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A Review of Hesitant Fuzzy Sets: Quantitative and Qualitative Extensions

  • Rosa M. RodríguezEmail author
  • Luis Martínez
  • Francisco Herrera
  • Vicenç Torra
Chapter
Part of the Studies in Fuzziness and Soft Computing book series (STUDFUZZ, volume 341)

Abstract

Since the concept of fuzzy set was introduced, different extensions and generalizations have been proposed to manage the uncertainty in different problems. This chapter is focused in a recent extension so-called hesitant fuzzy set. Many researchers have paid attention on it and have proposed different extensions both in quantitative and qualitative contexts. Several concepts, basic operations and its extensions are revised in this chapter.

Keywords

Hesitant fuzzy set Operations Extensions 

Notes

Acknowledgements

This work is partially funded by the Spanish research projects TIN2011-27076-C03-03, TIN2015-66524-P, the Spanish Ministry of Economy and Finance Postdoctoral Training (FPDI-2013-18193) and ERDF.

References

  1. 1.
    Atanassov, K.T.: Intuitionistic fuzzy sets. Fuzzy Sets Syst. 20, 87–96 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bedregal, B., Reiser, R., Bustince, H., López-Molina, C., Torra, V.: Aggregating functions for typical hesitant fuzzy elements and the action of automorphisms. Inf. Sci. 256(1), 82–97 (2014)CrossRefzbMATHGoogle Scholar
  3. 3.
    Beg, I., Rashid, T.: TOPSIS for hesitant fuzzy linguistic term sets. Int. J. Intell. Syst. 28, 1162–1171 (2013)CrossRefGoogle Scholar
  4. 4.
    Chen, N., Xu, Z.S.: Properties of interval-valued hesitant fuzzy sets. J. Intell. Fuzzy Syst. 27(1), 143–158 (2014)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Chen, N., Xu, Z.S., Xia, M.M.: Interval-valued hesitant preference relations and their applications to group decision making. Knowl. Based Syst. 37(1), 528–540 (2013)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chen, S.M., Hong, J.A.: Multicriteria linguistic decision making based on hesitant fuzzy linguistic term sets and the aggregation of fuzzy sets. Inf. Sci. 286, 63–74 (2014)CrossRefGoogle Scholar
  7. 7.
    Chen, Y., Penga, X., Guanb, G., Jiangb, H.: Approaches to multiple attribute decision making based on the correlation coefficient with dual hesitant fuzzy information. J. Intell. Fuzzy Syst. 26(5), 2547–2556 (2014)MathSciNetGoogle Scholar
  8. 8.
    Chiclana, F., Herrera, F., Herrera-Viedma, E.: Integrating multiplicative preference relations in a multipurpose decision-making model based on fuzzy preference relations. Fuzzy Sets Syst. 122(2), 277–291 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Crawford, G., Williams, C.: A note on the analysis of subjective judgment matrices. J. Math. Psychol. 29(4), 387–405 (1985)CrossRefzbMATHGoogle Scholar
  10. 10.
    Dong, Y., Chen, X., Herrera, F.: Minimizing adjusted simple terms in the consensus reaching process with hesitant linguistic assessments in group decision making. Inf. Sci. 297, 95–117 (2015)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Dubois, D., Gottwald, S., Hajek, P., Kacprzyk, J., Prade, H.: Terminological difficulties in fuzzy set theory—the case of “intuitionistic fuzzy sets”. Fuzzy Sets Syst. 156(3), 485–491 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Farhadinia, B.: Information measures for hesitant fuzzy sets and interval-valued hesitant fuzzy sets. Inf. Sci. 240, 129–144 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Goguen, J.A.: L-fuzzy sets. J. Math. Anal. Appl. 18(1), 145–174 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hesamian, G., Shams, M.: Measuring similarity and ordering based on hesitant fuzzy linguistic term sets. J. Intell. Fuzzy Syst. 28(2), 983–990 (2015)MathSciNetGoogle Scholar
  15. 15.
    Huang, H.C., Yang, X.: Pairwise comparison and distance measure of hesitant fuzzy linguistic term sets. Math. Probl. Eng. 1–8, 2014 (2014)Google Scholar
  16. 16.
    Ju, Y., Yang, S., Liu, X.: Some new dual hesitant fuzzy aggregation operators based on Choquet integral and their applications to multiple attribute decision making. J. Intell. Fuzzy Syst. 27(6), 2857–2868 (2014)MathSciNetGoogle Scholar
  17. 17.
    Ju, Y., Zhang, W., Yang, S.: Some dual hesitant fuzzy hamacher aggregation operators and their applications to multiple attribute decision making. J. Intell. Fuzzy Syst. 27(5), 2481–2495 (2014)MathSciNetGoogle Scholar
  18. 18.
    Klir, G.J., Yuan, B.: Fuzzy Sets and Fuzzy Logic: Theory and Applications. Prentice-Hall PTR (1995)Google Scholar
  19. 19.
    Lee, L.W., Chen, S.M.: Fuzzy decision making based on likelihood-based comparison relations of hesitant fuzzy linguistic term sets and hesitant fuzzy linguistic operators. Inf. Sci. 294, 513–529 (2015)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Li, L.G., Peng, D.H.: Interval-valued hesitant fuzzy hamacher synergetic weighted aggregation operators and their application to shale gas areas selection. Math. Probl. Eng. 1–25, 2014 (2014)MathSciNetGoogle Scholar
  21. 21.
    Li, Y.B., Zhang, J.P.: Approach to multiple attribute decision making with hesitant triangular fuzzy information and their application to customer credit risk assessment. J. Intell. Fuzzy Syst. 26(6), 2853–2860 (2014)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Liao, H., Xu, Z.S., Zeng, X.J.: Distance and similarity measures for hesitant fuzzy linguistic term sets and their application in multi-criteria decision making. Inf. Sci. 271, 125–142 (2014)MathSciNetCrossRefGoogle Scholar
  23. 23.
    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
  24. 24.
    Lin, R., Zhao, X., Wei, G.: Models for selecting an ERP system with hesitant fuzzy linguistic information. J. Intell. Fuzzy Syst. 26(5), 2155–2165 (2014)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Liu, H., Rodríguez, R.M.: A fuzzy envelope for hesitant fuzzy linguistic term set and its application to multicriteria decision making. Inf. Sci. 258, 266–276 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Martínez, L., Liu, J., Yang, J.B., Herrera, F.: A multigranular hierarchical linguistic model for design evaluation based on safety and cost analysis. Int. J. Intell. Syst. 20(12), 1161–1194 (2005)CrossRefzbMATHGoogle Scholar
  27. 27.
    Mendel, J.M., John, R.I.: Type-2 fuzzy sets made simple. IEEE Trans. Fuzzy Syst. 10(2), 117–127 (2002)CrossRefGoogle Scholar
  28. 28.
    Meng, F., Chen, X.: An approach to interval-valued hesitant fuzzy multi-attribute decision making with incomplete weight information based on hybrid Shapley operators. Informatica 25(4), 617–642 (2014)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Pei, Z., Yi, L.: A note on operations of hesitant fuzzy sets. Int. J. Comput. Intell. Syst. 8(2), 226–239 (2015)CrossRefGoogle Scholar
  30. 30.
    Peng, D.H., Wang, T.D., Gao, C.Y., Wang, H.: Continuous hesitant fuzzy aggregation operators and their application to decision making under interval-valued hesitant fuzzy setting. Sci. World J. 1–20, 2014 (2014)Google Scholar
  31. 31.
    Qian, G., Wang, H., Feng, X.: Generalized hesitant fuzzy sets and their application in decision support system. Knowl. Based Syst. 37(1), 357–365 (2013)CrossRefGoogle Scholar
  32. 32.
    Rodríguez, R.M., Martínez, L., Herrera, F.: Hesitant fuzzy linguistic term sets for decision making. IEEE Trans. Fuzzy Syst. 20(1), 109–119 (2012)CrossRefGoogle Scholar
  33. 33.
    Rodríguez, R.M., Martínez, L., Herrera, F.: A group decision making model dealing with comparative linguistic expressions based on hesitant fuzzy linguistic term sets. Inf. Sci. 241(1), 28–42 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Rodríguez, R.M., Martínez, L., Torra, V., Xu, Z.S., Herrera, F.: Hesitant fuzzy sets: state of the art and future directions. Int. J. Intell. Syst. 29(6), 495–524 (2014)CrossRefGoogle Scholar
  35. 35.
    Saaty, L.T.: The Analytic Hierarchy Process. McGraw-Hill (1980)Google Scholar
  36. 36.
    Saaty, R.W.: The analytic hierarchy process-what it is and how it is used. Math. Model. 9(3–5), 161–176 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Sahu, S.K., Sahu, N., Thakur, R.S., Thakur, G.S.: Hesitant fuzzy linguistic term set based document classification. In: Proceedings of the International Conference on Communication Systems and Network Technologies, pp. 586–590, Gwalior, India (2013)Google Scholar
  38. 38.
    Shi, J., Meng, C., Liu, Y.: Approach to multiple attribute decision making based on the intelligence computing with hesitant triangular fuzzy information and their application. J. Intell. Fuzzy Syst. 27(2), 701–707 (2014)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Singh, P.: A new method for solving dual hesitant fuzzy assignment problems with restrictions based on similarity measure. Appl. Soft Comput. 24, 559–571 (2014)CrossRefGoogle Scholar
  40. 40.
    Torra, V.: Hesitant fuzzy sets. Int. J. Intell. Syst. 25(6), 529–539 (2010)zbMATHGoogle Scholar
  41. 41.
    Torra, V.: Artificial intelligence research and development, chapter on the derivation of weights using the geometric mean approach for set-valued matrices. In: Frontiers in Artificial Intelligence and Applications, pp. 193–201 (2014)Google Scholar
  42. 42.
    Torra, V.: Derivation of priorities and weights for set-valued matrices using the geometric mean approach. Appl. Artif. Intell. 29(5), 500–513 (2015)CrossRefGoogle Scholar
  43. 43.
    Torra, V., Narukawa, Y.: Modeling decisions: information fusion and aggregation operators. Springer, Heidelberg (2007)Google Scholar
  44. 44.
    Torra, V., Narukawa, Y.: On hesitant fuzzy sets and decision. In: Proceedings of the 18th IEEE International Conference on Fuzzy Systems, pp. 1378–1382 (2009)Google Scholar
  45. 45.
    Türksen, I.B.: Interval valued fuzzy sets based on normal forms. Fuzzy Sets Syst. 20, 191–210 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Wang, C., Li, Q., Zhou, X.: Multiple attribute decision making based on generalized aggregation operators under dual hesitant fuzzy environment. J. Appl. Math. 1–12, 2014 (2014)MathSciNetGoogle Scholar
  47. 47.
    Wang, C., Li, Q., Zhou, X., Yang, T.: Hesitant triangular fuzzy information aggregation operators based on bonferroni means and their application to multiple attribute decision making. Sci. World J. 1–15, 2014 (2014)Google Scholar
  48. 48.
    Wang, H.: Extended hesitant fuzzy linguistic term sets and their aggregation in group decision making. Int. J. Comput. Intell. Syst. 8(1), 14–33 (2015)CrossRefGoogle Scholar
  49. 49.
    Wang, H., Xu, Z.S.: Some consistency measures of extended hesitant fuzzy linguistic preference relations. Inf. Sci. 297, 316–331 (2015)MathSciNetCrossRefGoogle Scholar
  50. 50.
    Wang, H., Zhao, X., Wei, G.: Dual hesitant fuzzy aggregation operators in multiple attribute decision making. J. Intell. Fuzzy Syst. 26(5), 2281–2290 (2014)MathSciNetzbMATHGoogle Scholar
  51. 51.
    Wang, J.Q., Wang, J., Chen, Q.H., Zhang, H.Y., Chen, X.H.: An outranking approach for multi-criteria decision-making with hesitant fuzzy linguistic term sets. Inf. Sci. 280, 338–351 (2014)MathSciNetCrossRefGoogle Scholar
  52. 52.
    Wang, J.Q., Wu, J.T., Wang, J., Zhang, H.Y., Chen, X.H.: Interval-valued hesitant fuzzy linguistic sets and their applications in multi-criteria decision-making problems. Inf. Sci. 288, 55–72 (2014)MathSciNetCrossRefGoogle Scholar
  53. 53.
    Wang, L., Ni, M., Zhu, L.: Correlation measures of dual hesitant fuzzy sets. J. Appl. Math. 1–12, 2013 (2013)Google Scholar
  54. 54.
    Wei, C., Ren, Z., Rodríguez, R.M.: A hesitant fuzzy linguistic TODIM method based on a score function. Int. J. Comput. Intell. Syst. 8(4), 701–712 (2015)CrossRefGoogle Scholar
  55. 55.
    Wei, C., Zhao, N., Tang, X.: Operators and comparisons of hesitant fuzzy linguistic term sets. IEEE Trans. Fuzzy Syst. 22(3), 575–585 (2014)CrossRefGoogle Scholar
  56. 56.
    Wei, G., Lin, R., Wang, H.: Distance and similarity measures for hesitant interval-valued fuzzy sets. J. Intell. Fuzzy Syst. 27(1), 19–36 (2014)MathSciNetzbMATHGoogle Scholar
  57. 57.
    Wei, G., Wang, H., Zhao, X., Lin, R.: Hesitant triangular fuzzy information aggregation in multiple attribute decision making. J. Intell. Fuzzy Syst. 26(3), 1201–1209 (2014)MathSciNetzbMATHGoogle Scholar
  58. 58.
    Wei, G., Zhao, X.: Induced hesitant interval-valued fuzzy Einstein aggregation operators and their application to multiple attribute decision making. J. Intell. Fuzzy Syst. 24(4), 789–803 (2013)MathSciNetzbMATHGoogle Scholar
  59. 59.
    Wei, G., Zhao, X., 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
  60. 60.
    Wei, G., Zhao, X., Lin, R.: Models for hesitant interval-valued fuzzy multiple attribute decision making based on the correlation coefficient with incomplete weight information. J. Intell. Fuzzy Syst. 26(4), 1631–1644 (2014)MathSciNetzbMATHGoogle Scholar
  61. 61.
    Xia, M.M., Xu, Z.S.: Hesitant fuzzy information aggregation in decision making. Int. J. Approx. Reason. 52, 395–407 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  62. 62.
    Xia, M.M., Xu, Z.S.: Managing hesitant information in GDM problems underfuzzy and multiplicative preference relations. Int. J. Uncertain. Fuzziness Knowl. Based Syst. 21(06), 865–897 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  63. 63.
    Yang, S., Ju, Y.: Dual hesitant fuzzy linguistic aggregation operators and their applications to multi-attribute decision making. J. Intell. Fuzzy Syst. 27(4), 1935–1947 (2014)MathSciNetzbMATHGoogle Scholar
  64. 64.
    Yavuz, M., Oztaysi, B., Cevik Onar, S., Kahraman, C.: Multi-criteria evaluation of alternative-fuel vehicles via a hierarchical hesitant fuzzy linguistic model. Expert Syst. Appl. 42(5), 2835–2848 (2015)CrossRefGoogle Scholar
  65. 65.
    Ye, J.: Correlation coefficient of dual hesitant fuzzy sets and its application to multiple attribute decision making. Appl. Math. Model. 38(2), 659–666 (2014)MathSciNetCrossRefGoogle Scholar
  66. 66.
    Yu, D.: Triangular hesitant fuzzy set and its application to teaching quality evaluation. J. Inf. Comput. Sci. 10(7), 1925–1934 (2013)CrossRefGoogle Scholar
  67. 67.
    Yu, D.: Some generalized dual hesitant fuzzy geometric aggregation operators and applications. Int. J. Uncertain. Fuzziness Knowl. Based Syst. 22(3), 367–384 (2014)CrossRefzbMATHGoogle Scholar
  68. 68.
    Yu, D., Li, D.F.: Dual hesitant fuzzy multi-criteria decision making and its application to teaching quality assessment. J. Intell. Fuzzy Syst. 27(4), 1679–1688 (2014)MathSciNetzbMATHGoogle Scholar
  69. 69.
    Zadeh, L.: Fuzzy sets. Inf. Control 8, 338–353 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  70. 70.
    Zhang, Y.: Research on the computer network security evaluation based on the DHFHCG operator with dual hesitant fuzzy information. J. Intell. Fuzzy Syst. 28(1), 199–204 (2015)MathSciNetGoogle Scholar
  71. 71.
    Zhang, Z., Wu, C.: A decision support model for group decision making with hesitant multiplicative preference relations. Inf. Sci. 282, 136–166 (2014)MathSciNetCrossRefGoogle Scholar
  72. 72.
    Zhang, Z., Wu, C.: Hesitant fuzzy linguistic aggregation operators and their applications to multiple attribute group decision making. J. Intell. Fuzzy Syst. 26(5), 2185–2202 (2014)MathSciNetzbMATHGoogle Scholar
  73. 73.
    Zhang, Z., Wu, C.: On the use of multiplicative consistency in hesitant fuzzy linguistic preference relations. Knowl. Based Syst. 72, 13–27 (2014)CrossRefGoogle Scholar
  74. 74.
    Zhao, X., Lin, R., Wei, G.: Hesitant triangular fuzzy information aggregation based on Einstein operations and their application to multiple attribute decision making. Expert Syst. Appl. 41(4, Part 1), 1086–1094Google Scholar
  75. 75.
    Zhong, G., Xu, L.: Models for multiple attribute decision making method in hesitant triangular fuzzy setting. J. Intell. Fuzzy Syst. 26(5), 2167–2174 (2014)MathSciNetzbMATHGoogle Scholar
  76. 76.
    Zhu, B., Xu, Z.S.: Regression methods for hesitant fuzzy preference relations. Technol. Econ. Dev. Econ. 19, S214–S227 (2013)Google Scholar
  77. 77.
    Zhu, B., Xu, Z.S.: Consistency measures for hesitant fuzzy linguistic preference relations. IEEE Trans. Fuzzy Syst. 22, 35–45 (2014)MathSciNetCrossRefGoogle Scholar
  78. 78.
    Zhu, B., Xu, Z.S.: Stochastic preference analysis in numerical preference relations. Eur. J. Oper. Res. 237(2), 628–633 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  79. 79.
    Zhu, B., Xu, Z.S., Xia, M.M.: Dual hesitant fuzzy sets. J. Appl. Math. 1–13, 2012 (2012)MathSciNetzbMATHGoogle Scholar
  80. 80.
    Zhu, J.Q., Fu, F., Yin, K.X., Luo, J.Q., Wei, D.: Approaches to multiple attribute decision making with hesitant interval-valued fuzzy information under correlative environment. J. Intell. Fuzzy Syst. 27(2), 1057–1065 (2014)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Rosa M. Rodríguez
    • 1
    Email author
  • Luis Martínez
    • 2
  • Francisco Herrera
    • 1
  • Vicenç Torra
    • 3
  1. 1.University of GranadaGranadaSpain
  2. 2.University of JaénJaénSpain
  3. 3.University of SkövdeSkövdeSweden

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