Evolutionary Multi-Objective Optimization in Uncertain Environments: Issues and Algorithms Book

1 Introduction 1

1.1 Multi-objective Optimization 1

1.1.1 Totally Conflicting, Non-conflicting, and Partially Conflicting Multi-objective Problems 2

1.1.2 Pareto Dominance and Optimality 3

1.1.3 Multi-objective Optimization Goals 5

1.2 Evolutionary Multi-objective Optimization 5

1.2.1 MOEA Framework 6

1.2.2 Basic MOEA Components 8

1.2.3 Benchmark Problems 16

1.2.4 Performance Metrics 18

1.3 Empirical Analysis and Performance Assessment Adequacy for EMO Techniques 21

1.3.1 Preliminary Discussions 21

1.3.2 Systematic Design for Empirical Assessment 25

1.3.3 Survey on Experimental Specifications 31

1.3.4 Conceptualizing Empirical Adequacy 33

1.3.5 Case Studies 36

1.4 Overview of This Book 38

1.5 Conclusion 39

Part I Evolving Solution Sets in the Presence of Noise

2 Noisy Eyolutionary Multi-objective Optimization 43

2.1 Noisy Multi-objective Optimization Problems 44

2.2 Performance Metrics for Noisy Multi-objective Optimization 45

2.3 Empirical Results of Noise Impact 46

2.3.1 General MOEA Behavior under Different Noise Levels 47

2.3.2 MOEA Behavior in the Objective Space 50

2.3.3 MOEA Behavior in Decision Space 53

2.4 Conclusion 54

3 Handling Noise in Evolutionary Multi-objective Optimization 55

3.1 Estimate Strength Pareto Evolutionary Algorithm 56

3.2 Multi-Objective Probabilistic Selection Evolutionary Algorithm 60

3.3 Noise Tolerant Strength Pareto Evolutionary Algorithm 63

3.4 Modified Non-dominated Sorting Genetic Algorithm II 65

3.5 Multi-objective Evolutionary Algorithm for Epistemic Uncertainty 67

3.6 Indicator-Based Evolutionary Algorithm for Multi-objective Optimization 70

3.7 Multi-Objective EvolutionaryAlgorithm with Robust Features 72

3.8 Comparative Study 80

3.9 Effects of the Proposed Features 92

3.10 Further Examination 97

3.11 Conclusion 98

4 Handling Noise in Evolutionary Neural Network Design 101

4.1 Singular Value Decomposition for ANN Design 102

4.1.1 Rank-Revealing Decomposition 102

4.1.2 Actual Rank of Hidden Neuron Matrix 103

4.1.3 Estimating the Threshold 106

4.1.4 Moore-Penrose Generalized Pseudoinverse 107

4.2 Hybrid Multi-Objective Evolutionary Neural Networks 107

4.2.1 Algorithmic Flow of HMOEN 107

4.2.2 Multi-Objective Fitness Evaluation 108

4.2.3 Variable-Length Representation for ANN Structure 109

4.2.4 SVD-Based Architectural Recombination 109

4.2.5 Micro-Hybrid Genetic Algorithm 112

4.3 Experimental Study 114

4.3.1 Experimental Setup 114

4.3.2 Analysis of HMOEN Performance 116

4.4 Conclusion 121

Part II Tracking Dynamic Multi-objective Landscapes

5 Dynamic Evolutionary Multi-objective Optimization 125

5.1 Dynamic Multi-objective Optimization Problems 126

5.2 Dynamic Multi-objective Problem Categorization 126

5.3 Dynamic Multi-objective Test Problems 128

5.3.1 TLK2 Dynamic Test Function 129

5.3.2 FDA Dynamic Test Functions 130

5.3.3 dMOP Test Functions 131

5.3.4 DSW Test Functions 133

5.3.5 JS Test Functions 134

5.4 Performance Metrics for Dynamic Multi-objective Optimization 135

5.4.1 Illustrating Performance Using Static Performance Measures 135

5.4.2 Time Averaging Static Performance Measures 136

5.5 Evolutionary Dynamic Optimization Techniques 138

5.5.1 Design Issues 138

5.5.2 Directional-Based Dynamic Evolutionary Multi-objective Optimization Algorithm 141

5.5.3 Dynamic Non-dominated Sorting Genetic Algorithm II 142

5.5.4 Dynamic Multi-objective Evolutionary Algorithm Based on an Orthogonal Design 144

5.5.5 Dynamic Queuing Multi-objective Optimizer 146

5.5.6 Multi-objective Immune Algorithm 148

5.6 Conclusion 152

6 A Coevolutionary Paradigm for Dynamic Multi-Objective Optimization 153

6.1 Competition, Cooperation, and Competitive-Cooperation in Coevolution 154

6.1.1 Competitive Coevolution 154

6.1.2 Cooperative Coevolution 155

6.1.3 Competitive-Cooperative Coevolution 158

6.2 Applying Competitive-Cooperation Coevolution for Multi-objective Optimization 160

6.2.1 Cooperative Mechanism 161

6.2.2 Competitive Mechanism 162

6.2.3 Implementation 164

6.3 Adapting COEA for Dynamic Multi-objective Optimization 165

6.3.1 Introducing Diversity via Stochastic Competitors 165

6.3.2 Handling Outdated Archived Solutions 167

6.4 Static Environment Empirical Study 168

6.4.1 Comparative Study of COEA 168

6.4.2 Effects of the Competitive Mechanism 172

6.4.3 Effects of Different Competition Schemes 174

6.5 Dynamic Environment Empirical Study 177

6.5.1 Comparative Study 177

6.5.2 Effects of Stochastic Competitors 182

6.5.3 Effects of Temporal Memory 182

6.6 Conclusion 185

Part III Evolving Robust Solution Sets

7 Robust Evolutionary Multi-objective Optimization 189

7.1 Robust Multi-objective Optimization Problems 189

7.2 Robust Measures 190

7.3 Robust Optimization Problems 191

7.4 Robust Continuous Multi-objective Test Problem Design 192

7.4.1 Robust Multi-objective Problem Categorization 192

7.4.2 Empirical Analysis of Existing Benchmark Features 194

7.5 Robust Continuous Multi-objective Test Problem Design 197

7.5.1 Basic Landscape Generation 199

7.5.2 Changing the Decision Space 202

7.5.3 Changing the Solution Space 202

7.5.4 Example of a Robust Multi-objective Test Suite 203

7.6 Vehicle Routing Problem with Stochastic Demand 207

7.6.1 Problem Features 208

7.6.2 Problem Formulation 210

7.7 Conclusion 211

8 Evolving Robust Solutions in Multi-Objective Optimization 213

8.1 Evolutionary Robust Optimization Techniques 214

8.1.1 Single-Objective Approach 214

8.1.2 Multi-objective Approach 215

8.1.3 Robust Multi-Objective Optimization Evolutionary Algorithm 216

8.2 Empirical Analysis 219

8.2.1 Fitness Inheritance for Robust Optimization 219

8.2.2 Evaluating GTCO Test Suite 219

8.2.3 Evaluating VRPSD Test Problems 225

8.3 Conclusion 227

9 Evolving Robust Routes 229

9.1 Overview of Existing Works 229

9.2 Hybrid Evolutionary Multi-Objective Optimization 230

9.2.1 Variable-Length Chromosome 231

9.2.2 Local Search Exploitation 232

9.2.3 Route-Exchange Crossover 232

9.2.4 Multi-mode Mutation 233

9.2.5 Route Simulation Method 235

9.2.6 Computing Budget 236

9.2.7 Algorithmic Flow of HMOEA 237

9.3 Simulation Results and Analysis 238

9.3.1 Performance of Hybrid Local Search 239

9.3.2 Comparison with a Deterministic Approach 241

9.3.3 Effects of Sample Size, H 244

9.3.4 Effects of M 246

9.4 Conclusion 247

10 Final Thoughts 249

References 253