Masahiro Inuiguchi

Possibilistic linear programming: a brief review of fuzzy mathematical programming and a comparison with stochastic programming in portfolio selection problem

In this paper, we review some fuzzy linear programming methods and techniques from a practical point of view. In the rst part, the general history and the approach of fuzzy mathematical programming are introduced. Using a numerical example, some models of fuzzy linear programming are described. In the second part of the paper, fuzzy mathematical programming approaches are compared to stochastic programming ones. The advantages and disadvantages of fuzzy mathematical programming approaches are exemplied in the setting of an optimal portfolio selection problem. Finally, some newly developed ideas and techniques in fuzzy mathematical programming are briey reviewed. c 2000 Elsevier Science B.V. All rights reserved.

Portfolio selection under independent possibilistic information

This paper deals with a portfolio selection problem with independently estimated possibilistic return rates. Under such a circumstance, a distributive investment has been regarded as a good solution in the traditional portfolio theory. However, the conventional possibilistic approach yields a concentrated investment solution. Considering the reason why a distributive investment is advocated, a new approach to the possibilistic portfolio selection is proposed.

Fuzzy Programming: A Survey of Recent Developments

In this paper, a survey of the major types of fuzzy programming is provided classifying into the following three categories; mathematical programming with vagueness, mathematical programming with ambiguity and mathematical programming with vagueness and ambiguity.

Variable-precision dominance-based rough set approach and attribute reduction

In this paper, a variable-precision dominance-based rough set approach (VP-DRSA) is proposed together with several VP-DRSA-based approaches to attribute reduction. The properties of VP-DRSA are shown in comparison to previous dominance-based rough set approaches. An advantage of VP-DRSA over variable-consistency dominance-based rough set approach in decision rule induction is emphasized. Some relations among the VP-DRSA-based attribute reduction approaches are investigated.

Fuzzy rough sets and multiple-premise gradual decision rules

We propose a new fuzzy rough set approach which, differently from most known fuzzy set extensions of rough set theory, does not use any fuzzy logical connectives (t-norm, t-conorm, fuzzy implication). As there is no rationale for a particular choice of these connectives, avoiding this choice permits to reduce the part of arbitrary in the fuzzy rough approximation. Another advantage of the new approach is that it is based on the ordinal properties of fuzzy membership degrees only. The concepts of fuzzy lower and upper approximations are thus proposed, creating a base for induction of fuzzy decision rules having syntax and semantics of gradual rules. The proposed approach to rule induction is also interesting from the viewpoint of philosophy supporting data mining and knowledge discovery, because it is concordant with the method of concomitant variations by John Stuart Mill. The decision rules are induced from lower and upper approximations defined for positive and negative relationships between credibility degrees of multiple premises, on one hand, and conclusion, on the other hand.

Goal programming problems with interval coefficients and target intervals

Abstract In conventional goal programming, the coefficients of objective functions and constraints, and target values are determined as crisp values. However, it is not frequent that the coefficients and the target values are known precisely. In such cases, the coefficients and target values should be represented by intervals reflecting the imprecision. This paper treats goal programming problems in which coefficients and target values are given the intervals. It is shown that four formulations of the problems can be considered. The properties of the four formulated problems are investigated. An example is given to demonstrate the differences between the four formulations.

Theory and methodologyMinimax regret solution to linear programming problems with an interval objective function

In this paper, a linear programming problem with an interval objective function is treated. First, the previous approaches to this problem are reviewed and the drawbacks are pointed out. To improve the drawbacks, a new approach to this problem is proposed by introducing the minimax regret criterion as used in decision theory. The properties of minimax regret solution are investigated. In order to obtain the minimax regret solution, a method of solution by a relaxation procedure is proposed. It is shown that the solution is obtained by repetitional use of the simplex method. A numerical example is given to illustrate the proposed solution method.

Robust optimization under softness in a fuzzy linear programming problem

Abstract In this paper, we discuss the softness and the robustness of the optimality in the setting of linear programming problems with a fuzzy objective function. A fuzzy goal defined on the deviation from the optimal value is introduced in order to define the soft-optimal solution. Fuzzy coefficients are regarded as possibility distributions. A necessity measure based on the possibility distribution is used for defining a necessarily optimal solution, i.e., a robust-optimal solution. Since a necessarily optimal solution does not exist in many cases, a necessarily soft-optimal solution is defined. A solution algorithm for the best necessarily soft-optimal solution is proposed.

A solution algorithm for fuzzy linear programming with piecewise linear membership functions

Abstract This paper deals with the fuzzy linear program with continuous piecewise linear membership functions. A fuzzy linear program with piecewise linear membership functions has been considered by Hannan and Nakamura. Hannans method can solve the problems only when all membership functions are concave in the range (0, 1) and Nakamuras method needs to use the linear programming technique repetitionally. We propose a technique to solve the problem using a standard linear programming when membership functions are strictly quasiconcave and the minimum operator is adopted for aggregating fuzzy goals. The repetitional use of linear programming technique is not required in our method.

Satisficing solutions and duality in interval and fuzzy linear programming

In this paper, we introduce a class of fuzzy linear programming problems and define the concepts of feasible and satisficing solutions--the necessary tools for dealing with such problems. In this way, we show that the class of crisp (classical) LP problems can be embedded into the class of FLP ones. Moreover, for FLP problems we define the concept of duality and prove the weak and strong duality theorems. Further, we define a class of interval linear programming problems as a special subclass of FLP problems and apply the previous results to this special case.