When one becomes familiar with and begins applying the basic principles presented in the book, they should be easily encouraged to go on to moreadvanced principles in DOE. Once successful with DOE, one will rarely need prodding to continue to learn more about this powerful tool. Every effort has been made to simplify the approach and minimize the complexity of the material. This book will use a simple statistical calculator rather than sophisticated software or spreadsheets. The book also assumes a basic knowledge of statistical techniques and process control. Inevitably, there are some points that might have been covered with more statistical depth. It is the author’s strong recommendation that anyone who completes this book immediately continue on to a more in-depth reference (some excellent ones are identified in the Bibliography of this book), both to learn about areas that are not covered here and to broaden the reader’s depth of knowledge of DOE in general. This book will be a useful reference when preparing for and taking many of the ASQ quality certification examinations, including the Certified Quality Technician (CQT), Certified Six Sigma Green Belt (CSSGB), Certified Quality Engineer (CQE), Certified Six Sigma Black Belt (CSSBB), and Certified Reliability Engineer (CRE).

xiii


Acknowledgments

I

would like to acknowledge the previous work of Larry B. Barrentine in An Introduction to Design of Experiments: A Simplified Approach. This book is an expansion of his efforts in an attempt to continue Barrentine’s method of presenting DOE studies in a simple, easy-to-follow style. Several sections of this book come directly from his previous work. I have made some changes to clarify and augment some of his points and present the topics in a consistent manner. I would like to thank those who have inspired, taught, and trained me throughout my academic and professional career. I also wish to recognize my friend and colleague, Scott Kochendoerfer, CQE, for lending his expertise in reviewing this book for accuracy and content. I would also like to express my sincere gratitude to James McLinn, Reliability Consultant at Ops A La Carte, and Stefan Mozar, Director and Adjunct Professor at Guangdong University of Technology, for reviewing the book and providing valuable feedback. Additionally, I would like to thank ASQ Quality Press, especially Matt Meinholz, Acquisitions Editor, and Paul Daniel O’Mara, Managing Editor, for their expertise and technical competence, which made this project a reality. Lastly, I would like to acknowledge the patience of my wife Dawn and my sons Jack and Sam, which allowed me time to research and write Practical Design of Experiments (DOE): A Guide for Optimizing Designs and Processes.

xv

Limit of Liability/Disclaimer of Warranty

T

he author has put forth his best efforts in compiling the content of this book; however, no warranty with respect to the material’s accuracy or completeness is made. Additionally, no warranty is made in regard to applying the recommendations made in this book to any business structure or environments. Businesses should consult regulatory, quality, and/or legal professionals prior to deciding on the appropriateness of advice and recommendations made within this book. The author shall not be held liable for loss of profit or other commercial damages resulting from the employment of recommendations made within this book, including special, incidental, consequential, or other damages.

xvi

1 Introduction

T

he English statistician Sir Ronald A. Fisher pioneered the development of design of experiments in the 1920s and 1930s, applying statistical techniques in the study of agriculture. During the 1940s, Robin L. Plackett and J. P. Burman introduced the idea of using smaller, more economical designs for experimentation (fractional factorials). The 1950s saw the introduction of response surface methodology (RSM), used in industrial experimentation by George E. P. Box and K. B. Wilson. During the same period of time, Genichi Taguchi introduced methods for improving the quality of manufactured goods, applying the loss function and signal-to-noise ratios to experimentation. The purpose of this introduction to the design of experiments (DOE) is to showcase the power and utility of this statistical tool while explaining how to plan and analyze an experiment. It is also an attempt to dispel the conception that DOE is reserved only for those with advanced mathematics training. It will be demonstrated that DOE is primarily a logic tool that can be easily grasped and applied, requiring only basic math skills. While software or spreadsheets would make the calculations more painless and provide greater versatility, it is necessary to understand what the software is doing. To this end, software is not used with this text; an inexpensive scientific calculator is used instead to insure that the basics are learned. This is by no means a complete study of the broad field of DOE. The intent is to introduce the basics, persuade the reader of the power of this tool, and then recommend resources for further study. The material covered will still be sufficient to support a large percentage of the experiments one may wish to perform. The prerequisites of this book are familiarity with the basic concepts of statistics, process stability, statistical process control (SPC), and measurement systems analysis (MSA). As in any process improvement activity, it is necessary to recognize that a process is made up of input variables, process variables, and output measures (see Figure 1.1). The intent is always to improve the output measure, which is labeled as the response. There is no direct control on the response variable; in the classical cause-andeffect approach, it is the effect . The causes are what dictate the response (dependent variable, output, Y ’s). To control the response, one must control the causes (independent variable, inputs, X ’s), which may be input variables and/or process variables involving the six elements shown in Figure 1.1. (These variables or causes will later be referred to as factors.) For example, there is no control setting in a sales process that allows one to set a sales level. To control sales, one must address those variables that cause sales to change, 1

2

Chapter One

Materials

Methods

Measurements Response(s)

Machines

People Independent inputs (X )

Figure 1.1

Environment Dependent outputs (Y )

Cause-and-effect diagram depicting inputs ( X ’s) and outputs (Y ’s).

for example, promotional literature, call frequency, pricing policies, credit policies, or personal sales techniques. A process may be very simple, or it may be a complex group of processes. In concert with this cause-and-effect, or systems, approach to the process, the concepts of process variation must be understood. Every response demonstrates variation. This variation results from (a) variation in the known input or process variables, (b) variation in the unknown process variables, and/or (c) variation in the measurement of the response variable. The combination of these sources results in the variation of that response. This variation is categorized by the classic SPC tools into two categories: (a) special cause variation—unusual responses compared to previous history; and (b) inherent variation—variation that has been demonstrated as typical of the process. A side note is needed here on terminology. Inherent , or typical, variation has a variety of labels that are often used interchangeably. In control charting, it is referred to as common cause variation. In control systems, it is called process noise. In DOE, it is called experimental error or random variation. To minimize confusion, it will be referred to in this text as either inherent variation or experimental error . Control charts are used to identify special cause variation and, hopefully, to identify the process variables or causes that led to such unusual responses. The presence of special causes within an experiment will create problems in reaching accurate conclusions. For this reason, DOE is more easily performed after the process has been stabilized using SPC tools. The presence of inherent variation also makes it difficult to draw conclusions. (In fact, that is one of the definitions of statistics: decision making in the presence of uncertainty or inherent variation.) If a process variable causes changes in the response that exceed the inherent variation, we state that the change is significant . Inherent variation can also be analyzed to determine whether the process will consistently meet a specification. The calculation of process capability is a comparison of the spread of the process with the specifications, resulting in test statistics such as C p and Cpk . Figure 1.2 illustrates the comparison of a process with its upper and lower specification limits. DOE is the simultaneous study of several process variables. By combining several variables in one study instead of creating a separate study for each, the amount of testing

Introduction

Lower specification limit

3

Upper specification limit

–3 ^ σ

+3 ^ σ

Statistical variation (6σ)

Part tolerance

Figure 1.2

Relationship between statistical control limits and product specifications.

Source: M. A. Durivage, Practical Engineering, Process, and Reliability, Statistics , Milwaukee: ASQ Quality Press, 2014. Used with permission.

required will be drastically reduced, and greater process understanding will result. This is in direct contrast to the typical one-factor-at-a-time (OFAT) approach, which limits understanding and wastes data. Additionally, OFAT studies cannot be assured of detecting the unique effects of combinations of factors (a condition later to be defined as an interaction). DOE includes the entire scope of experimentation, including defining the output measure(s) that one desires to improve, the candidate process variables that one will change, procedures for experimentation, actual performance of the experiment, and analysis and interpretation of the results. The objectives of the experimenter in a DOE are to learn how to: • Maximize the response • Minimize the response • Adjust the response to a nominal value • Reduce process variation • Make the process robust (that is, make the response insensitive to uncontrollable changes in the process variables) • Determine which variables are important to control and which are not

The basic experimental procedure is a series of basic logical steps that must be addressed as one prepares to launch a DOE: 1. What is the process to be studied? How broadly or narrowly is it defined? A flowchart is a good tool for this analysis.

4

Chapter One

2. What is the response? What needs to be improved? Should there be more than one response? Note: Additional responses are free, costing only the measurements! 3. What is the measurement precision? Is there bias in the measurement system? Has a measurement systems analysis been completed? Is it adequate? 4. Generate candidate factors. This is best done with a small team using brainstorming after a review of all available data and information on the process and response variables. A cause-and-effect diagram and a flowchart of the process are useful tools to use while brainstorming. The trick is to be innovative, to think outside usual boundaries, and yet not try to reinvent proven technology. Provide opportunities for surprises! The team should be knowledgeable about the issues and follow the rules for brainstorming. 5. Determine the levels for the factors selected for the DOE. In screening experiments, the rule is to have levels broadly spaced but not to the point of being foolhardy. In refining experiments, levels will be much tighter and will require more replication. 6. Select the experimental design. This is the set of treatments or runs that will be performed. This also includes deciding on the amount of replication. Finally, the randomized order of the trials is determined. (Randomization is the insurance policy against misleading conclusions due to outside influence during the experiment.) 7. Establish a plan to control (or at least monitor) extraneous variables. 8. Perform the experiment according to the design. The DOE must be carried out per its design. Identify trial materials carefully. Keep good notes. 9. Analyze, draw conclusions, and assess process impact. What process variables can be changed—and how—to improve the process? 10. Verify and document the new process as defined by the experiment. 11. Propose the next study for continuation of this project, or declare the project complete. Make sure that all reports that go beyond the team are in language and terminology that are easily understood. Note: It is extremely important that prior to and after performing DOE a line clearance is executed to prevent mix-ups and/or comingling of products, packaging, and labeling. The nine steps for analysis of effects are shown in Figure 1.3. This will be the basic flow for all experiments presented in this book. There will be times when some of the steps cannot be completed. For instance, steps 3, 4, 5, and 6 are not used when conducting experiments without repetitions, replicates, attribute data, ordered categorical data, and Taguchi’s signal-to-noise (S/N) ratios. In these cases, a half-normal plot can be used to determine the significant effects. Some of the examples in the book that use steps 3, 4,

Introduction

5

1. Calculate the absolute values of the effects 2. Create a Pareto chart of the absolute values of the effects 3. Calculate the standard deviation of the experiment, s e 4. Calculate the standard deviation of the effects, s Eff 5. Determine the t -statistic 6. Calculate the decision limits 7. Determine the significant effects 8. Graph the main effects and any significant interactions 9. Model the significant effects, significant interactions, and individual effects from significant interactions

Figure 1.3

Nine steps for analysis of effects.

5, and 6 will also have an associated half-normal plot for illustrative purposes. It should be noted that the use of statistical decision limits is the preferred method. For ease of instruction, a review of some the basic statistical tools and techniques is presented, followed by small experiments, which are followed by large experiments. In the real world, one would prefer to start with large experiments and progress to smaller ones in order to identify variables that affect the response variables. Terms and definitions are covered as they arise. The terminology used in DOE is often different from the equivalent terms in SPC, and is presented to assure easier readability. The initial example is used to define most of the unique terminology and many of the analytical techniques. It is suggested that the reader review the Glossary following the Appendixes. Disclaimer: All examples in this book are fictitious, and therefore the results should not be used to make product or process decisions.


Index

A

critical-to-quality (CTQ) characteristics, 97 cumulative percentage points (Appendix D), 153–54 z-scores of (Appendix E), 155–56

absolute values of the effects, 29–31, 41, 47–49, 66, 72–74, 81, 104, 110–11, 118 accuracy, versus precision, 97 aliases, 89 alpha (a) risk, 8–9 analysis of effects, nine steps for, 4–5 bond strength example, 29–38 chemical processing yield example, 47–55 corrosion study example, 41–46 ordered categorical data example, 81–86 plastic welding example (with transformation), 71–78 plastic welding example (without transformation), 66–71 Taguchi L4 orthogonal array example, 103–8 Taguchi L8 orthogonal array example, 108–16 Taguchi L9 orthogonal array example, 118–22 analysis of variance (ANOVA), 19–26 reliability of results, 19–20

D data, missing, 129 Dean and Dixon outlier test, 7–8 critical values of (Appendix A), 137 decision limits (DL), 34, 42, 45–46, 50, 68, 74, 83 dependent variable (Y ), 1 design of experiments (DOE) analytical considerations in, 94 basics of managing, 131–32 common problems and questions, 129–31 history of, 1 introduction to, 1–5 larger designs, 94 maximum number of factors, 130 miscellaneous designs, 94–95 objectives of, 3, 55 obstacles to application of, 132–33 procedural considerations in, 129–34 diamond factor, 60

B beta (b) risk, 8–9 blocking, 28–29, 130–31 Box, George E. P., 1, 55 Burman, J. P., 1

E effect, 1, 30 effect heredity, 92 effect plots, interpreting, 17–18 effect sparsity, 92 evolutionary operation (EVOP), 55, 95 experimental design, 28, 29 experimental er ror. See inherent variation experimental procedure, basic, 3– 4 experimental results, verifying, 36 experiments with qualitative (attribute d ata) responses, 65–86

C cause-and-effect diagram, 2 causes, 1–2 coefficient calculation, 35–36, 44–45, 53–54, 70–71, 75–78, 85–86, 106–7, 113–15, 119– 21, 124, 126 common cause variation. See inherent variation confounding, 88–89, 91–92 control chart, 2 control factors, 97

183

184

Index

ordered categorical data example, 78–86 plastic welding example (with transformation), 71–78 plastic welding example (without transformation), 65–71 with three factors, 47–64 chemical processing yield example, 47–55 with two factors, 27–46 bond strength example, 27–38 corrosion study example, 40– 46 unreplicated, analysis with, 60–64

interaction plots, interpreting, 17–18 interactions, 31, 88–93 significant, 35

L larger is better, S/N ratio, 101 left-skewed distribution, 13 long-tailed distribution, 14 loss function, 97

F

M

factors, 1 maximum number in DOE, 130 F -distribution, percentages of (Appendix B), 139–49 Fisher, Ronald A., 1, 97 fold-over design. See reflection fraction defective, S/N ratio, 102 fractional factorial designs, 91–92, 94 versus Plackett-Burman designs, 130 selected (Appendix H), 161–64 full-factorial designs, 28 selected (Appendix H), 161–63

main effects, graphing, 35, 43–44, 51, 69, 74, 84, 105, 112, 118 measurement systems analysis (MSA), 65, 78, 79 mixture designs, 95, 123–27 selected (Appendix K), 171–72 three-factor simplex example, 125–27

N

gage repeatability and reproducibility (GR&R) study, 65, 78, 79 geometric designs, 91, 130 golden factor, 60

noise factors, 97 nominal is b est, S/N ratio, 101 nongeometric designs, 91, 130 nonlinear models, 38–40 normal distribution probability points—area above Z (Appendix G), 159 normal distribution probability points—area below Z (Appendix F), 157 normal probability plots, 13–15 normality, “pencil test” of, 14–15

H

O

half-normal probability plot, 4–5, 15–16, 34–35, 43 hypothesis statement for one-way ANOVA, 20 for two-way ANOVA, 22–23 hypothesis testing, 8–12 apportionment of risk in, 9 conclusion statements, 11

one-factor-at-a-time (OFAT) experiment, 3 one-tail (lower-tailed) test, 10 one-tail (upper-tailed) test, 9 one-way ANOVA, 19–21 ordered categorical, S/ N ratio, 102 orthogonal designs, 97–99 Taguchi L4 array example, 103–8 Taguchi L8 array example, 108–16 Taguchi L9 array example, 116–22 outliers, detecting, 7–8

G

I independent variable ( X ), 1, 27 individual effects from significant interactions, 35, 44–46, 51–55, 69–71, 74–78, 84–86, 113–15, 119–22 inherent variation, 2, 32–33, 59, 129

P Pareto chart, 32, 41, 49, 66, 74, 81, 94, 104, 111, 118 “pencil test” of normality, 14–15

Plackett, Robin L., 1 Plackett-Burman screening designs, 91–92, 94 versus fractional factorial designs, 130 selected (Appendix I), 165–66 precision, versus accuracy, 97 probability plots, normal, 13–15 process, definition, 1 process noise. See inherent variation

R ramp time, 47, 59, 63 random variation. See inherent variation randomization, 28–29, 131 refining design, 38 reflection, 92–93 versus replication, 129 repeat run, versus replication, 28 replication, 131 versus reflection, 129 versus repeat run, 28 residual analysis, 60–64 resolution, 89–9 0 response, 1 response surface designs, 95 response surface methodology (RSM), 1 response variable, 27 right-skewed distribution, 13 risk, apportionment of, in hypothesis testing, 9 robust design, 97

S sample data, testing for a difference between two variances using, 11–12 screening designs, 37, 87–94, 131 short-tailed distribution, 13 signal-to-noise (S/N) ratios, 97, 101–2 significant effects, 34–35, 43, 44–46, 51–55, 68, 69–71, 74–78, 84–86, 104–5, 106–8, 112, 113–15, 118, 119–22 significant interactions, 35, 43–44, 44 –46, 51–55, 69–71, 74–78, 84 –86, 105–8, 112–15, 118– 22, 129 smaller is better, S/N ratio, 101–2 software considerations, in DOE, 133–34 spreadsheets, for DOE, 133–34 standard deviation of the effects (s Eff ), 33, 42, 50 standard deviation of the experiment (se), 33, 41–42, 49 statistical tools and techniques, 7–18 Student’s t -distribution, percentage points of (Appendix C), 151–52 Student’s t -test, 15

Index

185

T Taguchi, Genichi, 1, 97 Taguchi designs, 94–95 selected (Appendi x J), 167–70 Taguchi experiments, 97–122 Taguchi loss function, 97 Taguchi orthogonal designs, 97–99 L4 array example, 103–8 L8 array example, 108–16 L9 array example, 116–22 test method validation (TMV), 65, 78, 79 three-component mixture design, 123 three-factor simplex mixture design example, 125–27 t -statistic, 33–34, 42, 50 two-tail test, 10–11 two-way ANOVA, 22–26 type I error, 8 type II er ror, 8 typical variation. See inherent variation

U unreplicated experiments, analysis with, 60–64

V variables, 1–2 variances, testing for a difference between two, using sample d ata, 11–12 variation, inherent, 2, 32–33, 59, 129 variation analysis, 129 chemical processing yield example, 55–60

W weighted probability scoring scheme (WPSS), 78 Wilson, K. B., 1

X XPULT Experimental Catapult, 135

Z zero is best, S/N ratio, 102 z-scores of the cumulative percentage points

(Appendix E), 155–56


Practical Design of Experiments (DOE)

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