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Ranking defects and solving countermeasures for Pythagorean fuzzy sets with hesitant degree

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Abstract

Pythagorean fuzzy set (PFS) is the most concerned and effective tool to describe fuzzy information in the research of machine learning and decision science, its unique representation ability and theoretical method will be gradually applied to the field of machine learning and even artificial intelligence, which lays a theoretical foundation for further solving complex scientific problems in machine learning. Actually, the PFS is not only a generalization of traditional intuitionistic fuzzy set (IFS), but also a more favorable tool of dealing with uncertain multi-attribute decision-making problem. In particular, it relaxes a certain constraint condition, which greatly increases the range of domain value. Firstly, through counter examples, some defects in the original sorting criteria of Pythagorean fuzzy numbers (PFNs) are pointed out. According to geometric interpretation and theoretical analysis, it is obtained that the reason for the defects is that the definitions of the score function and accuracy function are unreasonable, and the hesitation of PFNs is not considered. Secondly, after abandoning the original score function and accuracy function, concepts of distance index and area of lower triangle are proposed by comparing with the maximum PFN (1, 0). Especially, by considering degrees of hesitation (non-zero) for PFNs, several calculation formulas of curve triangle centroid coordinates corresponding to each PFN are given by using the centroid formula in physics, and then a new centroid distance index is obtained. Finally, two ranking methods for PFNs are established based on centroid distance, including synthetic index ranking and centroid distance ranking. Some examples are given to verify the effectiveness of the proposed methods. The example results show that both ranking methods can overcome the shortcomings in existing approaches, and the centroid distance ranking is better than the synthetic index ranking.

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Authors and Affiliations

  1. School of Science, Hunan Institute of Technology, Hengyang, 421002, China

    Gang Sun

  2. School of Management, Tianjin Normal University, Tianjin, 300387, China

    Xiaoping Li

  3. Department of Mathematics and Physical, North China Electric Power University, Beijing, 102206, China

    Degang Chen

Authors
  1. Gang Sun
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  2. Xiaoping Li
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  3. Degang Chen
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Correspondence to Xiaoping Li.

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This Work Has Been Supported By National Natural Science Foundation of China (Grant No. 61463019), Natural Science Foundation of Hunan Province (Grant No. 2019JJ40062).

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Sun, G., Li, X. & Chen, D. Ranking defects and solving countermeasures for Pythagorean fuzzy sets with hesitant degree. Int. J. Mach. Learn. & Cyber. 13, 1265–1281 (2022). https://doi.org/10.1007/s13042-021-01446-x

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