Fuzzy Φ-convexity and fuzzy decision making

The notion of convexity and its various generalization is important for quantitative and qualitative studies in operations research or applied mathematics. It has also been considered by many authors in fuzzy set theory. In this paper, we study the concept of Φ-convex and Φ-quasiconvex fuzzy sets which was proposed byChen et al. [1], and develop some useful extrema properties of these fuzzy sets. We prove that any local maximizer of a Φ-convex fuzzy set is also a global maximizer, and that any strict local maximizer of a Φ-quasiconvex fuzzy set is also a global maximizer. We also study the class of strictly Φ-convex (respectively, strictly Φ-quasiconvex) fuzzy sets that is more restricted than the class of Φ-convex (respectively, Φ-quasiconvex) fuzzy sets. We prove for both families of strictly Φ-convex and strictly Φ-quasiconvex fuzzy sets that every local maximizer is also the unique global maximizer. In addition, some applications to fuzzy decision making are discussed.