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

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Abstract

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.

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References

  1. Atanassov K. Intuitionistic fuzzy sets. Fuzzy Sets Syst. 1986;20(1):87–96.

    Article  Google Scholar 

  2. Chen SM, Tan JM. Handling multicriteria fuzzy decision-making problems based on vague set theory. Fuzzy Sets Syst. 1994;67(2):163–72.

    Article  MathSciNet  Google Scholar 

  3. Hong DH, Choi CH. Multicriteria fuzzy decision-making problems based on vague set theory. Fuzzy Sets Syst. 2000;114(1):103–13.

    Article  Google Scholar 

  4. Szmidt E, Kacprzyk J. Distance between intuitionistic fuzzy sets. Fuzzy Sets Syst. 2000;114(3):505–18.

    Article  MathSciNet  Google Scholar 

  5. Szmidt E, Kacprzyk J. A new concept of a similarity measure for intuitionistic fuzzy sets and its use in group decision making. Lect Notes Comp Sci. 2005;3558(1):272–82.

    Article  Google Scholar 

  6. Sahu R, Dash SR, Das S. Career selection of students using hybridized distance measure based on picture fuzzy set and rough set theory. Decis Mak Appl Manag Eng. 2021;4(1):104–26.

    Article  Google Scholar 

  7. Petrovic I, Kankaras M. A hybridized IT2FS-DEMATEL-AHP-TOPSIS multicriteria decision making approach: Case study of selection and evaluation of criteria for determination of air traffic control radar position. Decis Mak Appl Manag Eng. 2020;3(1):146–64.

    Article  Google Scholar 

  8. Yager RR. Pythagorean fuzzy subsets. In: Proceeding Joint IFSA World Congress and NAFIPS Annual Meeting, Edmonton, Canada; 2013. p. 57–61.

  9. Yager RR, Abbasov AM. Pythagorean membeship grades complex numbers and decision making. Int J Intell Syst. 2013;28:436–52.

    Article  Google Scholar 

  10. Yager RR. Pythagorean membership grades in multicriteria decision making. IEEE Trans Fuzzy Syst. 2014;22(4):958–65.

    Article  Google Scholar 

  11. Zhang XL, Xu ZS. Extension of TOPSIS to multiple criteria decision making with Pythagorean fuzzy sets. Int J Intell Syst. 2014;29:1061–78.

    Article  MathSciNet  Google Scholar 

  12. Zhang XL. A novel approach based on similarity measure for Pythagorean fuzzy multiple criteria group deision making. Int J Intell Syst. 2016;31(6):593–611.

    Article  Google Scholar 

  13. Peng XD, Yang Y. Some results for Pythagorean fuzzy sets. Int J Intell Syst. 2015;30(11):1133–60.

    Article  MathSciNet  Google Scholar 

  14. Peng XD, Dai JG. Approaches to Pythagorean fuzzy stochastic multi-criteria decision making based on prospect theory and regret theory with new distance measure and score function. Int J Intell Syst. 2017;32(11):1187–214.

    Article  Google Scholar 

  15. Gou XJ, Xu ZS, Ren PJ. The properties of continuous Pythagorean fuzzy information. Int J Intell Syst. 2016;31(5):401–24.

    Article  Google Scholar 

  16. Garg H. A novel correlation coefficients between Pythagorean fuzzy sets and its applications to decision-making processes. Int J Intell Syst. 2016;31(12):1234–52.

    Article  Google Scholar 

  17. Garg H. Confidence levels based Pythagorean fuzzy aggregation operators and its application to decision-making process. Comput Math Organ Theory. 2017;23(4):546–71.

    Article  Google Scholar 

  18. Garg H. A new improved score function of an interval-valued Pythagorean fuzzy set based TOPSIS method. Int J Uncertain Quantif. 2017;7(5):463–74.

    Article  Google Scholar 

  19. Ren PJ, Xu ZS, Gou XJ. Pythagorean fuzzy TODIM approach to multi-criteria decision making. Appl Soft Comput. 2016;42:246–59.

    Article  Google Scholar 

  20. Zhang XL. Multicriteria Pythagorean fuzzy decision analysis: a hierarchical QUALIFLEX approach with the closeness based ranking. Inf Sci. 2016;330:104–24.

    Article  Google Scholar 

  21. Wei GW. Similarity measure of Pythagorean fuzzy sets based on the cosine function and their applications. Int J Intell Syst. 2018;33(3):634–52.

    Article  Google Scholar 

  22. Wei GW, Lu M. Pythagorean fuzzy Maclaurin symmetric mean operators in multiple attribute decision making. Int J Intell Syst. 2018;33(5):1043–70.

    Article  MathSciNet  Google Scholar 

  23. Peng XD, Dai J, Garg H. Exponential operation and aggregation operator for q-rung orthopair fuzzy set and their decision-making method with a new score function. Int J Intell Syst. 2018;33(11):2255–82.

    Article  Google Scholar 

  24. Peng XD. Algorithm for pythagorean fuzzy multi-criteria decision making based on WDBA with new score function. Fundam Inform. 2019;165(2):99–137.

    Article  MathSciNet  Google Scholar 

  25. Peng XD, Selvachandran G. Pythagorean fuzzy set: state of the art and future directions. Artif Intell Rev. 2017;1:1–55.

    Google Scholar 

  26. Rahman K, Abdullah S, Ahmad R. Pythagorean fuzzy Einstein geometric operators and their application to multiple attribute decision making. J Intell Fuzzy Syst. 2017;33:635–47.

    Article  Google Scholar 

  27. Rahman K, Khan MSA, Ullah M, Fahmi A. Multiple attribute group decision making for plant location selection with Pythagorean fuzzy weighted geometric aggregation operator. Nucleus. 2017;54:66–74.

    Google Scholar 

  28. Khan MSA, Abdullah S, Ali MY, et al. Pythagorean hesitant fuzzy sets and their application to group decision making with incomplete weight information. J Intell Fuzzy Syst. 2017;33:3971–85.

    Article  Google Scholar 

  29. Wang GJ, Tao YJ, Li YH. TOPSIS evaluation system of logistics transportation based on an ordered representation of the polygonal fuzzy set. Int J Fuzzy Syst. 2020;22(5):1565–81.

    Article  Google Scholar 

  30. Wang GJ, Duan Y. TOPSIS approach for multi-attribute decision making problems based on n-intuitionistic polygonal fuzzy sets description. Comput Ind Eng. 2018;124(10):573–81.

    Article  Google Scholar 

  31. Li XP, Tao YJ, Li YH. Decision making method for evaluating logistics companies based on the ordered representation of the polygonal fuzziness. J Intell Fuzzy Syst. 2020;39(3):3151–66.

    Article  Google Scholar 

  32. Li XP, Li YH, Tao YJ. Representation and aggregation of multi-source information of modern smart cities based on the intuitionistic polygonal fuzzy set. Int J Fuzzy Syst. 2021;23(4):967–83.

    Article  MathSciNet  Google Scholar 

  33. Li DQ, Zeng WY. Distance Measure of Pythagorean Fuzzy Sets. Int J Intell Syst. 2018;33(2):348–61.

    Article  MathSciNet  Google Scholar 

  34. Zhou F, Chen TY. Multiple criteria group decision analysis using a Pythagorean fuzzy programming model for multidimensional analysis of preference based on novel distance measures. Comput Ind Eng. 2020;148:1–19.

    Article  Google Scholar 

  35. Wang L, Garg H. Algorithm for multiple attribute decision-making with interactive archimedean norm operations under Pythagorean fuzzy uncertainty. Int J Comput Intell Syst. 2021;14(1):503–27.

    Article  Google Scholar 

  36. Huang C, Lin MW, Xu ZS. Pythagorean fuzzy MULTIMOORA method based on distance measure and score function: its application in multicriteria decision making process. Knowl Inf Syst. 2020;62(11):4373–406.

    Article  Google Scholar 

  37. Li HM, Lv LI, Li F, et al. A novel approach to emergency risk assessment using FMEA with extended MULTIMOORA method under interval-valued Pythagorean fuzzy environment. Int J Intell Comput Cybern. 2020;13(1):41–65.

    Article  MathSciNet  Google Scholar 

  38. Lin MW, Huang C, Chen RQ, et al. Directional correlation coefficient measures for Pythagorean fuzzy sets: their applications to medical diagnosis and cluster analysis. Complex Intell Syst. 2021;7(7):1025–43.

    Article  Google Scholar 

  39. Lin MW, Huang C, Xu ZS. TOPSIS method based on correlation coefficient and entropy measure for linguistic Pythagorean fuzzy sets and its application to multiple attribute decision making. Complexity. 2019;1–16.

  40. Lin MW, Chen YQ, Chen RQ. Bibliometric analysis on Pythagorean fuzzy sets during 2013–2020. Int J Intell Comput Cybern. 2021;14(2):104–21.

    Article  Google Scholar 

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Funding

This work has been supported by Natural Science Foundation of Hunan Province (Grant No. 2019JJ40062) and National Natural Science Foundation of China (Grant No. 61463019/61374009), and by Research Foundation of Education Bureau of Hunan Province (Grant No. 20C0565).

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Authors and Affiliations

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

    Gang Sun & Mingxin Wang

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

    Xiaoping Li

Authors
  1. Gang Sun
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  2. Mingxin Wang
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  3. Xiaoping Li
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Correspondence to Xiaoping Li.

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Sun, G., Wang, M. & Li, X. Centroid Coordinate Ranking of Pythagorean Fuzzy Numbers and its Application in Group Decision Making. Cogn Comput 14, 602–623 (2022). https://doi.org/10.1007/s12559-021-09976-w

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  • DOI: https://doi.org/10.1007/s12559-021-09976-w

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