Decision AidingSolving a multiobjective possibilistic problem through compromise programming☆
Introduction
In many complex decision situations data available may not be sufficient to define the parameters of a real problem in an exact or objective form. It is possible to handle imprecision through the possibility theory.
Possibility theory was proposed by Zadeh (1978) and developed by Dubois and Prade (1988); in it, fuzzy parameters are associated with possibility distributions in the same way that random variables are associated with probability distributions. Possibility distributions are represented as normal convex fuzzy sets, such as L–R fuzzy numbers. Since the 1980s, the possibility theory has become more and more important in the decision field and several methods have been developed to solve possibilistic programming problems (see Buckley, 1988, Buckley, 1989; Lai and Hwang, 1992; Julien, 1994; Arenas et al., 1998a, Arenas et al., 1998b, Arenas et al., 1999a, Arenas et al., 1999b, Arenas et al., 2001; Jiménez et al., 2000; Saati et al., 2001).
We shall consider here a multiobjective possibilistic linear programming problem (FP-MOLP) in which all the parameters are fuzzy. We suppose that they are represented by fuzzy numbers described by their possibility distribution estimated by the analyst from the information supplied by the Decision Maker (Tanaka, 1987).
The uncertain and/or imprecise nature of the problem's parameters involves two main problems: feasibility and optimality. Feasibility may be handled by comparing fuzzy numbers. In this paper we use a fuzzy relationship to compare fuzzy numbers (Jiménez, 1996) that verifies suitable properties and that, besides, is computationally efficient to solve linear problems because it preserves its linearity. Since this fuzzy preference relation does not admit degrees of indifference, we have defined––in Section 2––the concept of β-indifference.
Optimality is handled through Compromise Programming (CP). CP is a well-known Multiple Criteria Decision Making approach developed by Yu (1985) and Zeleny (1973). The basic idea in CP is the identification of an ideal solution as a point where each attribute under consideration achieves its optimum value. Zeleny states that alternatives that are closer to the ideal are preferred to those that are farther from it because being as close as possible to the perceived ideal is the rationale of human choice.
As a natural extension of the concept of an ideal solution for a crisp multiobjective linear programming problem, we are going to introduce the concept of fuzzy ideal solution. The accuracy between the fuzzy ideal solution and the objective values is evaluated handling fuzzy parameters through their expected intervals and using some of the interval results developed in this work. In Section 3 we define the discrepancy set and the discrepancy between intervals and also between fuzzy numbers. From these we will transform the initial multiobjective possibilistic linear programming problem (FP-MOLP) into a family of crisp problems.
Section snippets
Feasibility in the multiobjective possibilistic linear programming problem
We shall consider the following multiobjective possibilistic programming problem:where xt=(x1,x2,…,xn) is the crisp decision vector, is composed of fuzzy vectors which are the fuzzy coefficients of the k considered objectives, is the fuzzy technological matrix and are fuzzy parameters. We suppose that all fuzzy parameters of the problem are given by
Optimality in the multiobjective possibilistic linear programming problem
In order to apply the CP approach to solve the problem, we need to obtain the fuzzy ideal solution of β-FP-MOLP problem. For this, it is necessary to solve the following mono-objective fuzzy linear programming problems:where r=1,…,k.
To solve each (β-FLP) problem we use the solving method proposed by Arenas et al. (1998a). This method gives fuzzy solutions in the objectives space defined by their possibility distribution. The method is based on the extension principle and
Numerical example
We shall consider the following multiobjective linear program with fuzzy parameters:where fuzzy coefficients are characterized by triangular fuzzy numbers.
We shall solve problem (23) setting w1=w2=w3=1. As a first step we ask the DM to fix the
Summary and conclusions
In this paper we have proposed a solving method based on compromise programming for a fuzzy multiobjective linear programming problem whose parameters are imprecise data. Normally, they are either given by the Decision Maker who has imprecise information and/or expresses his considerations subjectively, or by statistical inference from past data whose reliability is doubtful. Therefore, it is reasonable to construct a model reflecting imprecise data or ambiguity in terms of fuzzy sets.
The
Acknowledgements
We are grateful to the referees for their useful comments and suggestions.
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This work was supported by the Spanish Department of Science and Technology (project BFM2000-0010). This support is gratefully acknowledged.