Fuzzy convexity and multiobjective convex optimization problems

K e y w o r d s F u z z y convexity; Fuzzy criterion sets, Pareto-optimal decision, Multiple objective optimization. 1. I N T R O D U C T I O N Since most of practical decision problems are fuzzy and approximate, fuzzy decision making becomes one of the most impor tan t practical approaches. However, the result ing problems are frequently complicated and difficult to solve. One effective way to overcome these difficulties is to explore the fuzzy convexity properties of the resulting problems. However, in order to formulate these desirable result ing problems, we must have a complete, or, at least, reasonable unders tanding about the basic convexity properties of fuzzy sets. For example, in an earlier paper [2], we formulated several fuzzy nonlinear programming problems based on the concept of fuzzy convexity. Different types of convexity and generalized convexity of fuzzy sets were studied by several authors, including Ammar and Metz [I], Ramik and Vlach [3], Sarkar [4], Syau and coworkers [2,5,6], and Yang [7-9], aiming at applications to fuzzy nonlinear programming. Supported by the National Science Council of the Republic of China under contract NSC 89-2213-E-212-055. This work was carried out while the first author was visiting the Department of Industrial and Manufacturing Systems Engineering, Kansas State University. *Author to whom all correspondence should be addressed. 0898-1221/06/$ see front matter (~) 2006 Elsevier Ltd. All rights reserved. Typeset by .AA/tS-TEX doi:10.1016/j.camwa.2006.03.017 352 Y.-R. SYAU AND E. S. LEE We shall restrict attention here to fuzzy sets on the n-dimensional Euclidean space R '~. The concept of convex fuzzy sets was introduced by Zadeh [10], in which a fuzzy set with membership function # : R ~ -+ [0, 1] was called convex if + (1 > m i n { , ( x ) , u(v)) (1.1) for all x, y E supp(#) = {t E R ~ : #(t) > 0} and A E [0, 1]. Consider the following fuzzy set with membership function, {1 # ( x ) = g, i f 0 < x < 2 ; 1 1, i f ~ < x _ < l , satisfying (1.1). Any point in [0, 1/2) is a local maximizer of # but not a global maximizer. This indicates that, in maximizing a fuzzy decision (for details, see [11]), such a convexity does not ensure tha t a local maximizer is also a global maximizer. Therefore, we will s tudy a more restrictive definition of fuzzy convexity due to Ammar and Metz [1], which ensures that a local maximizer is also a global maximizer as will be shown later. Another important property of the more restrictive convex fuzzy sets is tha t any convex combination of such convex fuzzy sets is also a convex fuzzy set. I t will be shown that this property is also laking in the usual convex fuzzy sets which will be called quasiconvex fuzzy sets in this paper. In this paper a fuzzy set with membership function # : R ~ ~ [0, 1] satisfying (1.1) will be called a quasiconvex fuzzy set; and a strictly quasiconvex fuzzy set if strict inequality holds for all x, y E supp(/~), x ¢ y and A E (0, 1). We shall say, a fuzzy set with membership function # : R n --~ [0, 1] is a convex fuzzy set if