Further characterizations of possibility-theoretical indices in fuzzy optimization

Abstract

In this paper, we investigate six types of fuzzy relations based on Zadeh's possibility theory. Such relations were originally proposed by Dubois and Prade as comparison indices for fuzzy numbers and were extended by Inuiguchi, Ichihashi, and Kume to general cases. The fuzzy relations we consider are defined for fuzzy sets in a vector space preordered by a convex cone. We reveal the connection of the fuzzy relations with set relations, order-like crisp relations known in the area of set optimization. Specifically, we prove some equalities and equivalences involving the fuzzy relations and set relations under three different assumptions. We finally consider a fuzzy optimization problem in a general form and define its solution concepts. Applying the above equivalences to the problem, we obtain its relationship to certain set optimization problems.