Fuzzy Programming and the Multicriteria Decision Problem

Abstract

This paper studies the use of fuzzy programming in determining undominated, and only un-dominated, solutions to multicriteria decision problems. The multicriteria problem is not fuzzy, and fuzzy programming is employed to generate the set of undominated solutions. Membership functions are defined in the usual way when the objective is to maximize all the objective functions in the multi-criteria decision problem. We first consider the product operator as a method of combining the membership functions. We show that the set of solutions to the fuzzy program is the Pareto optimal set for all multicriteria decision problems. We also discuss an interactive application and a solution algorithm for solving the fuzzy program. We next discuss the minimum operator as a procedure for combining the membership functions. We show that the set of solutions to the fuzzy program always contains the set of undominated solutions, but some solutions to the fuzzy program may be dominated. We then study arbitrary methods G of combining the membership functions. We show that the set of solutions to the fuzzy program is the Pareto optimal set for all multicriteria decision problems if and only if G has the dominance and the zero properties , We then apply these results to some new methods of combining membership functions that have recently appeared.