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

K e y w o r d s F u z z y number, Possibility and necessity indices, a-optimal solution, Embedding theorem, Banach space. 1. I N T R O D U C T I O N The concept of fuzzy set was introduced by Zadeh [1]. Since then, many applications of fuzzy sets have been widely developed. One of them is the fuzzy optimization in operations research. The randomness occurring in the optimization problems is categorized as the stochastic optimization problems [2-5]. However, the imprecision (fuzziness) occurring in the optimization problems is categorized as the fuzzy optimization problems. Inuiguchi and Rarm~ [6] give a brief review of fuzzy optimization and a comparison with stochastic optimization in portfolio selection problem. BeLlman and Zadeh [7] inspired the development of fuzzy optimization by providing the aggregation operators, which combined the fuzzy goals and fuzzy decision space. After this motivation and inspiration, there came a lot of articles dealing with the fuzzy optimization problems. Zimmermann [8-10] applied fuzzy sets theory to the linear programming problems and linear multiobjective programming problems. The collection of papers on fuzzy optimization edited by Slowinski [11] and Delgado et al. [12] gives the main stream of this topic. Lai and Hwang [13,14] also give an insightful survey. Purl and Ralescu [15] and Kaleva [16] have proven that the set of all fuzzy numbers can be embedded into a Banach space isometrically and isomorphically. Wu and Ma [17] provide a 0895-7177/04/$ see front matter © 2004 Elsevier Ltd. All rights reserved. Typeset by ~4~¢S-TEX doi:10.1016/j.mcm.2003.12.008