Probabilistic Design for Optimization and Robustness for Engineers

BRYAN DODSON, Executive Engineer, SKF, USA PATRICK C. HAMMETT, Lead Faculty Six Sigma Program, Integrative Systems & Design, College of Engineering, University of Michigan, Ann Arbor, USA RENÉ KLERX, Principal Statistician, SKF, The Netherlands

Preface ix

Acknowledgments xi

1 New product development process 1

1.1 Introduction 1

1.2 Phases of new product development 2

1.2.1 Phase I—concept planning 3

1.2.2 Phase II—product planning 4

1.2.3 Phase III—product engineering design and verification 6

1.2.4 Phase IV—process engineering 9

1.2.5 Phase V—manufacturing validation and ramp-up 10

1.3 Patterns of new product development 11

1.4 New product development and Design for Six Sigma 13

1.4.1 DfSS core objectives 13

1.4.2 DfSS methodology 15

1.4.3 Embedded DfSS 16

1.5 Summary 17

Exercises 17

2 Statistical background for engineering design 19

2.1 Expectation 19

2.2 Statistical distributions 24

2.2.1 Normal distribution 24

2.2.2 Lognormal distribution 27

2.2.3 Weibull distribution 30

2.2.4 Exponential distribution 32

2.3 Probability plotting 34

2.3.1 Probability plotting—lognormal distribution 35

2.3.2 Probability plotting—normal distribution 36

2.3.3 Probability plotting—Weibull distribution 37

2.3.4 Probability plotting—exponential distribution 39

2.3.5 Probability plotting with confidence limits 40

2.4 Summary 43

Exercises 44

3 Introduction to variation in engineering design 46

3.1 Variation in engineering design 46

3.2 Propagation of error 47

3.3 Protecting designs against variation 48

3.4 Estimates of means and variances of functions of several variables 51

3.5 Statistical bias 59

3.6 Robustness 59

3.7 Summary 60

Exercises 61

4 Monte Carlo simulation 63

4.1 Determining variation of the inputs 63

4.2 Random number generators 64

4.3 Validation 66

4.4 Stratified sampling 70

4.5 Summary 74

Exercises 75

5 Modeling variation of complex systems 76

5.1 Approximating the mean, bias, and variance 77

5.2 Estimating the parameters of non-normal distributions 81

5.3 Limitations of first-order Taylor series approximation for variance 84

5.4 Effect of non-normal input distributions 91

5.5 Nonconstant input standard deviation 93

5.6 Summary 93

Exercises 95

6 Desirability 98

6.1 Introduction 98

6.2 Requirements and scorecards 99

6.2.1 Types of requirements 100

6.2.2 Design scorecard 101

6.3 Desirability—single requirement 103

6.3.1 Desirability—one-sided limit 104

6.3.2 Desirability—two-sided limit 106

6.3.3 Desirability—nonlinear function 107

6.4 Desirability—multiple requirements 109

6.4.1 Maxi-min total desirability index 114

6.5 Desirability—accounting for variation 115

6.5.1 Determining desirability—using expected yields 115

6.5.2 Determining desirability—using non-mean responses 116

6.6 Summary 118

Exercises 118

7 Optimization and sensitivity 123

7.1 Optimization procedure 123

7.2 Statistical outliers 128

7.3 Process capability 129

7.4 Sensitivity and cost reduction 133

7.4.1 Reservoir flow example 134

7.4.2 Reservoir flow initial solution 135

7.4.3 Reservoir flow initial solution verification 136

7.4.4 Reservoir flow optimized with normal horsepower distribution 138

7.4.5 Reservoir flow optimized with normal horsepower distribution verification 140

7.4.6 Reservoir flow horsepower variation sensitivity 141

7.4.7 Reservoir flow horsepower lognormal probability plot 143

7.4.8 Reservoir flow horsepower Cpk optimization using a lognormal distribution 144

7.5 Summary 149

Exercises 150

8 Modeling system cost and multiple outputs 153

8.1 Optimizing for total system cost 153

8.2 Multiple outputs 158

8.2.1 Optimization 159

8.2.2 Computing nonconformance 159

8.3 Large-scale systems 164

8.4 Summary 166

Exercises 167

9 Tolerance analysis 170

9.1 Introduction 170

9.2 Tolerance analysis methods 174

9.2.1 Historical tolerancing 174

9.2.2 Worst-case tolerancing 175

9.2.3 Statistical tolerancing 175

9.3 Tolerance allocation 178

9.4 Drift, shift, and sorting 179

9.5 Non-normal inputs 182

9.6 Summary 182

Exercises 182

10 Empirical model development 185

10.1 Screening 185

10.2 Response surface 193

10.2.1 Central composite designs 194

10.3 Taguchi 200

10.4 Summary 200

Exercises 201

11 Binary logistic regression 202

11.1 Introduction 202

11.2 Binary logistic regression 205

11.2.1 Types of logistic regression 205

11.2.2 Binary versus ordinary least squares regression 206

11.2.3 Binary logistic regression and the logit model 208

11.2.4 Binary logistic regression with multiple predictors 211

11.2.5 Binary logistic regression and sample size planning 211

11.2.6 Binary logistic regression fuel door example 212

11.2.7 Binary logistic regression—significant binary input 213

11.2.8 Binary logistic regression—nonsignificant binary input 214

11.2.9 Binary logistic regression—continuous input 214

11.2.10 Binary logistic regression—multiple inputs 215

11.3 Logistic regression and customer loss functions 217

11.4 Loss function with maximum (or minimum) response 220

11.5 Summary 223

Exercises 223

12 Verification and validation 225

12.1 Introduction 225

12.2 Engineering model V&V 228

12.3 Design verification methods and tools 230

12.3.1 Design verification reviews 230

12.3.2 Virtual prototypes and simulation 231

12.3.3 Physical prototypes and early production builds 232

12.3.4 Confirmation testing comparing alternatives 232

12.3.5 Confirmation tests comparing the design to acceptance criteria 233

12.4 Process validation procedure 233

12.5 Summary 238

References 239

Bibliography 242

Answers to selected exercises 246

Index 251