Introduction to non-convex fuzzy geometry

Abstract

This study is an extension of the recently developed theory on analytical fuzzy geometry with fuzzy numbers and fuzzy points. In the existing study, both the fuzzy numbers and fuzzy points are convex and normal fuzzy sets in nature. However, in reality, considering fuzzy objects as convex and normal sets is an oversimplified assumption. Here, the nature of non-convex and non-normal fuzzy sets is studied. In the first part of this paper, the concept of same points has been extended for non-convex and non-normal fuzzy sets. A computationally efficient method for extended addition, subtraction, multiplication, and division of two generalized fuzzy sets is proposed using the extended concept of same points. The proposed procedure drastically reduces the computational complexity of extended arithmetic operations on two generalized fuzzy sets drastically.In the second part of this paper, the genesis and morphology of fuzzy points are studied. The conjunction of non-convex fuzzy sets generates a new class of study which is termed as non-convex or generalized fuzzy geometry where the generalized fuzzy point is the key concept. The effect of different t-norms as a conjunction operator on fuzzy sets is studied with geometrical visualization. The extended arithmetic operations between generalized fuzzy points have been defined using the extended concept of same points developed here. The formation of the locus of a non-convex fuzzy point as a fuzzy line segment has also been introduced.