A novel unified ranking method of Pythagorean fuzzy numbers characterized by the residual sector area

Abstract Pythagorean fuzzy number (PFN) is not only an extension of traditional intuitionistic fuzzy number (IFN), but also can deal with the decision-making problem of multi-attribute index information in a wider range. In recent years, it has been rapidly developed and popularized in the field of decision science. In this paper, it was pointed out that there were many defects in some existing score functions and ranking criteria of PFNs through counter examples, and the main causes of these defects were analyzed with some examples. All IFNs and PFNs are unified in the Pythagorean fuzzy environment, by calculating the residual sector area(RSA) of the geometric figure corresponding to PFN and its degree of hesitation, a new score function and ranking criterion are proposed, the rationality of the ranking criterion are proved, and the basic properties of the score function are discussed. In addition, a new score function and its ranking method were proposed in the Pythagorean fuzzy environment, which ended the chaotic situation of independent ranking of IFNs and PFNs. It only needs a score function to rank all PFNs perfectly, especially for some equivalent PFNs, which can further realize accurate comparison, so as to overcome the contradiction between other methods and traditional IFN’s methods. Therefore, the proposed method has deep practical background and theoretical significance, and is more universal, practical and scientific. However, the new score function has not been applied to the decision-making problem of multi-attribute information. Maybe it needs to improve and continue to explore other excellent properties. This also provides a theoretical basis for the study of the wide application of Pythagorean fuzzy sets. (1) The ranking methods of IFNs and PFNs are contradictory. Since PFN is a generalization of IFN, the method given in [14,15] should be suitable for the method in [2,3], such as, let (0.5,0.3)) () 0.5 0.3 0.16 0.15 0.