Multiobjective Optimization

Yann Collette • Patrick Siarry

Multiobjective Optimization Principles and Case Studies

Springer

Contents

Forewords Part I Principle of multiobjective optimization methods 1

Introduction : multiobjective optimization and domination 1.1 What is a multiobjective optimization problem ? 1.2 Vocabulary and definitions 1.3 Classification of optimization problems 1.4 Multiobjective optimization 1.5 Multiplicity of solutions 1.6 Domination 1.6.1 Introduction and definitions 1.6.2 Example 1.6.3 Sizing of a beam 1.7 Illustration of the interest of multiobjective optimization 1.8 Relations derivated from domination 1.8.1 Introduction 1.8.2 Lexicographic optimality 1.8.3 Extremal optimality 1.8.4 Maximal optimality 1.8.5 Cone optimality 1.8.6 A-domination 1.8.7 Domination to the Geoffrion sens 1.8.8 Conclusion 1.9 Tradeoff surface 1.10 Convexity 1.11 Tradeoff surface representation 1.12 Resolution methods of multiobjective optimization problems 1.13 Annotated bibliography

17 17 18 19 20 21 21 21 24 28 30 33 33 35 35 36 37 40 40 41 42 44 44 45 46

VI

Contents

2

Scalar methods 2.1 The Weighted sum of objective functions method 2.2 Keeney-Raiffa method 2.3 Distance to a reference objective method 2.4 Compromise method 2.5 Hybrids methods 2.6 The goal attainment method 2.7 Goal programming method 2.8 Lexicographic method 2.9 Proper equality constraint method 2.10 Proper inequality constraints method 2.11 Lin-Tabak algorithm 2.12 Lin-Giesy algorithm 2.13 Annotated bibliography

47 47 54 54 58 63 64 69 71 72 76 77 78 79

3

Interactive methods 3.1 Introduction 3.2 Surrogate worth tradeoff method 3.3 Fandel method 3.4 STEP method 3.5 Jahn method 3.6 Geoffrion method 3.7 Simplex method 3.8 Annotated bibliography

81 81 81 84 91 94 97 101 104

4

Fuzzy methods 4.1 Introduction to fuzzy logic 4.1.1 Drawing a parallel with classical logic 4.1.2 Membership function 4.2 Sakawa method 4.3 Reardon method

105 105 105 106 107 112

5

Methods which use a metaheuristic 5.1 What's a metaheuristic ? 5.2 Task decomposition in optimization 5.3 Generality 5.4 Simulated annealing 5.4.1 P.A.S.A method (Pareto Archived Simulated Annealing) 5.4.2 M.O.S.A method (Multiple Objective Simulated Annealing) . . . 5.5 Tabu search 5.6 Genetic algorithms 5.7 Multiobjective optimization and genetic algorithms 5.7.1 "non aggregative" method 5.7.2 The "aggregative" methods 5.8 Annotated bibliography

115 115 115 116 118 119 121 125 125 130 131 133 142

Contents Decision aid methods 6.1 Introduction 6.2 Definitions 6.2.1 Order and equivalence relations 6.2.2 Preference relations 6.2.3 Definition of a criteria 6.2.4 Analysis 6.3 Various methods 6.3.1 Introduction Notations The representation 6.3.2 The ELECTRE I method 6.3.3 The ELECTRE IS method 6.3.4 The ELECTRE II method 6.3.5 The ELECTRE III method 6.3.6 The ELECTRE IV method 6.3.7 The ELECTRE TRI method 6.3.8 The PROMETHEE I method 6.3.9 The PROMETHEE II method 6.4 Annotated bibliography

VII 143 143 145 145 146 147 147 148 148 148 149 151 159 160 168 176 178 181 186 186

Part II Evaluation of methods and choice criterion 7

Performances measurement 7.1 Introduction 7.2 Error ratio 7.3 Generational distance 7.4 The STDGD metric 7.5 Maximal error to the tradeoff surface 7.6 Hyperarea and hyperarea ratio 7.7 Spacing 7.8 The HRS metric 7.9 Overall non dominated vector generation 7.10 Progress measure 7.11 Generational non dominated vector generation 7.12 Non dominated vectors addition 7.13 Waves 7.14 Zitzler metrics 7.14.1 The relative metrics 7.14.2 The absolute metrics 7.15 Laumanns metric 7.16 Annotated bibliography

189 189 190 191 193 194 197 199 201 201 202 204 204 205 205 206 208 209 210

VIII

Contents

8

Test functions of multiobjective optimization methods 8.1 Introduction 8.2 Deb test problems 8.2.1 The non convex Deb function 8.2.2 The discontinuous Deb function 8.2.3 The multifrontal Deb function 8.2.4 The non uniform Deb function 8.3 The Hanne test problems 8.3.1 The linear Hanne function 8.3.2 The convex Hanne function 8.3.3 The non convex Hanne function 8.3.4 The discontinuous Hanne function 8.3.5 The Hanne function with several efficiency areas 8.4 Annotated bibliography

211 211 211 212 214 215 217 218 218 221 223 224 227 227

9

Attempt to classify multiobjective optimization methods 9.1 Introduction 9.2 "Mathematical" classification of optimization methods 9.2.1 Introduction 9.2.2 The Erghott classification formula 9.2.3 Conclusion 9.3 Hierarchical classification of the multiobjective optimization methods 9.3.1 Introduction 9.3.2 A hierarchical graph A hierarchy for the treatment of the multiobjective problem . . A hierarchy for interactions A hierarchy for the optimization methods How to use this hierarchy 9.3.3 Classification of some methods 9.3.4 How to choose a method

229 229 230 230 230 233 233 233 233 233 236 237 238 239 241

Part III Cases study 10 Case study n°l: qualification of scientific software 10.1 Introduction 10.2 Description of the problem 10.3 To represent the tradeoff surface 10.4 Conclusion

247 247 247 249 254

11 Case study n°2: study of the extension of a telecommunication network 11.1 Network 11.2 Choice criteria 11.2.1 Parameter used to modelize the disponibility 11.2.2 Link between the disponibility and the cost

255 256 256 256 257

Contents

11.3

11.4

11.5 11.6

11.2.3 Costs The investments costs Maintenance/management costs Costs related to economical activity of the network (profitability, penalities) Study of a network extension 11.3.1 Problematic 11.3.2 Modelisation Variables Criteria Constraints 11.3.3 Needed data Frozen infrastructure of the network Variable infrastructure of the network Demands matrix Method of resolution 11.4.1 Definition of an admissible solution 11.4.2 Algorithm Initialization 1: encoding Initialization 2 : construction of an initial solution Evaluation 1 : computation of the current solution Evaluation 2 : localisation of this solution with respect to the current Pareto frontier Selection Breeding Stopping criterion Global algorithm Results Conclusion

12 Case study n°3: multicriteria decision tools to deal with bids . . . 12.1 Introduction 12.2 First generation model 12.2.1 Goal of the model 12.2.2 The criteria of the first generation model 12.2.3 Evolution of the first generation model 12.3 Understand the insufficiency of the first generation model 12.3.1 Examples of non discriminative criteria 12.3.2 Inefficient criterion due to the marking method 12.3.3 Criterion which are in fact constraints 12.4 Second generation model 12.4.1 Principle of the rebuilding of the model 12.4.2 Abandon of the uniqueness of the model principle 12.4.3 Architecture of the model families 12.4.4 Criteria

IX

257 257 258 258 258 258 259 259 259 259 260 260 260 260 261 261 262 262 262 263 264 265 265 266 267 267 268 271 271 271 271 272 274 275 276 278 280 280 281 282 282 283

X

Contents 12.4.5 Tuning of the model 12.4.6 Performance of the model 12.5 Conclusion

285 286 286

Conclusion

289

References

293

Index

315