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2000 | Book

Introduction Read chapter Read first chapter

Large Scale Interactive Fuzzy Multiobjective Programming

Decomposition Approaches

Author: Prof. Dr. Masatoshi Sakawa

Publisher: Physica-Verlag HD

Book Series : Studies in Fuzziness and Soft Computing

Included in: Professional Book Archive

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About this book

Simultaneous considerations of multiobjectiveness, fuzziness and block angular structures involved in the real-world decision making problems lead us to the new field of interactive multiobjective optimization for large scale programming problems under fuzziness. The aim of this book is to introduce the latest advances in the new field of interactive multiobjective optimization for large scale programming problems under fuzziness on the basis of the author's continuing research. Special stress is placed on interactive decision making aspects of fuzzy multiobjective optimization for human-centered systems in most realistic situations when dealing with fuzziness. The book is intended for graduate students, researchers and practitioners in the fields of operations research, industrial engineering, management science and computer science.

Table of Contents

Frontmatter
1. Introduction
Abstract
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 solutions 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.
Masatoshi Sakawa
2. Mathematical Preliminaries
Abstract
This chapter is devoted to mathematical preliminaries, which will be used in the remainder of this book. Starting with several basic definitions involving fuzzy sets, operations on fuzzy sets, especially fuzzy numbers, are outlined. Bellman and Zadeh’s approach to decision making in a fuzzy environment, called fuzzy decision, is then examined. Fundamental notions and methods of multiobjective, interactive multiobjective, and interactive fuzzy multiobjective linear programming are briefly reviewed. A brief discussion of genetic algorithms is also given.
Masatoshi Sakawa
3. The Dantzig-Wolfe Decomposition Method
Abstract
In this chapter, for convenience in our subsequent discussions in Chapters 4, 5 and 6, the Dantzig-Wolfe decomposition method for large scale linear programming problems with block angular structures is explained in detail. The basic procedure and some of its variants are also introduced.
Masatoshi Sakawa
4. Large Scale Fuzzy Linear Programming
Abstract
This chapter treats large scale linear programming problems with block angular structures for which the Dantzig-Wolfe decomposition method has been successfully applied. Considering the vague or fuzzy nature of human judgments, both the fuzzy goal and fuzzy constrains of the decision maker are introduced. Having determined the corresponding membership functions, following the convex fuzzy decision for combining them, under suitable conditions, it is shown that the formulated problem can be reduced to a number of independent linear subproblems and the satisficing solution for the decision maker is directly obtained just only by solving the subproblems. Moreover, even if the appropriate conditions are not satisfied, it is shown that the Dantzig-Wolfe decomposition method is applicable.
Masatoshi Sakawa
5. Large Scale Fuzzy Multiobjective Linear Programming
Abstract
As a multiobjective generalization of the previous chapter, this chapter treats large scale multiobjective linear programming problems with block angular structures. The fuzzy goals of the decision maker are quantified by eliciting the corresponding linear membership functions. Following the fuzzy decision of Bellman and Zadeh for combining them, fuzzy multiobjective linear programming, using the Dantzig-Wolfe decomposition method, is presented for obtaining the satisficing solution for the decision maker. Interactive fuzzy multiobjective linear programming for deriving the satisficing solution for the decision maker by updating the reference membership levels is also presented by utilizing the Dantzig-Wolfe decomposition method.
Masatoshi Sakawa
6. Large Scale Multiobjective Linear Programming with Fuzzy Numbers
Abstract
In this chapter, in contrast to the large scale multiobjective linear programming problems with block angular structures discussed thus far, by considering the experts’ imprecise or fuzzy understanding of the nature of the parameters in the problem-formulation process, large scale multiobjective linear programming problems with block angular structures involving fuzzy numbers are formulated. Through the introduction of extended Pareto optimality concepts, interactive decision making methods, using the Dantzig-Wolfe decomposition method, both without and with the fuzzy goals of the decision maker, for deriving a satisficing solution for the decision maker from the extended Pareto optimal solution set are presented together with detailed numerical examples.
Masatoshi Sakawa
7. Genetic Algorithms with Decomposition Procedures
Abstract
This chapter presents a detailed treatment of genetic algorithms with decomposition procedures as developed for large scale 0–1 knapsack problems with block angular structures. Through the introduction of a triple string representation and the corresponding decoding algorithm, it is shown that a potential solution satisfying not only block constraints but also coupling constraints can be obtained for each individual. The chapter also includes several numerical experiments.
Masatoshi Sakawa
8. Large Scale Fuzzy Multiobjective 0–1 Programming
Abstract
This chapter can be viewed as the multiobjective version of Chapter 7 and treats large scale multiobjective 0–1 knapsack problems with block angular structures by incorporating the fuzzy goals of the decision maker. On the basis of the genetic algorithm with decomposition procedures presented in Chapter 7, interactive fuzzy multiobjective 0–1 programming as well as fuzzy multiobjective 0–1 programming, both proposed by the authors, are explained in detail together with several numerical experiments.
Masatoshi Sakawa
9. Large Scale Interactive Multiobjective 0–1 Programming with Fuzzy Numbers
Abstract
In this chapter, as the 0–1 version of Chapter 6, large scale multiobjective multidimensional 0–1 knapsack problems with block angular structures involving fuzzy numbers are formulated. Along the same line as in Chapter 6, interactive decision making methods using the genetic algorithm with decomposition procedures, both without and with the fuzzy goals of the decision maker, for deriving a satisficing solution efficiently from an extended Pareto optimal solution set are presented. Several numerical experiments are also given.
Masatoshi Sakawa
10. Further Research Directions
Abstract
In the preceding chapters, on the basis of the author’s continuing research works, we have discussed the latest advances in the new field of interactive multiobjective optimization for large scale linear and 0–1 programming problems with the block angular under fuzziness. However, many other interesting related topics, including multiobjective linear fractional and nonlinear programming problems with block angular structures under fuzziness, still remain to be further explored. In this chapter, being limited by space, we only outline the latest major research results of these topics because of their close relationship to our discussions in the preceding chapters.
Masatoshi Sakawa
Backmatter
Metadata
Title
Large Scale Interactive Fuzzy Multiobjective Programming
Author
Prof. Dr. Masatoshi Sakawa
Copyright Year
2000
Publisher
Physica-Verlag HD
Electronic ISBN
978-3-7908-1851-2
Print ISBN
978-3-662-00386-2
DOI
https://doi.org/10.1007/978-3-7908-1851-2