Fuzzy optimization problemsbased on the embedding theorem and possibility and necessity measures

Abstract

The embedding theorem shows that the set of all fuzzy numbers can be embedded into a Banach space. Inspired by this embedding theorem, we propose a solution concept of fuzzy optimization problem based on the possibility and necessity measures by solving a biobjective optimization problem. This biobjective optimization problem is obtained by applying the embedding function to the original fuzzy optimization problem. We then also consider the fuzzy optimization problem with fuzzy coefficients (i.e., the coefficients are assumed to be fuzzy numbers). Under a setting of core value of fuzzy number, we show that the optimal solution of its corresponding crisp optimization problem (the usual optimization problem) is also a 1-optimal solution of the original fuzzy optimization problem.