Frank Aggregation Operators and Their Application to Probabilistic Hesitant Fuzzy Multiple Attribute Decision-Making


The fuzzy aggregation information plays an important role in the group decision support system under the interval-valued hesitant fuzzy information and interval-valued probabilistic hesitant fuzzy information. Therefore, in this paper, we develop a new approach of the interval-valued hesitant fuzzy Frank aggregation (IVHFFA) and interval-valued probabilistic hesitant fuzzy aggregation (IVPHFFA) operators. First, we define some operational laws of IVHEs and IVPHFEs by using Frank t-norm and t-conorm. Furthermore, we develop a series of IVHFFA and IVPHFFA operators based on these operational laws under the IVHF and IVPHF information. Also, discuss some fundamental properties and relations of the proposed aggregation operators for IVHF and IVPHF information. In order to implement the proposed aggregation operators of IVHF and IVPHF information in group decision-making problem, we construct general algorithms for multi-attribute group decision-making problem based on the proposed IVHFFA and IVPHFFA. Finally, from the comparative and sensitivity analysis, the proposed fuzzy decision-making model is more effective and reliable as compared with existing method.

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  1. 1.

    Bedregal, B., Beliakov, G., Bustine, H., Calvo, T., Mesiar, R.: D Patermain: A class of fuzzy multisets with a xed number of memberships. Inf. Sci. 189, 117 (2012)

    Article  Google Scholar 

  2. 2.

    Badi, I., Ballem, M.: Supplier selection using therough BWM-MAIRCA model: A case study in pharmaceutical supplying in Libya. Decis. Maki. Appl. Manag. Eng. 1(2), 2560–6018 (2018)

    Google Scholar 

  3. 3.

    Chen, N., Xu, Z.: Properties of interval-valued hesitant fuzzy sets. J. Intell. Fuzzy Syst. 27(1), 143–158 (2014)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Chen, N., Xu, Z., Xia, M.: Correlation coefficients of hesitant fuzzy sets and their applications to clustering analysis. Appl. Math. Model. 37(4), 2197–2211 (2013)

    MathSciNet  Article  Google Scholar 

  5. 5.

    De, A., Das, S., Kar, S.: Multiple attribute decision making based on probabilistic interval-valued intuitionistic fuzzy set and extended TOPSIS method. J. Intell. Fuzzy Syst. 37(4), 5229–5248 (2019)

    Article  Google Scholar 

  6. 6.

    Gupta, K., Kumar, S.: Hesitant probabilistic fuzzy set based time series forecasting method. Granular Comput. 4(4), 739–758 (2019)

    Article  Google Scholar 

  7. 7.

    Jagannath, R., Adhikary K., Kar S., Pamucar D.: A rough strength relational DEMATEL model for analysing the key success factors of hospital service quality. Decis. Mak. Appl. Manag. Eng. 1, 121–142 (2018)

    Article  Google Scholar 

  8. 8.

    Joshi, D.K., Beg, I., Kumar, S.: Hesitant probabilistic fuzzy linguistic sets with applications in multi-criteria group decision making problems. Mathematics 6(4), 47 (2018)

    Article  Google Scholar 

  9. 9.

    Jiang, F., Ma, Q.: Multi-attribute group decision making under probabilistic hesitant fuzzy environment with application to evaluate the transformation efficiency. Appl. Intell. 48(4), 953–965 (2018)

    Article  Google Scholar 

  10. 10.

    Krishankumar, R., Ravichandran, K.S., Kar, S., et al.: Interval-valued probabilistic hesitant fuzzy set for multi-criteria group decision-making. Soft Comput. 23, 10853–10879 (2019)

    Article  Google Scholar 

  11. 11.

    Lin M.M., Zhan Q., Xu Z., Chen R.: Group decision making with probabilistic hesitant multiplicative preference relations based on consistency and consensus. IEEE Access, 6, 63329-63344 (2018)

  12. 12.

    Li, D.Q., Zeng, W.Y., Zhao, Y.B.: Note on distance measure of hesitant fuzzy sets. Inf. Sci. 321, 103–115 (2015)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Li, J., Wang, Z.X.: A programming model for consistency and consensus in group decision making with probabilistic hesitant fuzzy preference relations. Int. J. Fuzzy Syst. 20(8), 2399–2414 (2018)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Li, J., Wang, J.Q., Hu, J.H.: Multi-criteria decision-making method based on dominance degree and BWM with probabilistic hesitant fuzzy information. Int. J. Mach. Learn. & Cyber. 10(7), 1671–1685 (2019)

    Article  Google Scholar 

  15. 15.

    Li, J., Wang, J.Q.: Multi-criteria decision-making with probabilistic hesitant fuzzy information based on expected multiplicative consistency. Neural Comput. Appl. 31, 8897–8915 (2019)

    Article  Google Scholar 

  16. 16.

    Li, J., Wang, Z.X.: Consensus building for probabilistic hesitant fuzzy preference relations with expected additive consistency. Int. J. Fuzzy Syst. 20(5), 1495–1510 (2018)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Meng, F., Wang, C., Chen, X., Zhang, Q.: Correlation coefficients of interval-valued hesitant fuzzy sets and their application based on the shapley function. Int. J. Intell. Syst. 31(1), 17–43 (2016)

    Article  Google Scholar 

  18. 18.

    Meng, V., Chen, X.H.: Correlation cofficient of hesitant fuzzy sets and their application based on fuzzy measures. Cognit. Comput. 7, 445–463 (2015)

    Article  Google Scholar 

  19. 19.

    Milosavljevic, M.A., Bursac, M.A., Trickovic, G.A.: Selection of the railroad container terminal in Serbia based on multi criteria decision-making methods. Decis. Mak. Appl. Manag. Eng. 1(2), 2560–6018 (2018)

    Article  Google Scholar 

  20. 20.

    Miyamoto, S.: Fuzzy multisets and their generalizations. Proc. Int. Conf. Membr. Comput. 2125, 225–236 (2000)

    MATH  Google Scholar 

  21. 21.

    Rodriguez, R.M., Martnez, L., Torra, V., Xu, Z.S., Herrera, F.: Hesitant fuzzy sets: State of the art and future directions. Int. J. Intell. Syst. 29, 495–524 (2014)

    Article  Google Scholar 

  22. 22.

    Sindhu, M.S., Rashid, T., Kashif, A., Guirao, J.L.G.: Multiple criteria decision making based on probabilistic interval-valued hesitant fuzzy sets by using LP methodology. Discrete Dyn. Nat. Soc. 2019, 1527612 (2019)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Shao, S., Zhang, X.: Measures of Probabilistic Neutrosophic Hesitant Fuzzy Sets and the Application in Reducing Unnecessary Evaluation Processes. Mathematics 7(7), 649 (2019)

    Article  Google Scholar 

  24. 24.

    Torra, V.: Hesitant fuzzy sets. Int. J. Intell. Syst. 25, 529–539 (2010)

    MATH  Google Scholar 

  25. 25.

    Tian, X., Xu, Z., Fujita, H.: Sequential funding the venture project or not? A prospect consensus process with probabilistic hesitant fuzzy preference information. Knowl. Based Syst. Inform. Sci. 467, 179–198 (2018)

    Google Scholar 

  26. 26.

    Wu, Z.X., Liu, X.D., Wang, Z.W., Zhang, S.T.: Dynamic emergency decision-making method with probabilistic hesitant fuzzy information based on GM (1, 1) and TOPSIS. IEEE Access 7, 7054–7066 (2018)

    Article  Google Scholar 

  27. 27.

    Wu, W., Li, Y., Ni, Z., Jin, F., Zhu, X.: Probabilistic interval-valued hesitant fuzzy information aggregation operators and their application to multi-attribute decision making. Algorithms 11, 120 (2018)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Xu, Z., Da, Q.-L.: An overview of operators for aggregating information. Int. J. Intell. Syst. 18, 953–969 (2003)

    Article  Google Scholar 

  29. 29.

    Xia, M., Xu, Z.: Hesitant fuzzy information aggregation in decision making. Int. J. Approx. Reason. 52, 395–407 (2011)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Xu, Z.S., Zhou, W.: Consensus building with a group of decision makers under the hesitant probabilistic fuzzy environment. Fuzz. Optim. Decis. Mak. 16, 481–503 (2017)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Yu, D.: Some hesitant fuzzy information aggregation operators based on Einstein operational laws. In. J. Intell. Syst. 29, 320–340 (2014)

    Article  Google Scholar 

  32. 32.

    Zhang, Z., Wu, C.: Weighted hesitant fuzzy sets and their application to multi-criteria decision making. Br. J. Math. Comput. Sci. 4, 1091–1123 (2014)

    Article  Google Scholar 

  33. 33.

    Zhou, W., Xu, Z.: Group consistency and group decision making under uncertain probabilistic hesitant fuzzy preference environment. Inform. Sci. 414, 276–288 (2017)

    Article  Google Scholar 

  34. 34.

    Zhou, L., Zhao, X., Wei, G.: Hesitant fuzzy Hamacher aggregation operators and their application to multiple attribute decision making. J. Intell. Fuzzy Syst. 26, 2689–2699 (2014)

    MathSciNet  Article  Google Scholar 

  35. 35.

    Zadeh, L.A.: Fuzzy Sets. Inf. control. 8, 338–353 (1965)

    Google Scholar 

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Correspondence to Ronnason Chinram.

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Yahya, M., Abdullah, S., Chinram, R. et al. Frank Aggregation Operators and Their Application to Probabilistic Hesitant Fuzzy Multiple Attribute Decision-Making. Int. J. Fuzzy Syst. (2020).

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  • Hesitant fuzzy set
  • Probabilistic hesitant fuzzy aggregation operator
  • Decision-making problem
  • Fuzzy decision support system