Convexity and upper semicontinuity of fuzzy sets

Since almost all practical problems are fuzzy and approximate, fuzzy decision making becomes one of the most important practical approaches. One off the important aspects for formulating and for solving fuzzy decision problems is the concept of convexity. In this paper, we investigate the interrelationships of several concepts of generalized convex fuzzy sets. We also prove that, in the upper semicontinuous case, the class of semistrictly quasi-convex fuzzy sets lies between the convex and quasi-convex classes. Aggregation or composition is an essential part for optimization or modeling, and some important composition rules for upper semicontinuous fuzzy sets are developed. We prove that a convex combination of upper semicontinuous fuzzy sets is an upper semicontinuous fuzzy set and the intersection of finitely many upper semicontinuous fuzzy sets is an upper semicontinuous fuzzy set. Finally, the criteria for the existence of fuzzy decision under upper semicontinuity conditions are derived and two examples in multiple objective programming are used to illustrate the approach