Sakawa / Nishizaki / Katagiri Fuzzy Stochastic Multiobjective Programming

1. Auflage 2011
ISBN: 978-1-4419-8402-9
Verlag: Springer US
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 159, 264 Seiten

Reihe: International Series in Operations Research & Management Science

Although studies on multiobjective mathematical programming under uncertainty have been accumulated and several books on multiobjective mathematical programming under uncertainty have been published (e.g., Stancu-Minasian (1984); Slowinski and Teghem (1990); Sakawa (1993); Lai and Hwang (1994); Sakawa (2000)), there seems to be no book which concerns both randomness of events related to environments and fuzziness of human judgments simultaneously in multiobjective decision making problems. In this book, the authors are concerned with introducing the latest advances in the field of multiobjective optimization under both fuzziness and randomness on the basis of the authors' continuing research works. Special stress is placed on interactive decision making aspects of fuzzy stochastic multiobjective programming for human-centered systems under uncertainty in most realistic situations when dealing with both fuzziness and randomness. Organization of each chapter is briefly summarized as follows:Chapter 2 is devoted to mathematical preliminaries, which will be used throughout the remainderof the book. Starting with basic notions and methods of multiobjective programming, interactivefuzzy multiobjective programming as well as fuzzy multiobjective programming is outlined.In Chapter 3, by considering the imprecision of decision maker's (DM's) judgment for stochasticobjective functions and/or constraints in multiobjective problems, fuzzy multiobjective stochasticprogramming is developed. In Chapter 4, through the consideration of not only the randomness of parameters involved inobjective functions and/or constraints but also the experts' ambiguous understanding of the realized values of the random parameters, multiobjective programming problems with fuzzy random variables are formulated. In Chapter 5, for resolving conflict of decision making problems in hierarchical managerial orpublic organizations where there exist two DMs who have different priorities in making decisions, two-level programming problems are discussed. Finally, Chapter 6 outlines some future research directions.

Autoren/Hrsg.

Sakawa, Masatoshi

Nishizaki, Ichiro

Katagiri, Hideki

Inhaltsverzeichnis

1;Preface;6
2;Contents;9
3;Introduction and Historical Remarks;11
3.1;1.1 Background;11
3.2;1.2 Description of contents;17
4;Fundamentals;20
4.1;2.1 Fuzzy programming;20
4.1.1;2.1.1 Fuzzy sets;20
4.1.2;2.1.2 Fuzzy goals and Fuzzy constraints;23
4.1.3;2.1.3 Linear programming problems with fuzzy parameters;25
4.1.3.1;2.1.3.1 Possibility-based model;25
4.1.3.2;2.1.3.2 Level set-based model;27
4.2;2.2 Stochastic programming;29
4.2.1;2.2.1 Random variables;29
4.2.2;2.2.2 Two-stage programming;30
4.2.3;2.2.3 Chance constraint programming;33
4.3;2.3 Multiobjective programming;35
4.3.1;2.3.1 Multiobjective programming problem;35
4.3.2;2.3.2 Interactive multiobjective programming;36
4.3.3;2.3.3 Fuzzy multiobjective programming;37
4.4;2.4 Two-level programming;41
4.4.1;2.4.1 Fuzzy programming for two-level programming;41
4.4.2;2.4.2 Stackelberg solution to two-level programming problem;45
4.5;2.5 Genetic algorithms;49
4.5.1;2.5.1 Fundamental elements in genetic algorithms;49
4.5.1.1;2.5.1.1 Representation of individuals;50
4.5.1.2;2.5.1.2 Fitness function and scaling;51
4.5.1.3;2.5.1.3 Genetic operators;51
4.5.2;2.5.2 Genetic algorithm for integer programming;53
4.5.2.1;Computational procedure of GADSLPRRSU;56
5;Fuzzy Multiobjective Stochastic Programming;57
5.1;3.1 Fuzzy multiobjective stochastic linear programming;57
5.1.1;3.1.1 Expectation and variance models;58
5.1.1.1;3.1.1.1 Expectation model;60
5.1.1.2;3.1.1.2 Variance model;64
5.1.1.3;3.1.1.3 Numerical example;67
5.1.2;3.1.2 Probability and fractile models;70
5.1.2.1;3.1.2.1 Probability model;70
5.1.2.2;3.1.2.2 Numerical example;78
5.1.2.3;3.1.2.3 Fractile model;79
5.1.2.4;3.1.2.4 Numerical example;83
5.1.3;3.1.3 Simple recourse model;85
5.1.3.1;Interactive fuzzy satisficing method for the simple recourse model;89
5.1.3.2;3.1.3.1 Numerical example;89
5.2;3.2 Extensions to integer programming;91
5.2.1;3.2.1 Expectation and variance models;92
5.2.1.1;3.2.1.1 Expectation model;93
5.2.1.2;3.2.1.2 Variance model;93
5.2.1.3;Interactive fuzzy satisficing method for the variance model with integer decision variables;95
5.2.1.4;3.2.1.3 Numerical example;96
5.2.2;3.2.2 Probability and fractile models;98
5.2.2.1;3.2.2.1 Probability model;98
5.2.2.2;Interactive fuzzy satisficing method for the probability model with integer decision variables;100
5.2.2.3;3.2.2.2 Numerical example;100
5.2.2.4;3.2.2.3 Fractile model;101
5.2.2.5;Interactive fuzzy satisficing method for the fractile model with integer decision variables;103
5.2.3;3.2.3 Simple recourse model;103
5.2.3.1;Interactive fuzzy satisficing method for the simple recourse model with integer decision variables;105
5.2.3.2;3.2.3.1 Numerical example;106
6;Multiobjective Fuzzy Random Programming;108
6.1;4.1 Multiobjective fuzzy random linear programming;108
6.1.1;4.1.1 Possibility-based expectation and variance models;111
6.1.1.1;4.1.1.1 Possibility-based expectation model;114
6.1.1.2;4.1.1.2 Possibility-based variance model;120
6.1.1.3;4.1.1.3 Numerical example;124
6.1.2;4.1.2 Possibility-based probability and fractile models;125
6.1.2.1;4.1.2.1 Possibility-based probability model;129
6.1.2.2;4.1.2.2 Possibility-based fractile model;134
6.1.2.3;4.1.2.3 Numerical example;137
6.1.3;4.1.3 Level set-based models;138
6.1.3.1;4.1.3.1 Level set-based expectation model;141
6.1.3.2;4.1.3.2 Level set-based variance model;144
6.1.3.3;4.1.3.3 Level set-based probability model;147
6.1.3.4;4.1.3.4 Level set-based fractile model;152
6.1.3.5;4.1.3.5 Numerical example;155
6.2;4.2 Extensions to integer programming;156
6.2.1;4.2.1 Possibility-based expectation and variance models;157
6.2.1.1;4.2.1.1 Possibility-based expectation model;157
6.2.1.2;4.2.1.2 Possibility-based variance model;159
6.2.1.3;4.2.1.3 Numerical example;160
6.2.2;4.2.2 Possibility-based probability and fractile models;161
6.2.2.1;4.2.2.1 Possibility-based probability model;162
6.2.2.2;4.2.2.2 Possibility-based fractile model;163
6.2.2.3;4.2.2.3 Numerical example;165
6.2.3;4.2.3 Level set-based models;166
6.2.3.1;4.2.3.1 Level set-based expectation model;167
6.2.3.2;4.2.3.2 Level set-based variance model;169
6.2.3.3;4.2.3.3 Level set-based probability model;171
6.2.3.4;4.2.3.4 Level set-based fractile model;173
6.2.3.5;4.2.3.5 Numerical example;174
7;Stochastic and Fuzzy Random Two-Level Programming;176
7.1;5.1 Cooperative two-level programming;176
7.1.1;5.1.1 Stochastic two-level linear programming;177
7.1.1.1;5.1.1.1 Expectation model;179
7.1.1.2;5.1.1.2 Variance model;179
7.1.1.3;5.1.1.3 Numerical example;182
7.1.2;5.1.2 Fuzzy random two-level linear programming;184
7.1.2.1;5.1.2.1 Possibility-based probability model;185
7.1.2.2;Interactive fuzzy programming in the possibility-based probability model;189
7.1.2.3;5.1.2.2 Level set-based fractile model;189
7.1.2.4;5.1.2.3 Numerical example;193
7.1.3;5.1.3 Extensions to integer programming;195
7.1.3.1;5.1.3.1 Expectation model for stochastic two-level integer programming problems;195
7.1.3.2;5.1.3.2 Variance model for stochastic two-level integer programming problems;198
7.1.3.3;5.1.3.3 Numerical example;200
7.1.3.4;5.1.3.4 Possibility-based probability model for fuzzy random two-level integer programming problems;201
7.1.3.5;5.1.3.5 Level set-based fractile model for fuzzy random two-level programming;205
7.2;5.2 Noncooperative two-level programming;207
7.2.1;5.2.1 Stochastic two-level linear programming;208
7.2.1.1;5.2.1.1 Expectation model;209
7.2.1.2;5.2.1.2 Variance model;209
7.2.1.3;5.2.1.3 Numerical example;211
7.2.2;5.2.2 Fuzzy random two-level linear programming;214
7.2.2.1;5.2.2.1 Possibility-based probability model;214
7.2.2.2;5.2.2.2 Level set-based fractile model;218
7.2.2.3;5.2.2.3 Numerical example;220
7.2.3;5.2.3 Extensions to integer programming;222
7.2.3.1;5.2.3.1 Expectation model for stochastic two-level integer programming problems;223
7.2.3.2;5.2.3.2 Variance model for stochastic two-level integer programming problems;224
7.2.3.3;5.2.3.3 Possibility-based probability model for fuzzy random two-level integer programming problems;225
7.2.3.4;5.2.3.4 Level set-based fractile model for fuzzy random two-level integer programming problems;227
7.2.3.5;5.2.3.5 Numerical example;229
8;Future Research Directions;231
8.1;6.1 Random fuzzy variable;231
8.2;6.2 Random fuzzy linear programming;233
8.2.1;6.2.1 Possibility-based probability model;234
8.2.2;6.2.2 Possibility-based fractile model;237
8.3;6.3 Multiobjective random fuzzy programming;238
8.3.1;6.3.1 Possibility-based probability model;239
8.3.2;6.3.2 Possibility-based fractile model;241
8.4;6.4 Random fuzzy two-level programming;243
8.4.1;6.4.1 Possibility-based probability model;244
8.4.2;6.4.2 Possibility-based fractile model;248
9;References;252
10;Index;267