New Methods of finding support planes of cut-level fuzzy power sets and Geometry of the convex fuzzy power sets

Abstract

Abstract The concepts of the convex fuzzy power sets and crisply defined membership functions in fuzzy set theory are important to deeply analyze to look further for more precise solutions of the fuzzy mathematical programming problems. The Extension principle was defined in terms of the level cuts of the membership functions. The theorem of the decomposition proves the existence of the fuzzy power set as the union of all level cuts of membership function. The Convexity of the level cut fuzzy set was determined at the points of the universe, where the membership function is approximated. The Theorems of the convex and quasi-convex fuzzy sets extend and expand the extension principle to the higher quality to look for more precise solutions of fuzzy problems beyond the support planes. The Fuzzy geometry and topological categories researched further to arrive to the newest Lemma for the cyclic quadrilateral convex fuzzy sets. The analytical and graphical representation of modeling with membership functions type I through types 6 are investigated to the point to be highly recommended in engineering problems by using heuristic methods and models.