On n-polygonal interval-valued fuzzy sets

Abstract

The structure of interval-valued fuzzy sets is complex in regard to their arithmetic operations. To reduce the computational complexity of arithmetic operations on interval-valued fuzzy sets, we propose the notion of an n-polygonal interval-valued fuzzy set considering two 2n+2-tuple ordered real numbers. Complete representation of n-polygonal interval-valued fuzzy sets and numbers is provided. Moreover, we demonstrate that the n-polygonal interval-valued fuzzy numbers can approximate the general interval-value fuzzy numbers with any precision. Next, we introduce arithmetic operations on n-polygonal interval-valued fuzzy numbers by capitalizing on the good characteristics of n-polygonal interval-valued fuzzy numbers. The properties of the introduced arithmetic operations are also addressed. In addition, with the aid of a concrete example, we verify the effectiveness of the approximation ability of the n-polygonal interval-valued fuzzy set. Furthermore, we study the properties of the topological space of n-polygonal interval-valued fuzzy numbers. This proves that this space is a complete, separable and local compact metric space when endowed with the newly defined distance between two n-polygonal interval-valued fuzzy numbers. By product, it shows that arithmetic operations introduced here on n-polygonal interval-valued fuzzy numbers are continuous. Finally, the practicability of n-polygonal interval-valued fuzzy numbers is verified by an example.