On a Solution Set to a Mathematical Programming Problem With a Fuzzy Set of Constraint Indices

Abstract

The present paper investigates a mathematical programming problem with a fuzzy set of constraint indices. To formalize the set of feasible alternatives, the concept of intersection of crisp sets with a fuzzy set of operands is used. The result of this operation is a type-2 fuzzy set (T2FS) (a fuzzy set whose membership function takes a fuzzy set of values). Furthermore, the corresponding membership function is given. It is shown that the T2FS of feasible alternatives can be represented in the form of the collection of T2FSs with constant secondary grades. The notion of the solution T2FS to the mathematical programming problem with a fuzzy set of constraint indices is proposed. The basic properties of the solution T2FS are investigated. The notion of a compromise maximizing solution is given. Several results are proved that make it possible to simplify its construction. Illustrative examples are given.