Centroid Coordinate Ranking of Pythagorean Fuzzy Numbers and its Application in Group Decision Making

Cognitive computing contains different cognitive characteristics, especially when dealing with group decision-making problems, it is considered as a cognitive-based human behavior, in which collecting and processing data from multiple resources is an important stage. Intuitionistic fuzzy number (IFN) and Pythagorean fuzzy number (PFN) are the most reliable tools to deal with fuzzy information by utilizing membership and non-membership, where the distance measure and similarity of IFNs or PFNs play an important role in dealing with incomplete information in order to achieve the final decision and PFN is a generalization of IFN. Motivated by these, some important concepts for PFNs are proposed by geometric methods to deal with group decision-making problems in this paper. Through counter examples, it is pointed out that the score function and accuracy function of PFNs are inconsistent with the traditional ranking rules of IFNs, the concepts of the centroid coordinate and hesitation factor are proposed by geometric distance. In addition, Pythagorean fuzzy distance measure (PFDM) through the centroid coordinate and hesitation factor are introduced, and proved the distance measure satisfies the axiomatic conditions of distance, a calculation example is given in the form of tables. A unified ranking method for IFNs and PFNs is given by comparing with the smallest PFN (0,1). The weight vector, positive or negative ideal solution is calculated by aggregating centroid coordinate matrices, a new TOPSIS method is given by using Pythagorean fuzzy weighted distance (PFWD) and relative closeness. These results show that the decision matrix and positive (negative) ideal solutions represented by the centroid coordinate and hesitation factor can reflect the fuzzy information more comprehensively. The proposed method not only has a wide range of application, but also reduces the loss of information and is easier to be implemented. It is only applied to the multi criteria decision making problem for the first time, it also has some other good properties that need to be further explored and supplemented. This provides a theoretical basis for studying the wide application of Pythagorean fuzzy sets.