International Journal of Fuzzy Systems

, Volume 17, Issue 4, pp 509–520 | Cite as

Hesitant Fuzzy Maclaurin Symmetric Mean Operators and Its Application to Multiple-Attribute Decision Making

  • Jindong Qin
  • Xinwang LiuEmail author
  • Witold Pedrycz


In this paper, we investigate the multiple-attribute decision-making (MADM) problems based on traditional Maclaurin symmetric mean operator operating under hesitant fuzzy environment. The Maclaurin symmetric mean (MSM) operator is a classical mean type aggregation operator and has strong modeling capability in modern information fusion theory, which has particular advantages for aggregating multi-dimension arguments. The prominent characteristic of the MSM operator is that it can capture the interrelationship among the multi-input arguments. Motivated by the idea of MSM operator, we develop the hesitant fuzzy Maclaurin symmetric mean (HFMSM) operator for aggregating the hesitant fuzzy information. Some desirable properties such as monotonicity, boundedness, idempotency are studied. Furthermore, we have discussed some special cases with respect to different parameter values of the HFMSM operator in detail. For the situations where the input arguments have different importance, we further develop the weighted hesitant fuzzy Maclaurin symmetric mean operator to aggregate hesitant fuzzy information. Based on which, an approach to MADM problems with hesitant fuzzy information is developed. Finally, a practical example with paper quality evaluation of sciencepaper online in China is provided to illustrate the practicality and effectiveness of the proposed method.


Hesitant fuzzy set Maclaurin symmetric mean Hesitant fuzzy Maclaurin symmetric mean (HFMSM) operator Weighted hesitant fuzzy Maclaurin symmetric mean (WHFMSM) operator Multiple-attribute decision making Paper quality evaluation of sciencepaper online 



The work is supported by the National Natural Science Foundation of China (NSFC) under Projects 71171048 and 71371049, Ph.D. Program Foundation of Chinese Ministry of Education 20120092110038, the Scientific Research and Innovation Project for College Graduates of Jiangsu Province CXZZ13_0138, the Scientific Research Foundation of Graduate School of Southeast University YBJJ1454,and the Scholarship from China Scholarship Council (No. 201406090096).


  1. 1.
    Lai, Y.-J., Liu, T.-Y., Hwang, C.L.: TOPSIS for MODM. Eur. J. Oper. Res. 76, 486–500 (1994)zbMATHCrossRefGoogle Scholar
  2. 2.
    Wang, Y.M., Parkan, C.: Multiple attribute decision making based on fuzzy preference information on alternatives: ranking and weighting. Fuzzy Sets Syst. 153, 331–346 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Herrera, F., Martínez, L.: A 2-tuple fuzzy linguistic representation model for computing with words. IEEE Trans. Fuzzy Syst. 8, 746–752 (2000)CrossRefGoogle Scholar
  4. 4.
    Pedrycz, W.: Why triangular membership functions? Fuzzy Sets Syst. 64, 21–30 (1994)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Mendel, J.M., John, R.I., Liu, F.: Interval type-2 fuzzy logic systems made simple. IEEE Trans. Fuzzy Syst. 14, 808–821 (2006)CrossRefGoogle Scholar
  6. 6.
    Atanassov, K.T.: Intuitionistic fuzzy sets. Fuzzy Sets Syst. 20, 87–96 (1986)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Torra, V.: Hesitant fuzzy sets. Int. J. Intell. Syst. 25, 529–539 (2010)zbMATHGoogle Scholar
  8. 8.
    Torra, V., Narukawa, Y.: On hesitant fuzzy sets and decision. In: Proceeding of the IEEE International Conference on Fuzzy Systems, pp. 1378–1382 (2009)Google Scholar
  9. 9.
    Chen, N., Xu, Z.S., Xia, M.M.: Correlation coefficients of hesitant fuzzy sets and their applications to clustering analysis. Appl. Math. Model. 37, 2197–2211 (2013)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Xu, Z.S., Xia, M.M.: Distance and similarity measures for hesitant fuzzy sets. Inf. Sci. 181, 2128–2138 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Xu, Z.S., Xia, M.M.: On distance and correlation measures of hesitant fuzzy information. Int. J. Intell. Syst. 26, 410–425 (2011)zbMATHCrossRefGoogle Scholar
  12. 12.
    Xu, Z.S., Xia, M.M.: Hesitant fuzzy entropy and cross-entropy and their use in multiattribute decision-making. Int. J. Intell. Syst. 27, 799–822 (2012)CrossRefGoogle Scholar
  13. 13.
    Peng, D.H., Gao, C.Y., Gao, Z.F.: Generalized hesitant fuzzy synergetic weighted distance measures and their application to multiple criteria decision-making. Appl. Math. Model. 37, 5837–5850 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Farhadinia, B.: Information measures for hesitant fuzzy sets and interval-valued hesitant fuzzy sets. Inf. Sci. 240, 129–144 (2013)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Yu, D., Zhang, W., Xu, Y.: Group decision making under hesitant fuzzy environment with application to personnel evaluation. Knowl. Based Syst. 52, 1–10 (2013)CrossRefGoogle Scholar
  16. 16.
    Zhu, B., Xu, Z.S., Xia, M.M.: Dual hesitant fuzzy sets. J. Appl. Math., Article ID 879629 (2012)Google Scholar
  17. 17.
    Chen, N., Xu, Z.S.: Hesitant fuzzy ELECTRE II approach: a new way to handle multi-criteria decision making problems. Inf. Sci. 292, 175–197 (2015)CrossRefGoogle Scholar
  18. 18.
    Liao, H.C., Xu, Z.S.: A VIKOR-based method for hesitant fuzzy multi-criteria decision making. Fuzzy Optim. Decis. Making 12, 373–392 (2013)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Zhu, B., Xu, Z.S.: Consistency measures for hesitant fuzzy linguistic preference relations. IEEE Trans. Fuzzy Syst. 22, 35–45 (2014)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Rodriguez, R.M., Martinez, L., Herrera, F.: Hesitant fuzzy linguistic term sets for decision making. IEEE Trans. Fuzzy Syst. 20, 109–119 (2012)CrossRefGoogle Scholar
  21. 21.
    Meng, F.Y., Chen, X.H., Zhang, Q.: Generalized hesitant fuzzy generalized Shapley–Choquet integral operators and their application in decision making. Int. J. Fuzzy Syst. 16, 400–410 (2014)Google Scholar
  22. 22.
    Tan, C.Q., Yi, W., Chen, X.H.: Hesitant fuzzy Hamacher aggregation operators for multicriteria decision making. Appl. Soft Comput. 26, 325–349 (2015)CrossRefGoogle Scholar
  23. 23.
    Xia, M.M., Xu, Z.S., Chen, N.: Some hesitant fuzzy aggregation operators with their application in group decision making. Group Decis. Negot. 22, 259–279 (2013)CrossRefGoogle Scholar
  24. 24.
    Xia, M.M., Xu, Z.S.: Hesitant fuzzy information aggregation in decision making. Int. J. Approx. Reason. 52, 395–407 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Yu, D.J.: Some hesitant fuzzy information aggregation operators based on Einstein operational laws. Int. J. Intell. Syst. 29, 320–340 (2014)CrossRefGoogle Scholar
  26. 26.
    Wei, G.W., Zhao, X.F., Lin, R.: Some hesitant interval-valued fuzzy aggregation operators and their applications to multiple attribute decision making. Knowl. Based Syst. 46, 43–53 (2013)CrossRefGoogle Scholar
  27. 27.
    Zhu, B., Xu, Z.S.: Hesitant fuzzy bonferroni means for multi-criteria decision making. J. Oper. Res. Soc. 64, 1831–1840 (2013)CrossRefGoogle Scholar
  28. 28.
    Zhu, B., Xu, Z.S., Xia, M.M.: Hesitant fuzzy geometric Bonferroni means. Inf. Sci. 205, 72–85 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  29. 29.
    Yu, D.J., Wu, Y.Y., Zhou, W.: Generalized hesitant fuzzy Bonferroni mean and its application in multi-criteria group decision making. J. Inf. Comput. Sci. 9, 267–274 (2012)Google Scholar
  30. 30.
    Zhou, W.: On hesitant fuzzy reducible weighted Bonferroni mean and its generalized form for multicriteria aggregation. J. Appl. Math., Article ID 954520 (2014)Google Scholar
  31. 31.
    Zhang, Z., Wang, C., Tian, D., Li, K.: Induced generalized hesitant fuzzy operators and their application to multiple attribute group decision making. Comput. Ind. Eng. 67, 116–138 (2014)CrossRefGoogle Scholar
  32. 32.
    Maclaurin, C.: A second letter to martin folkes, esq.; concerning the roots of equations, with demonstration of other rules of algebra. Philos. Trans. R. Soc. Lond. Ser. A 36, 59–96 (1729)CrossRefGoogle Scholar
  33. 33.
    Detemple, D.W., Robertson, J.M.: On generalized symmetric means of two variables. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 634, 236–238 (1979)MathSciNetGoogle Scholar
  34. 34.
    Beliakov, G., James, S.: On extending generalized Bonferroni means to Atanassov orthopairs in decision making contexts. Fuzzy Sets Syst. 211, 84–98 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  35. 35.
    Liu, P., Liu, Z., Zhang, X.: Some intuitionistic uncertain linguistic Heronian Mean operators and their application to group decision making. Appl. Math. Comput. 230, 570–586 (2014)CrossRefGoogle Scholar
  36. 36.
    Bapat, R.B.: Symmetric function means and permanents. Linear Algebra Appl. 182, 101–108 (1993)zbMATHMathSciNetCrossRefGoogle Scholar
  37. 37.
    Abu-Saris, R., Hajja, M.: On Gauss compounding of symmetric weighted arithmetic means. J. Math. Anal. Appl. 322, 729–734 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  38. 38.
    Gao, P.: On a conjecture on the symmetric means. J. Math. Anal. Appl. 337, 416–424 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  39. 39.
    Qin, J.D., Liu, X.W.: An approach to intuitionistic fuzzy multiple attribute decision making based on Maclaurin symmetric mean operators. J. Intell. Fuzzy Syst. 27, 2177–2190 (2014)MathSciNetGoogle Scholar

Copyright information

© Taiwan Fuzzy Systems Association and Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.School of Economics and ManagementSoutheast UniversityNanjingPeople’s Republic of China
  2. 2.Department of Electrical and Computer EngineeringUniversity of AlbertaEdmontonCanada

Personalised recommendations