Bipolar Pythagorean Fuzzy Sets and Their Application in Multi-attribute Decision Making Problems

Abstract

In this paper, the idea of the bipolar Pythagorean fuzzy sets (BPFSs) and its activities, which is a generalization of fuzzy sets, bipolar fuzzy sets (BFSs), intuitionistic fuzzy sets and bipolar intuitionistic fuzzy sets is proposed, with the goal that it can deal with dubious data all the more flexibly during the process of decision making. The key objective of this research paper has presented another variant of the Pythagorean fuzzy sets so called BPFSs. In bipolar Pythagorean fuzzy sets, membership degrees are satisfying the condition $$0 \le \left( {\mu_{p}^{ + } \left( x \right)} \right)^{2}$$ + $$\left( {v_{p}^{ + } \left( x \right)} \right)^{2} \le 1$$ and $$0 \le \left( {\mu_{p}^{ - } \left( x \right)} \right)^{2}$$ + $$\left( {v_{p}^{ - } \left( x \right)} \right)^{2} \le 1$$ instead of $$0 \le \left( {\mu_{p} \left( x \right)} \right)^{2}$$ + $$\left( {v_{p} \left( x \right)} \right)^{2} \le 1$$ as is in Pythagorean fuzzy sets and $$0 \le \mu_{p} \left( x \right)$$ + $$v_{p} \left( x \right) \le 1$$ as is in the intuitionistic fuzzy sets. Here, negative membership degree means the certain counter-property comparing to a bipolar Pythagorean fuzzy set. Also, the BPFSs weighted average operator and the BPFSs weighted geometric operator to aggregate the BPFSs is developed here. Further a multi attribute decision making technique is developed and the proposed aggregation operators are used. Finally, a numerical methodology for execution of the proposed system is introduced.