ORDINAL OPTIMIZATION

SOFT OPTIMIZATION FOR HARD PROBLEMS

Yu-Chi Ho Harvard University Massachusetts, USA Tsinghua University Beijing, China

Qian-Chuan Zhao Tsinghua University Beijing, China

Qing-Shan Jia Tsinghua University Beijing, China

Sprin ger

Table of Contents

Preface - - - - - - - xiü

Acknowledgements— - --- - - —xv

I Introduction 1

II Ordinal Optimization Fundamentals -—7

1 Two basic ideas of Ordinal Optimization (OO) 7 2 Definitions, terminologies, and concepts for OO — 9 3 A simple demonstration of OO 13 4 The exponential convergence of order and goal softening 15

4.1 Large deviation theory - 16 4.2 Exponential convergence w.r.t. order - — 21 4.3 Proof of goal softening 26

4.3.1 Blind pick -- 26 4.3.2 Horse race — 28

5 Universal alignment probabilities 37 5.1 Blind pick selection rule - 38 5.2 Horse race selection rule— 39

6 Deterministic complex optimization problem and Kolmogorov equivalence 48

7 Example applications 51 7.1 Stochastic Simulation modeis 51 7.2 Deterministic complex modeis 53

8 Preview of remaining chapters —54

III Comparison of Selection Rules 57

1 Classification of selection rules 60 2 Quantify the efficiency of selection rules -69

viii Table of Contents

2.1 Parameter settings in experiments for regression functions 73 2.2 Comparison of selection rules 77

3 Examples of search reduction - 80 3.1 Example: Picking with an approximate model 80 3.2 Example: A buffer resource allocation problem 84

4 Some properties of good selection rules 88 5 Conclusion 90

IV Vector Ordinal Optimization 93

1 Defmitions, terminologies, and concepts for VOO 94 2 Universal alignment probability 99 3 Exponential convergence w.r.t. order 104 4 Examples of search reduction 106

4.1 Example: When the Observation noise contains normal distribution 106

4.2 Example: The buffer allocation problem 108

V Constrained Ordinal Optimization 113

1 Determination of selected set in COO 115 1.1 Blind pick with an imperfect feasibility model 115 1.2 Impact of the quality of the feasibility model on BPFM 119

2 Example: Optimization with an imperfect feasibility model 122 3 Conclusion 124

VI Memory Limited Strategy Optimization 125

1 Motivation (the need to find good enough and simple strategies) 126 2 Good enough simple strategy search based on OO 128

2.1 Building crude model 128 2.2 Random sampling in the design space of simple strategies 133

3 Conclusion 135

VII Additional Extensions of the OO Methodology 137

1 Extremely large design space 138 2 Parallel implementation of OO 143

2.1 The concept of the Standard clock 144

Table of Contents ix

2.2 Extension to non-Markov cases using second order approximations — — 147 2.2.1 Second order approximation 148 2.2.2 Numerical testing —152

3 Effect of correlated Observation noises 154 4 Optimal Computing Budget Allocation and Nested Partition- 159

4.1 OCBA — - — - — - 160 4.2 NP — 164

5 Performance order vs. Performance value— 168 6 Combination with other optimization algorithms 175

6.1 Using other algorithms as selection rules in 0 0 177 6.1.1 GA+OO —- 177 6.1.2 SA+OO 183

6.2 Simulation-based parameter optimization for algorithms 186 6.3 Conclusion -188

VIII Real World Application Examples- 189

1 Scheduling problem for apparel manufacturing —190 1.1 Motivation 191 1.2 Problem formulation -192

1.2.1 Demand modeis 193 1.2.2 Production facilities 195 1.2.3 Inventory dynamic 196 1.2.4 Summary —-197

1.3 Application of ordinal optimization- 198 1.3.1 Random sampling of designs 199 1.3.2 Crude model —- — - — - 200

1.4 Experimental results 202 1.4.1 Experiment 1: 100 SKUs—- —-202 1.4.2 Experiment 2: 100 SKUs with consideration

on satisfaction rate —204 1.5 Conclusion 206

2 The turbine blade manufacturing process optimization problem 207 2.1 Problem formulation 208 2.2 Application of 0 0 - — 213 2.3 Conclusion— — 219

3 Performance optimization for a remanufacturing System— 220 3.1 Problem formulation of constrained optimization —220 3.2 Application of COO 224

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3.2.1 Feasibility model for the constraint 224 3.2.2 Grude model for the Performance— 224 3.2.3 Numerical results - 225

3.3 Application of VOO - 227 3.4 Conclusion - —232

4 Witsenhausen problem 232 4.1 Application of 0 0 to find a good enough control law 234

4.1.1 Crude model - - — 235 4.1.2 Selection of promising subsets 237

4.2 Application of 0 0 for simple and good enough control laws—245 4.3 Conclusion - - - 251

Appendix A Fundamentals of Simulation and Performance Evaluation 253

1 Introduction to Simulation 253 2 Random numbers and variables generation 255

2.1 The linear congruential method 255 2.2 The method of inverse transform 257 2.3 The method of rejection 258

3 Sampling, the central limit theorem, and confidence intervals 260 4 Nonparametric analysis and order statistics 262 5 Additional problems of simulating DEDS 262 6 The alias method of choosing event types 264

Appendix B Introduction to Stochastic Processes and Generalized Semi-Markov Processes as Models for Discrete Event Dynamic Systems and Simulations -267

1 Elements of stochastic sequences and processes 267 2 Modeling of discrete event Simulation using stochastic sequences 271

Appendix C Universal Alignment Tables for the Selection Rules in Chapter III 279

Table of Contents XI

Appendix D Exercises — 291

1 True/False questions— 291 2 Multiple-choice questions 293 3 General questions — 297

References- 305

Index — - - — — 315