Large Scale Interactive Fuzzy Multiobjective Programming. Decomposition Approaches

223 Pages·2000·5.154 MB·English

by  Masatoshi Sakawa

Masatoshi Sakawa
Large Seale
Interaetive Fuzzy
Multiobjeetive
Programming
Decomposition Approaches
Prof. Dr. Masatoshi Sakawa
Department of Industrial
and Systems Engineering
Faculty of Engineering
Hiroshima University
1-4-1 Kagamiyama
Higashi-Hiroshima 739-8527
Japan
E-mail: [email protected]
ISSN 1434-9922
Cataloging-in-Publication Data applied for
Die Deutsche Bibliothek - CIP-Einheitsaufnabme
Sakawa. Masatoshi: Large scale interactive fuzzy multiobjective programming: de
composition approaches: with 25 tables / Masatoshi Sakawa. - Heidelberg; New
York: Physica-Verl.. 2000
(Studies in fuzziness and soft computing; Vol. 48)
ISBN 978-3-662-00386-2 ISBN 978-3-7908-1851-2 (eBook)
DOI 10.1007/978-3-7908-1851-2
Physica-Verlag is a company in the BertelsmannSpringer pubJishing group.
© Physica-Verlag Heidelberg 2000
Softcover reprint of the hardcover 1s t edition 2000
Hardcover Design: Erich Kirchner. Heidelberg
SPIN 10763587 88/2202-5 4 3 2 I 0 - Printed on acid-free paper
Preface
The main characteristics of the real-world decision making problems facing
humans today are large scale and have multiple objectives including eco
nomic, environmental, social, and technical issues.
Hence, actual decision making problems formulated as mathematical pro
gramming problems involve very large numbers of variables and constraints.
Due to the high dimensionality of the problems, it becomes difficult to obtain
optimal solutions for such large scale programming problems. Fortunately,
however, most of the large scale programming problems arising in applica
ti on almost always have a special structure that can be exploited. One fa
miliar structure is the block angular structure to the constraints that can
be used to formulate the subproblems for reducing both the processing time
and memory requirements. With this observation, after the publication of the
Dantzig-Wolfe decomposition method, both the dual and primal decomposi
tion methods for solving large scale nonlinear programming problems with
block angular structures have been proposed. Observe that the term large
scale programming problems frequently means mathematical programming
problems with block angular structures involving large numbers of variables
and constraints.
Furthermore, it seems natural that the consideration of many objectives in
the actual decision making process requires multiobjective approaches rather
than a single objective. One of the major systems-analytic multiobjective ap
proaches to decision making under constraints is multiobjective programming
as a generalization of traditional single-objective programming. For such mul
tiobjective programming problems, it is significant to realize that multiple ob
jectives are often noncommensurable and conflict with each other. With this
observation, in multiobjective programming problems, the notion of Pareto
optimality or efficiency has been introduced instead of the optimality concept
for single-objective problems. However, decisions with Pareto optimality or
efficiency are not uniquely determined; the final decision must be selected by
adecision maker, which well represents the subjective judgments, from the
set of Pareto optimal or efficient solutions.
However, recalling the vagueness or fuzziness inherent in human judg
ments, two types of inaccuracies in human judgments should be incorporated
in multiobjective programming problems. One is the experts' ambiguous un
derstanding of the nature of the parameters in the problem-formulation pro
cess, and the other is the fuzzy goal of the decision maker for each of the
objective functions. For handling and tackling such kinds of imprecisions or
vaguenesses in human beings, it is not hard to imagine that the conventional
multiobjective programming approaches, such as a deterministic or even a
probabilistic approach, cannot be applied.
Naturally, simultaneous considerations of block angular structures, multi
objectiveness and fuzziness involved in the real-world decision making prob
lems lead us to the new field of interactive multiobjective optimization for
large scale programming problems under fuzziness. In this book, the au
thor is concerned with introducing the latest advances in the new field of
interactive multiobjective optimization for large scale linear and 0-1 pro
gramming problems under fuzziness on the basis of the author's continuing
research. As further research directions, some of the most important related
results, including interactive multiobjective optimization for linear fractional
and nonlinear programming problems with block angular structures are also
presented. Special stress is placed on interactive decision making aspects of
fuzzy multiobjective optimization for human-centered systems in most real
istic situations when dealing with fuzziness.
The intended readers of this book are senior undergraduate students,
graduate students, researchers and practitioners in the fields of operations
research, industrial engineering, management science, computer science, and
other engineering disciplines that deal with the subjects of interactive multi
objective optimization for large scale programming problems under fuzziness.
In order to master all the material discussed in this book, the readers would
probably be required to have some background in linear algebra and mathe
matical programming. However, by skipping the mathematical details, much
can be learned about large scale interactive fuzzy multiobjective program
ming for human-centered systems in most realistic settings without prior
mathematical sophistication.
The author would like to express his sincere appreciation to Professor
Janusz Kacprzyk of Polish Academy of Sciences, whose arrangements and
warm encouragement made it possible for this book to be written. Special
thanks should also be extended to Professor Yoshikazu Sawaragi, chairman of
the Japan Institute of Systems Research and emeritus professor of Kyoto Uni
versity, Department of Applied Mathematics and Physics, for his invariant
stimulus and encouragement ever since the author's student days at Kyoto
U niversity. The author is also thankful to Dr. Masahiro Inuiguchi of Osaka
University and Dr. Kazuya Sawada of Matsushita Electric Works, Ltd. for
their contributions to Chapters 4 and 5, Dr. Kosuke Kato of Hiroshima Uni
versity for his contribution to Chapters 7, 8 and 9, and Section 10.1, and
Dr. Hitoshi Yano of Nagoya City University for his contribution to Section
10.2. Further thanks are due to Dr. Kosuke Kato of Hiroshima University for
reviewing parts of the manuscript and far his helpful comments and sugges
tions. The author also wishes to thank all of his undergraduate and graduate
students at Hiroshima University. Special thanks go to his former gradu
ate students Ryuuji Mizouchi, Hideki Mohara, Keiichi Kubota and Toshihiro
Ikegame of Hiroshima University for their invaluable assistance through dis
cussions and computer works. Finally, the author would like to thank Dr.
Martina Bihn, Physica-Verlag, Heidelberg, for her assistance in the publica
tion of this book.
Hiroshima, December 1999 Masatoshi Sakawa
Contents
1. Introduetion.............................................. 1
1.1 Introduction and historical remarks . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Organization of the book . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 6
2. Mathematieal Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1 Fuzzy sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Fuzzy numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . .. . . . . . .. 14
2.3 Fuzzy decision ......................................... 15
2.4 Multiobjective linear programming ....................... 19
2.5 Interactive multiobjective linear programming. . . . . . . . . . . . .. 21
2.6 Interactive fuzzy multiobjective linear programming ........ 24
2.7 Genetic algorithms ..................................... 32
2.7.1 Outline of genetic algorithms ...................... 32
2.7.2 Genetic algorithms with double strings . . . . . . . . . . . . .. 35
3. The Dantzig-Wolfe Deeomposition Method . . . . . . . . . . . . . .. 41
3.1 Linear programming problems with block angular structures. 41
3.2 Development of the decomposition algorithm. . . . . . . . . . . . . .. 45
3.3 Initial feasible basic solution . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 48
3.4 Unbounded subproblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 48
3.5 Restricted master problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 50
4. Large Seale Fuzzy Linear Programming. ........ .... .. .. .. 53
4.1 Linear programming problems with block angular structures. 53
4.2 Fuzzy goal and fuzzy constraints ......................... 54
4.3 Fuzzy linear programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 56
4.4 Numerical example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . .. 62
5. Large Scale Fuzzy Multiobjective Linear Programming . .. 65
5.1 Multiobjective linear programming problems with block an-
gular structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 65
5.2 FUzzy goals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 66
5.3 FUzzy multiobjective linear programming . . . . . . . . . . . . . . . . .. 68
5.4 Interactive fuzzy multiobjective linear programming ........ 78
6. Large Scale Multiobjective Linear Programming with Fuzzy
Numbers ................................................. 85
6.1 Introduction........................................... 85
6.2 Interactive multiobjective linear programming with fuzzy
numbers ............................................... 86
6.2.1 Problem formulation and solution concepts ...... . . .. 86
6.2.2 Minimax problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 90
6.2.3 Interactive programming . . . . . . . . . . . . . . . . . . . . . . . . .. 92
6.2.4 Numerical example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 94
6.3 Interactive fuzzy multiobjective linear programming with
fuzzy numbers ......................................... 98
6.3.1 FUzzy goals and solution concepts .............. . . .. 98
6.3.2 Minimax problems ................................ 100
6.3.3 Interactive fuzzy programming ..................... 104
6.3.4 Numerical example ............................... 107
6.3.5 Numerical experiments ............... '. ............ 109
6.4 Conclusion ............................................ 115
7. Genetic Algorithms with Decomposition Procedures ...... 117
7.1 Introduction ........................................... 117
7.2 Multidimensional 0-1 knapsack problems with block angular
structures ............................................. 118
7.3 Genetic algorithms with decomposition procedures .......... 119
7.3.1 Coding and decoding ............................. 119
7.3.2 Fitness and scaling ............................... 122
7.3.3 Reproduction .................................... 122
7.3.4 Crossover ....................................... 123
7.3.5 Mutation ........................................ 125
7.3.6 Genetic algorithms with decomposition procedures .... 126
7.4 Numerical experiments .................................. 128
8. Large Scale Fuzzy Multiobjective 0-1 Programming ....... 135
8.1 Multiobjective multidimensional 0-1 knapsack problems with
block angular structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 135
8.2 Fuzzy goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 136
8.3 Fuzzy multiobjective 0-1 programming .................... 137
8.3.1 Fuzzy programming through genetic algorithms .. . . .. 137
8.3.2 Numerical experiments ............................ 139
8.4 Interactive fuzzy multiobjective 0-1 programming ........... 144
8.4.1 Interactive fuzzy programming through genetic algo-
rithms .......................................... 144
8.4.2 Numerical experiments ............................ 147
9. Large Scale Interactive Multiobjective 0-1
Programming with Fuzzy Numbers ....................... 149
9.1 Introduction ........................................... 149
9.2 Interactive multiobjective 0-1 programming with fuzzy
numbers ............................................... 150
9.2.1 Problem formulation and solution concepts .......... 150
9.2.2 Interactive programming through genetic algorithms .. 153
9.2.3 Numerical experiments ............................ 156
9.3 Interactive fuzzy multiobjective 0-1 programming with fuzzy
numbers ............................................... 160
9.3.1 Fuzzy goals and solution concepts .................. 160
9.3.2 Interactive fuzzy programming through genetic algo-
rithms .......................................... 161
9.3.3 Numerical experiments ............................ 164
9.4 Conclusion ............................................ 170
10. Further Research Directions .............................. 173
10.1 Large scale fuzzy multiobjective linear fractional
programming .......................................... 173
10.1.1 Introduction ..................................... 173
10.1.2 Problem formulation .............................. 175
10.1.3 Minimax problems ................................ 177
10.1.4 Interactive fuzzy multiobjective programming ........ 181
10.1.5 Numerical example ............................... 183
10.1.6 Conclusion ...................................... 186
10.2 Large scale fuzzy multiobjective nonlinear programming ..... 187
10.2.1 Multiobjective nonlinear programming problems with
block angular structures ........................... 188
10.2.2 Dual decomposition method with fuzzy goals ........ 188
10.2.3 Primal decomposition method with fuzzy goals ....... 196
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
Index ......................................................... 215
1. Introd uction
1.1 Introduction and historical remarks
The increasing complexity of modern-day society has brought new problems
involving very large numbers of variables and constraints. Due to the high
dimensionality of the problems, it becomes difficult to obtain optimal solu
tions for such large scale programming problems. Fortunately, however, most
of the large scale programming problems arising in application almost always
have a special structure that can be exploited. One familiar structure is the
block angular structure to the constraints that can be used to formulate the
subproblems.
From such a point of view, in the early 1960s, Dantzig and Wolfe [13, 14)
introduced the elegant and attractive decomposition method for linear pro
gramming problems. The Dantzig-Wolfe decomposition method, when ap
plied to large scale linear programming problems with block angular struc
tures, implies that the entire problem can be solved by solving a coordinated
sequence of independent subproblems, and the process of coordination is
shown to be finite.
After the publication of the Dantzig-Wolfe decomposition method, the
subsequent works on large scale linear and nonlinear programming problems
with block angular structures have been numerous [13, 14, 32, 34, 39, 40,
41,42,58,69, 104, 123, 137). Among the nonlinear extensions of the decom
position method, the dual decomposition method proposed by Lasdon [57)
and the primal decomposition method proposed by Geoffrion (26) are well
known for solving large scale nonlinear programming problems with block
angular structures. Abrief and unified survey of major approaches to large
scale mathematical programming proposed before 1970 can be found in the
papers by Geoffrion [24, 25). More comprehensive discussions of the major
large scale mathematical programming proposed through the early 1970s can
also be found in Lasdon [58) and Wismer (137).