Bipolar Spherical Fuzzy Sets and Their Application in Multi-Attribute Decision Making Problems

In this paper, the concept of the bipolar spherical fuzzy set (BSFS) and its operations is developed. It is proposed as a generalization of fuzzy sets, bipolar fuzzy sets, picture fuzzy sets, and spherical fuzzy sets, allowing uncertain information to be handled more flexibly in the decision-making process. The key objective of this research paper is to introduce a new version of the spherical fuzzy set so called bipolar spherical fuzzy set (BSFS). The bipolar spherical fuzzy set is a new extension of spherical fuzzy sets and picture fuzzy sets. In bipolar spherical fuzzy sets, membership degrees are satisfying the condition \( 0 \leq (s^+)^2(x) + (i^+)^2(x) + (d^+)^2(x) \leq 1 \) and \( 0 \leq (s^-)^2(x) + (i^-)^2(x) + (d^-)^2(x) \leq 1 \) instead of \( 0 \leq s^2(x) + i^2(x) + d^2(x) \leq 1 \) as is in spherical fuzzy sets and \( 0 \leq s(x) + i(x) + d(x) \leq 1 \) as is in the picture fuzzy set. Here, negative membership degree denotes the implicit counter-property corresponding to a bipolar spherical fuzzy set. Also, the BSF-set weighted average operator and BSF-set weighted geometric operator are given to aggregate the BSF-sets. Further a multi attribute decision making (MADM) technique is developed and the proposed aggregation operators are utilized. Finally, a numerical approach for implementation of the proposed technique is developed.