Geometric score function of Pythagorean fuzzy numbers determined by the reliable information region and its application to group decision-making

Pythagorean fuzzy number (PFN) is not only a generalization of general intuitionistic fuzzy number (IFN), but also it can deal with multi-attribute group decision-making problems in a wider range, and it has profound geometric meaning and background. The main works of this article are to establish a unified score function through geometric method on some shortcomings of existing ranking criteria for PFNs, so as to clarify the confusion caused by the widely popular PFNs and IFNs independent ranking for a long time, and apply the proposed method to multi-attribute group decision-making problems, which supplies a new thought for further study of large group decision-making and data statistics problems. Firstly, some conflicts between the two existing rankings for PFNs and IFNs are pointed out, and the major reason for the contradiction are analyzed through counterexamples. Secondly, all IFNs are unified into the PFNs environment according to the coordinate transformation, some fuzzy information distribution contained in each PFN is divided into the reliable information region (RIR) and hesitant information region (HIR) by geometric method. A new geometric score function and ranking method are proposed by the area of reliable information region and hesitation degree in the Pythagorean fuzzy environment, the rationality of the ranking criterion is proved, and the basic properties of geometric score function are obtained by partial derivative and monotonicity. Finally, the new ranking method is applied to the practical problem of multi-attribute information group decision -making through case analysis, it is showed that the proposed method not only overcomes some shortcomings of other methods, but also ends the long-term confusion between IFNs and PFNs ranking separately, and some of the method are of data.