Abstract

Convey human being's information to a mathematical formula and vice versa by a proper tool, is considered a first step to solve the problem of assigning a membership degree (suitable value) of an object to a set by decision-makers. Different information with its periodic circumstances highlighted the need for an extra variable to be built into the mathematical tools. This extra variable adds a significant role to represent a special type of information that has the same data but with a different meaning in different time/levels/phases. However, the idea of extending the range of membership degree to unit disk in the complex plane solved this problem. In this research, we choose bipolar multi-fuzzy information to be generalized to the complex realm using the Cartesian form "x+iy". Therefore, the formal definition of complex bipolar multi-fuzzy sets (CBMFS) is introduced with a range of membership lies in the unit square. The advantage of CBMFS is that the real and imaginary parts of CBMFSs can represent bipolar multi-fuzzy information. Also, some basic operations and numerical examples on CBMFS are presented and studied its properties. Moreover, the relations of CBMFS with each of bipolar fuzzy sets (BFS) and complex bipolar fuzzy sets (CBFS), respectively, are illustrated. Finally, two types of distances and ∂-equality under CBMFS are presented. Also, some illustrative examples and properties of complex bipolar multi-fuzzy distance and ∂-equality are obtained.