An Interactive Fuzzy Satisficing Method for Multiobjective Linear Fractional Programming Problems with Fuzzy Parameters

Abstract

In this paper, we focus on multiobjective linear fractional programming problems with fuzzy parameters and present a new interactive fuzzy satisficing method for obtaining the satisficing solution of the decision maker on the basis of the linear programming method. The fuzzy parameters in the description of the objective functions and the constraints, which reflect the experts’ ambiguous understanding of the nature of the parameters in the problem-formulation process, are characterized by fuzzy numbers. The concept of α-multiobjective linear fractional programming and α-Pareto optimality is introduced based on the α-level sets of the fuzzy numbers. Through the interaction with the decision maker (DM), the fuzzy goals of the DM for each of the objective functions in α-multiobjective linear fractional programming are quantified by eliciting the corresponding membership functions. After determining the membership functions, in order to generate a candidate for the satisficing solution which is also α-Pareto optimal, if the DM specifies the degree α of the α-level sets and the reference membership values, the minimax problem is solved by combined use of the bisection method and the linear programming method, and the DM is supplied with the corresponding α-Pareto optimal solution together with the trade-off rates among the values of the membership functions and the degree α. By considering the current values of the membership functions and α as well as the trade-off rates, the DM acts on this solution by updating his/her reference membership values and/or the degree α. In this way, the satisficing solution for the DM can be derived efficiently from among an α-Pareto optimal solution set. On the basis of the proposed method, a time-sharing computer program is written and an illustrative numerical example is demonstrated.

This research was partially supported by the Kajima Foundation’s Research Grant.