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Geometric Ranking of Pythagorean Fuzzy Numbers Based on Upper Curved Trapezoidal Area Characterization Score Function

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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 information in a wider range. In recent years, it has been rapidly developed and popularized in the field of decision science. Firstly, it is pointed out that there are some defects in the existing score function and ranking criteria of PFNs through counter examples, and the main causes of these defects are analyzed according to the geometric method. Secondly, IFNs are unified into the Pythagorean fuzzy environment, the new unified score function and geometric ranking criterion of IFNs and PFNs are proposed by the corresponding upper curved trapezoidal area and hesitation factor, and then the rationality of geometric ranking method and some basic properties of the score function are studied. Finally, the comparison of other ranking methods shows that the proposed method only needs a score function and its own hesitation to rank all PFNs uniformly, especially the accurate comparisons among equivalent PFNs are realized. This not only overcomes the contradiction and confusion caused by the respective ranking of traditional IFNs and PFNs, but also provides a theoretical basis for further expanding the fuzzy decision-making method of Pythagorean fuzzy sets.

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References

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

    Article  MATH  Google Scholar 

  2. Chen, S.M., Tan, J.M.: Handling multicriteria fuzzy decision-making problems based on vague set theory. Fuzzy Sets Syst. 67(2), 163–172 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  3. Hong, D.H., Choi, C.H.: Multicriteria fuzzy decision-making problems based on vague set theory. Fuzzy Sets Syst. 114(1), 103–113 (2000)

    Article  MATH  Google Scholar 

  4. Xu, Z.S.: Some similarity measures of intuitionistic fuzzy sets and their applications to multiple attribute decision making. Fuzzy Optim. Decis. Making 6, 109–121 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Pedrycz, W., Song, M.L.: Analytic hierarchy process (AHP) in group decision making and its optimization with an allocation of information granularity. IEEE Trans. Fuzzy Syst. 19, 527–539 (2011)

    Article  Google Scholar 

  6. Chen, S.M., Cheng, S.H., Lan, T.C.: Multicriteria decision making based on the TOPSIS method and similarity measures between intuitionistic fuzzy values. Inf. Sci. 367–368, 279–295 (2016)

    Article  Google Scholar 

  7. Yager, R.R., Abbasov, A.M.: Pythagorean membeship grades, complex numbers and decision making. Int. J. Intell. Syst. 28, 436–452 (2013)

    Article  Google Scholar 

  8. Yager, R.R.: Pythagorean membership grades in multicriteria decision making. IEEE Trans. Fuzzy Syst. 22(4), 958–965 (2014)

    Article  Google Scholar 

  9. Zhang, X.L., Xu, Z.S.: Extension of TOPSIS to multiple criteria decision making with Pythagorean fuzzy sets. Int. J. Intell. Syst. 29(12), 1061–1078 (2014)

    Article  Google Scholar 

  10. Peng, X.D., Yang, Y.: Some results for Pythagorean fuzzy sets. Int. J. Intell. Syst. 30(11), 1133–1160 (2015)

    Article  MathSciNet  Google Scholar 

  11. Ren, P.J., Xu, Z.S., Gou, X.J.: Pythagorean fuzzy TODIM approach to multi-criteria decision making. Appl. Soft Comput. 42(2), 246–259 (2016)

    Article  Google Scholar 

  12. Zhang, X.L.: Multicriteria pythagorean fuzzy decision analysis: a hierarchical QUALIFLEX approach with the closeness index-based ranking methods. Inf. Sci. 33(1), 104–124 (2016)

    Article  Google Scholar 

  13. Wan, S.P., Jin, Z., Wang, F., et al.: A new ranking method for Pythagorean fuzzy numbers. 12th International Conference on Intelligent Systems and Knowledge Engineering, 2017, Nov 24–26, Nanjing, China, pp. 1–6.

  14. Peng, X.D., Dai, J.: 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. 32(1), 1187–1214 (2017)

    Article  Google Scholar 

  15. Li, D.Q., Zeng, W.Y.: Distance measure of Pythagorean fuzzy sets. Int. J. Intell. Syst. 33(2), 348–361 (2018)

    Article  MathSciNet  Google Scholar 

  16. Peng, X.D.: Algorithm for pythagorean fuzzy multi-criteria decision making based on WDBA with new score function. Fund. Inform. 165(2), 99–137 (2019)

    MathSciNet  MATH  Google Scholar 

  17. Huang, C., Lin, M.W., Xu, Z.S.: Pythagorean fuzzy MULTIMOORA method based on distance measure and score function: its application in multicriteria decision making process. Knowl. Inf. Syst. 62(11), 4373–4406 (2020)

    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. 7(5), 463–474 (2017)

    Article  Google Scholar 

  19. Garg, H.: A novel improved accuracy function for interval valued Pythagorean fuzzy sets and its applications in decision making process. Int. J. Intell. Syst. 31(12), 1247–1260 (2017)

    Article  Google Scholar 

  20. Garg, H.: A linear programming method based on an improved score function for interval-valued Pythagorean fuzzy numbers and its application to decision-making. Int. J. Uncertainty Fuzz. Knowl.-Based Syst. 26(1), 67–80 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  21. Wan, S.P., Jin, Z., Dong, J.Y.: Pythagorean fuzzy mathematical programming method for multi-attribute group decision making with Pythagorean fuzzy truth degrees. Knowl. Inf. Syst. 55(2), 437–466 (2018)

    Article  Google Scholar 

  22. Peng, X.D., 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. 33(11), 2255–2282 (2018)

    Article  Google Scholar 

  23. Peng, X.D., Ma, X.L.: Pythagorean fuzzy multi-criteria decision making method based on CODAS with new score function. J. Intell. Fuzzy Syst. 38(3), 3307–3318 (2020)

    Article  MathSciNet  Google Scholar 

  24. Rahman, K., Abdullah, S., Ahmad, R.: Pythagorean fuzzy Einstein geometric operators and their application to multiple attribute decision making. J. Intell. Fuzzy Syst. 33, 635–647 (2017)

    Article  MATH  Google Scholar 

  25. Ullah, K., Mahmood, A., Ali, Z., et al.: On some distance measures of complex Pythagorean fuzzy sets and their applications in pattern recognition. Complex Intell. Syst. 6(1), 15–27 (2020)

    Article  Google Scholar 

  26. Firozja, M.A., Agheli, B., Jamkhaneh, E.B.: A new similarity measure for Pythagorean fuzzy sets. Complex Intell. Syst. 6(1), 67–74 (2020)

    Article  Google Scholar 

  27. Khan, M.S.A., Abdullah, S., Ali, M.Y., et al.: Extension of TOPSIS method based on Choquet integral under interval-valued Pythagorean fuzzy environment. J. Intell. Fuzzy Syst. 34, 267–282 (2018)

    Article  Google Scholar 

  28. Wang, G.J., Tao, Y.J., Li, Y.H.: TOPSIS evaluation system of logistics transportation based on an ordered representation of the polygonal fuzzy set. Int. J. Fuzzy Syst. 22(5), 1565–1581 (2020)

    Article  Google Scholar 

  29. Li, X.P., Tao, Y.J., Li, Y.H.: Decision making method for evaluating logistics companies based on the ordered representation of the polygonal fuzziness. J. Intell. Fuzzy Syst. 9(3), 3151–3166 (2020)

    Article  Google Scholar 

  30. Li, X.P., Li, Y.H., Tao, Y.J.: Representation and aggregation of multi-source information of modern smart cities based on the intuitionistic polygonal fuzzy set. Int. J. Fuzzy Syst. 23(4), 967–983 (2021)

    Article  MathSciNet  Google Scholar 

  31. Wang, L., Wang, H.: An integrated qualitative group decision-making method for assessing health-care waste treatment technologies based on linguistic terms with weakened hedges. Appl. Soft Comput. (2022). https://doi.org/10.1016/j.asoc.2022.108435

    Article  Google Scholar 

  32. Sun, G., Li, X.P., Chen, D.G.: Ranking defects and solving countermeasures for Pythagorean fuzzy sets with hesitant degree. Int. J. Mach. Learn. Cybern. 13(5), 1265–1281 (2022)

    Article  Google Scholar 

  33. Sun, G., Wang, M.X., Li, X.P.: Centroid coordinate ranking of Pythagorean fuzzy numbers and its application in group decision making. Cogn. Comput. 14(2), 602–623 (2022)

    Article  Google Scholar 

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Funding

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

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Authors and Affiliations

  1. Department of Mathematics, Teacher’s College, Liaodong University, Dandong, 118003, Liaoning, China

    Yanhong Li

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

    Gang Sun

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

    Xiaoping Li

Authors
  1. Yanhong Li
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  2. Gang Sun
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  3. Xiaoping Li
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Correspondence to Gang Sun or Xiaoping Li.

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Li, Y., Sun, G. & Li, X. Geometric Ranking of Pythagorean Fuzzy Numbers Based on Upper Curved Trapezoidal Area Characterization Score Function. Int. J. Fuzzy Syst. 24, 3564–3583 (2022). https://doi.org/10.1007/s40815-022-01359-z

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  • DOI: https://doi.org/10.1007/s40815-022-01359-z

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