# Statistical Robust Design: An Industrial Perspective

## Books

A UNIQUELY PRACTICAL APPROACH TO ROBUST DESIGN FROM A STATISTICAL AND ENGINEERING PERSPECTIVE

Variation in environment, usage conditions, and the manufacturing process has long presented a challenge in product engineering, and reducing variation is universally recognized as a key to improving reliability and productivity. One key and cost-effective way to achieve this is by robust design – making the product as insensitive as possible to variation.

With Design for Six Sigma training programs primarily in mind, the author of this book offers practical examples that will help to guide product engineers through every stage of experimental design: formulating problems, planning experiments, and analysing data. He discusses both physical and virtual techniques, and includes numerous exercises and solutions that make the book an ideal resource for teaching or self-study.

• Presents a practical approach to robust design through design of experiments.

• Offers a balance between statistical and industrial aspects of robust design.

• Includes practical exercises, making the book useful for teaching.

• Covers both physical and virtual approaches to robust design.

• Supported by an accompanying website (www.wiley/com/go/robust) featuring MATLAB® scripts and solutions to exercises.

• Written by an experienced industrial design practitioner.

This book’s state of the art perspective will be of benefit to practitioners of robust design in industry, consultants providing training in Design for Six Sigma, and quality engineers. It will also be a valuable resource for specialized university courses in statistics or quality engineering.

Preface ix

1 What is robust design? 1

1.1 The importance of small variation 1

1.2 Variance reduction 2

1.3 Variation propagation 4

1.4 Discussion 5

1.4.1 Limitations 6

1.4.2 The outline of this book 7

Exercises 8

2 DOE for robust design, part 1 11

2.1 Introduction 11

2.1.1 Noise factors 11

2.1.2 Control factors 12

2.1.3 Control-by-noise interactions 12

2.2 Combined arrays: An example from the packaging industry 13

2.2.1 The experimental array 15

2.2.2 Factor effect plots 15

2.2.3 Analytical analysis and statistical significance 17

2.2.4 Some additional comments on the plotting 20

2.3 Dispersion effects 21

Exercises 23

Reference 25

3 Noise and control factors 27

3.1 Introduction to noise factors 27

3.1.1 Categories of noise 28

3.2 Finding the important noise factors 33

3.2.1 Relating noise to failure modes 33

3.2.2 Reducing the number of noise factors 34

3.3 How to include noise in a designed experiment 40

3.3.1 Compounding of noise factors 40

3.3.2 How to include noise in experimentation 45

3.3.3 Process parameters 48

3.4 Control factors 48

Exercises 49

References 51

4 Response, signal, and P diagrams 53

4.1 The idea of signal and response 53

4.1.1 Two situations 54

4.2 Ideal functions and P diagrams 55

4.2.1 Noise or signal factor 56

4.2.2 Control or signal factor 56

4.2.3 The scope 58

4.3 The signal 63

4.3.1 Including a signal in a designed experiment 64

Exercises 65

5 DOE for robust design, part 2 69

5.1 Combined and crossed arrays 69

5.1.1 Classical DOE versus DOE for robust design 69

5.1.2 The structure of inner and outer arrays 70

5.2 Including a signal in a designed experiment 74

5.2.1 Combined arrays with a signal 74

5.2.2 Inner and outer arrays with a signal 81

5.3 Crossed arrays versus combined arrays 89

5.3.1 Differences in factor aliasing 91

5.4 Crossed arrays and split-plot designs 94

5.4.1 Limits of randomization 94

5.4.2 Split-plot designs 95

Exercises 98

References 99

6 Smaller-the-better and larger-the-better 101

6.1 Different types of responses 101

6.2 Failure modes and smaller-the-better 102

6.2.1 Failure modes 102

6.2.2 STB with inner and outer arrays 103

6.2.3 STB with combined arrays 106

6.3 Larger-the-better 106

6.4 Operating window 108

6.4.1 The window width 110

Exercises 113

References 115

7 Regression for robust design 117

7.1 Graphical techniques 117

7.2 Analytical minimization of (g′(z))2 120

7.3 Regression and crossed arrays 121

7.3.1 Regression terms in the inner array 127

Exercises 128

8 Mathematics of robust design 131

8.1 Notational system 131

8.2 The objective function 132

8.2.1 Multidimensional problems 136

8.2.2 Optimization in the presence of a signal 138

8.2.3 Matrix formulation 139

8.2.4 Pareto optimality 141

8.3 ANOVA for robust design 144

8.3.1 Traditional ANOVA 144

8.3.2 Functional ANOVA 146

8.3.3 Sensitivity indices 149

Exercises 152

References 153

9 Design and analysis of computer experiments 155

9.1 Overview of computer experiments 156

9.1.1 Robust design 157

9.2 Experimental arrays for computer experiments 161

9.2.1 Screening designs 161

9.2.2 Space filling designs 163

9.2.3 Latin hypercubes 165

9.2.4 Latin hypercube designs and alphabetical optimality criteria 166

9.3 Response surfaces 167

9.3.1 Local least squares 168

9.3.2 Kriging 169

9.4 Optimization 171

9.4.1 The objective function 171

9.4.2 Analytical techniques or Monte Carlo 173

Exercises 175

References 176

10 Monte Carlo methods for robust design 177

10.1 Geometry variation 177

10.1.1 Electronic circuits 179

10.2 Geometry variation in two dimensions 179

10.3 Crossed arrays 192

11 Taguchi and his ideas on robust design 195

11.1 History and origin 195

11.2 The experimental arrays 197

11.2.1 The nature of inner arrays 197

11.2.2 Interactions and energy thinking 199

11.2.3 Crossing the arrays 200

11.3 Signal-to-noise ratios 200

11.4 Some other ideas 203

11.4.1 Randomization 203

11.4.2 Science versus engineering 204

11.4.3 Line fitting for dynamic models 204

11.4.4 An aspect on the noise 206

11.4.5 Dynamic models 207

Exercises 208

References 208

Appendix A Loss functions 209

A.1 Why Americans do not buy American television sets 209

A.2 Taguchi’s view on loss function 211

A.3 The average loss and its elements 211

A.4 Loss functions in robust design 214

Exercises 215

References 217

Appendix B Data for chapter 2 219

Appendix C Data for chapter 5 223

Appendix D Data for chapter 6 231

Index 233

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