Skip to main content
SpringerLink
Log in

Weighted Interval-Valued Hesitant Fuzzy Sets and Its Application in Group Decision Making

  • Published:
International Journal of Fuzzy Systems Aims and scope Submit manuscript

Abstract

The interval-valued hesitant fuzzy set, which allows decision makers to use several interval numbers to assess a variable, is a useful tool to deal with situations in which people are hesitant in providing their interval-valued assessments. In this paper, we introduce the concept of weighted interval-valued hesitant fuzzy set, in which different weights are designed to these possible membership degrees, and the weights indicate that the decision maker has different confidence in giving every possible assessment of the membership degree. Then we define some basic operations such as union, intersection, complement, multiplication and power operation of weighted interval-valued hesitant fuzzy sets and weighted interval-valued hesitant fuzzy elements, discuss their operation properties, and propose the score function of the weighted interval-valued hesitant fuzzy element to compare two weighted hesitant fuzzy elements. Furthermore, we introduce the concept of hesitance degree of weighted interval-valued hesitant fuzzy element, present four aggregation operators such as the weighted interval-valued hesitant fuzzy-weighted averaging operator, the weighted interval-valued hesitant fuzzy-weighted geometric operator, the generalized weighted interval-valued hesitant fuzzy-weighted averaging operator and the generalized weighted interval-valued hesitant fuzzy-weighted geometric operator to aggregate weighted interval-valued hesitant fuzzy information, and build the mathematical model of multi-criteria group decision making based on the expert weights (known and unknown). Finally, a numerical example is given to illustrate the effectiveness and feasibility of our proposed method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bedregal, B., Reiser, R., Bustince, H., Lopez-Molina, C., Torra, V.: Aggregation functions for typical hesitant fuzzy elemennts and the action of automorphisms. Inf. Sci. 255, 82–99 (2014)

    Article  MATH  Google Scholar 

  2. Chen, N., Xu, Z.S., Xia, M.M.: Interval-valued hesitant preference relations and their applications to group decision making. Knowl. Based Syst. 37, 528–540 (2013)

    Article  Google Scholar 

  3. 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)

    Article  MathSciNet  MATH  Google Scholar 

  4. Facchinetti, G., Ricci, R.G., Muzzioli, S.: Note on ranking fuzzy triangular numbers. Int. J. Intell. Syst. 13, 613–622 (1998)

    Article  Google Scholar 

  5. Farhadinia, B.: Information measures for hesitant fuzzy sets and interval-valued hesitant fuzzy sets. Inf. Sci. 240, 129–144 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Farhadinia, B.: A novel method of ranking hesitant fuzzy values for multiple attribute decision making problems. Int. J. Intell. Syst. 28, 752–767 (2013)

    Article  Google Scholar 

  7. Farhadinia, B.: Distance and similarity measures for higher order hesitant fuzzy sets. Knowl. Based Syst. 55, 43–48 (2014)

    Article  MATH  Google Scholar 

  8. Gitinavard, H., Makui, A., Jabbarzadeh, A.: Interval-valued hesitant fuzzy method based on group decision analysis for estimating weights of decision makers. J. Ind. Syst. Eng. 9, 96–110 (2016)

    Google Scholar 

  9. Herrera, F., Herrera-Viedma, E.: Linguistic decision analysis: steps for solving decision problems under linguistic information. Fuzzy Sets Syst. 115, 67–82 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ju, Y., Liu, X., Yang, S.: Interval-valued dual hesitant fuzzy aggregation operators and their applications to multiple attribute decision making. J. Intell. Fuzzy Syst. 27, 1203–1218 (2014)

    MathSciNet  MATH  Google Scholar 

  11. 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 operations. Inf. Sci. 294, 513–529 (2015)

    Article  MATH  Google Scholar 

  12. Li, D.Q., Zeng, W.Y., Li, J.H.: New distance and similarity measures on hesitant fuzzy sets and their applications in multiple criteria decision making. Eng. Appl. Artif. Intell. 40, 11–16 (2015)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. Li, D.Q., Zeng, W.Y., Yin, Q.: Ranking method of interval numbers based on the Boolean matrix. Soft Comput. 22(12), 4113–4122 (2018)

    Article  MATH  Google Scholar 

  15. Liao, H.C., Xu, Z.S., Xia, M.M.: Multiplicative consistency of hesitant fuzzy preference relation and its application in group decision making. Int. J. Inf. Technol. Decis. Mak. 13, 47–76 (2014)

    Article  Google Scholar 

  16. Liao, H.C., 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)

    Article  MathSciNet  MATH  Google Scholar 

  17. Liao, H.C., Xu, Z.S., Zeng, X.J.: Novel correlation coefficients between hesitant fuzzy sets and their application in decision making. Knowl. Based Syst. 82, 115–127 (2015)

    Article  Google Scholar 

  18. Liao, H.C., Xu, Z.S.: Approaches to manage hesitant fuzzy linguistic information based on the cosine distance and similarity measures for HFLTSs and their application in qualitative decision making. Expert Syst. Appl. 42, 5328–5336 (2015)

    Article  Google Scholar 

  19. Liu, J., Zhou, N., Zhuang, L.H., Li, N., Jin, F.F.: Interval-valued hesitant fuzzy multiattribute group decision making based on improved Hamacher aggregation operators and continuous entropy. Math. Probl. Eng. 2017, 1–20 (2017)

    Google Scholar 

  20. Meng, F.Y., Chen, X.H., Zhang, Q.: Multi-attribute decision analysis under a linguistic hesitant fuzzy environment. Inf. Sci. 267, 287–305 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. Mu, Z.M., Zeng, S.Z., Baležentis, T.: A novel aggregation principle for hesitant fuzzy elements. Knowl. Based Syst. 84, 134–143 (2015)

    Article  Google Scholar 

  22. Onar, S.C., Oztaysi, B., Kahraman, C.: Strategic decision selection using hesitant fuzzy TOPSOS and interval type-2 fuzzy AHP: a case study. Int. J. Comput. Intell. Syst. 7, 1002–1021 (2014)

    Article  Google Scholar 

  23. Park, J.H., Gwak, M.G., Kwun, Y.C.: Uncertain linguistic harmonic mean operators and their applications to multiple attribute group decision making. Computing 93, 47–64 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  24. Pavlačka, O.: On various approaches to normalization of interval and fuzzy weights. Fuzzy Sets Syst. 243, 110–130 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  25. 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)

    Article  MathSciNet  MATH  Google Scholar 

  26. 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. 2014, 1–20 (2014)

    Google Scholar 

  27. Qian, G., Wang, H., Feng, X.: Generalized hesitant fuzzy sets and their application in decision support system. Knowl. Based Syst. 37, 357–365 (2013)

    Article  Google Scholar 

  28. Quirós, P., Alonso, P., Bustince, H., Díaz, I., Montes, S.: An entropy measure definition for finite interval-valued hesitant fuzzy sets. Knowl. Based Syst. 84, 121–133 (2015)

    Article  Google Scholar 

  29. Rodríguez, R.M., Martínez, L., Herrera, F.: Hesitant fuzzy linguistic term sets for decision making. IEEE Trans. Fuzzy Syst. 20, 109–119 (2012)

    Article  Google Scholar 

  30. 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, 28–42 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  31. 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, 495–524 (2014)

    Article  Google Scholar 

  32. Sevastjanov, P., Dymova, L.: Generalised operations on hesitant fuzzy values in the framework of Dempster–Shafer theory. Inf. Sci. 311, 39–58 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  34. Torra, V., Narukawa, Y.: On hesitant fuzzy sets and decision. In: The 18th IEEE International Conference on Fuzzy Systems, Jeju Island, Korea, pp. 1378–1382 (2009)

  35. Wang, J., Lan, J.B., Ren, P.Y., Luo, Y.Y.: Some programming models to derive priority weights from additive interval fuzzy preference relation. Knowl. Based Syst. 27, 69–77 (2012)

    Article  Google Scholar 

  36. 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)

    Article  MathSciNet  MATH  Google Scholar 

  37. Wei, C.P., Zhao, N., Tang, X.J.: Operators and comparisons of hesitant fuzzy linguistic term sets. IEEE Trans. Fuzzy Syst. 22, 575–585 (2014)

    Article  Google Scholar 

  38. Wei, G.W.: Hesitant fuzzy prioritized operators and their application to multiple attribute decision making. Knowl. Based Syst. 31, 176–182 (2012)

    Article  Google Scholar 

  39. 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)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  41. Xia, M.M., Xu, Z.S., Chen, N.: Some hesitant fuzzy aggregation operators with their aplication in group decision making. Group Decis. Negot. 22, 259–279 (2013)

    Article  Google Scholar 

  42. Xiong, S.H., Chen, Z.S., Li, Y.L., Chin, K.S.: On extending power-geometric operators to interval-valued hesitant fuzzy sets and their applications to group decision making. Int. J. Inf. Technol. Decis. Mak. 15, 1055–1114 (2016)

    Article  Google Scholar 

  43. Xu, Z.S., Da, Q.L.: The uncertain OWA operator. Int. J. Intell. Syst. 17, 569–575 (2002)

    Article  MATH  Google Scholar 

  44. Xu, Z.S., Xia, M.M.: Distance and similarity measures for hesitant fuzzy sets. Inf. Sci. 181, 2128–2138 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  45. Xu, Z.S., Xia, M.M.: On distance and correlation measures of hesitant fuzzy information. Int. J. Intell. Syst. 26, 410–425 (2011)

    Article  MATH  Google Scholar 

  46. Xu, Z.S., Zhang, X.L.: Hesitant fuzzy multi-attribute decision making based on TOPSIS with incomplete weight information. Knowl. Based Syst. 52, 53–64 (2013)

    Article  Google Scholar 

  47. Ye, J.: Correlation coefficient of dual hesitant fuzzy sets and its application to multiple attribute decision making. Appl. Math. Model. 38, 659–666 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  48. Ye, J.: Interval-valued hesitant fuzzy prioritized weighted aggregation operators for multiple attribute decision making. J. Algorithm Comput. Technol. 8, 179–192 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  49. 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 

  50. Zadeh, L.A.: Fuzzy sets. Inf. Control 8, 338–356 (1965)

    Article  MATH  Google Scholar 

  51. Zeng, W.Y., Li, D.Q., Yin, Q.: Distance and similarity measures of hesitant fuzzy sets and their application in pattern recognition. Pattern Recognit. Lett. 84, 267–271 (2016)

    Article  Google Scholar 

  52. Zhang, N., Wei, G.W.: Extension of VIKOR method for decision making problem based on hesitant fuzzy set. Appl. Math. Model. 37, 4938–4947 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  53. Zhang, Z.M.: Hesitant fuzzy power aggregation operators and their application to multiple attribute group decision making. Inf. Sci. 234, 150–181 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  54. Zhu, B., Xu, Z.S., Xia, M.M.: Hesitant fuzzy geomeric Bonferroni means. Inf. Sci. 182, 72–85 (2012)

    Article  MATH  Google Scholar 

  55. Zhu, B., Xu, Z.S.: Consistency measures for Hesitant fuzzy linguistic preference relations. IEEE Trans. Fuzzy Syst. 22, 35–45 (2014)

    Article  Google Scholar 

Download references

Acknowledgements

The authors are grateful to the anonymous reviewers, for their excellent comments and valuable suggestions, and the Editor-in-Chief, Professor Shun-Feng Su, for his kind help, that help us improve this paper.

Author information

Authors and Affiliations

  1. College of Information Science and Technology, Beijing Normal University, Beijing, 100875, People’s Republic of China

    Wenyi Zeng, Deqing Li & Qian Yin

  2. School of Computer and Artificial Intelligence, Xiamen Institute of Technology, Xiamen, 361021, People’s Republic of China

    Deqing Li

Authors
  1. Wenyi Zeng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Deqing Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Qian Yin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Deqing Li or Qian Yin.

Additional information

This work is supported by Grants from the National Natural Science Foundation of China (10971243, 61472043).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zeng, W., Li, D. & Yin, Q. Weighted Interval-Valued Hesitant Fuzzy Sets and Its Application in Group Decision Making. Int. J. Fuzzy Syst. 21, 421–432 (2019). https://doi.org/10.1007/s40815-018-00599-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40815-018-00599-2

Keywords

Search

Navigation