On formalization of a general fuzzy mathematical problem

Abstract

The problem considered in this paper concerns decision-making on the basis of information presented in a fuzzy form. It differs from the usual mathematical programming problems of maximizing a given function over a given set of alternatives mainly in that the values of such function are fuzzy and are represented by fuzzy sets. The question is how to choose alternatives from the given set which give in some sense “the best” fuzzy value of this function. The fuzzy-valued function in question may be thought of as a fuzzy utility function specifying fuzzy utility estimates to the alternatives, or as a performance function of a system with reaction to choices of alternatives (controls) allowing only fuzzy description. The values of this function have the form of fuzzy subsets of some universal set of estimates (or reactions) and a binary preference relation (generally fuzzy) is assumed to be specified in this set. In the paper this relation is extended from the elements of the universal set onto fuzzy subsets of this set. Some properties of this induced relation are studied. This relation is used to extract a fuzzy subset of nondominated alternatives and maximal elements of this subset are suggested as rational choices. It is shown that under some assumptions alternatives exist which are in fact unfuzzily dominated thus serving as unfuzzy solutions to a fuzzily formulated problem.