Skip to main content
Log in

Aggregating Intuitionistic Fuzzy Preference Relations with Symmetrical Intuitionistic Fuzzy Bonferroni Mean Operators in Group Decision Making

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

Abstract

As a useful aggregation technique, the Bonferroni mean can capture the interrelationship between input arguments and has been a hot research topic, especially, in intuitionistic fuzzy environment. In this paper, it is pointed out by an example that the existing intuitionistic fuzzy Bonferroni mean (IFBM) operators fail to satisfy the need in group decision making with intuitionistic fuzzy preference relations (IFPRs). Then, symmetrical intuitionistic fuzzy Bonferroni mean (SIFBM) operator and weighted SIFBM operator are developed to settle the above issue and some desirable properties of them are provided. Furthermore, an acceptable group multiplicative consistency of the IFPRs is introduced and a novel algorithm is established to jointly and stepwisely reach the acceptable group multiplicative consistency and consensus of IFPRs in group decision making. Finally, numerical examples are given to illustrate the effectiveness of the proposed method and comparisons with the existing methods are made to demonstrate the advantages of the proposed method.

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

Access this article

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. Chiclana, F., Herrera-Viedma, E., Herrera, F., Alonso, S.: Induced ordered weighted geometric operators and their use in the aggregation of multiplicative preference relations. Int. J. Intell. Syst. 19, 233–255 (2004)

    Article  MATH  Google Scholar 

  2. Saaty, T.L.: The analytic hierarchy process. McGraw-Hill, New York (1980)

    MATH  Google Scholar 

  3. Cabrerizo, F.J., Urena, R., Pedrycz, W., Herrera-Viedma, E.: Building consensus in group decision making with an allocation of information granularity. Fuzzy Sets Syst. 255, 115–127 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chiclana, F., Herrera, F., Herrera-Viedma, E.: Integrating three representation models in fuzzy multipurpose decision making based on fuzzy preference relations. Fuzzy Sets Syst. 97, 33–48 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  5. Herrera-Viedma, E., Herrera, F., Chiclana, F., Luque, M.: Some issues on consistency of fuzzy preference relations. Eur. J. Oper. Res. 154, 98–109 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  6. Orlovsky, A.: Decision-making with a fuzzy preference relation. Fuzzy Sets Syst. 1, 155–167 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  7. Tanino, T.: Fuzzy preference orderings in group decision making. Fuzzy Sets Syst. 12, 117–131 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  8. Dong, Y., Xiao, J., Zhang, H., Wang, T.: Managing consensus and weights in iterative multiple-attribute group decision making. Appl. Soft Comput 48, 80–90 (2016)

    Article  Google Scholar 

  9. Xu, Z.S.: Intuitionistic fuzzy aggregation operators. IEEE Trans. Fuzzy Syst. 15, 1179–1187 (2007)

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  11. Szmidt, E., Kacprzyk, J.: A consensus-reaching process under intuitionistic fuzzy preference relations. Int. J. Intell. Syst. 18, 837–852 (2003)

    Article  MATH  Google Scholar 

  12. Liao, H.C., Xu, Z.S.: Consistency of the fused intuitionistic fuzzy preference relation in group intuitionistic fuzzy analytic hierarchy process. Appl. Soft Comput. 35, 812–826 (2015)

    Article  Google Scholar 

  13. Wan, S., Xu, G., Dong, J.: A novel method for group decision making with interval-valued Atanassov intuitionistic fuzzy preference relations. Inform. Sci. 372, 53–71 (2016)

    Article  MATH  Google Scholar 

  14. Xu, G., Wan, S., Wang, F., Dong, J., Zeng, Y.: Mathematical programming methods for consistency and consensus in group decision making with intuitionistic fuzzy preference relations. Knowl. Based Syst. 98, 30–43 (2016)

    Article  Google Scholar 

  15. Beliakov, G., Bustince, H., Goswami, D.P., Mukherjee, U.K., Pal, N.R.: On averaging operators for Atanassov’s intuitionistic fuzzy sets. Inform. Sci. 181, 1116–1124 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Behret, H.: Group decision making with intuitionistic fuzzy preference relations. Knowl. Based Syst. 70, 33–43 (2014)

    Article  Google Scholar 

  17. Chu, J., Liu, X., Wang, Y., Chin, K.: A group decision making model considering both the additive consistency and group consensus of intuitionistic fuzzy preference relationst. Comput. Ind. Eng. 101, 227–242 (2016)

    Article  Google Scholar 

  18. Wang, Z.J., Wang, Y., Li, K.W.: An acceptable consistency-based framework for group decision making with intuitionistic preference relations. Group Decis. Negot. 25, 181–202 (2016)

    Article  Google Scholar 

  19. Wu, J., Chiclana, F.: Multiplicative consistency of intuitionistic reciprocal preference relations and its application to missing values estimation and consensus building. Knowl. Based Syst. 71, 187–200 (2014)

    Article  Google Scholar 

  20. Ma, Z.M., Xu, Z.S.: Hyperbolic scales involving appetites-based intuitionistic multiplicative preference relations for group decision making. Inform. Sci. 451–452, 310–325 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  21. Xia, M., Xu, S.: Entropy/cross entropy-based group decision making under intuitionistic fuzzy environment. Inform. Fus. 13, 31–47 (2012)

    Article  Google Scholar 

  22. Ma, Z.M., Yang, W.: Symmetric intuitionistic fuzzy weighted mean operators based on weighted Archimedean t-norms and t-conorms for multi-criteria decision making. Informatica 31, 89–112 (2020)

    Article  MathSciNet  Google Scholar 

  23. Beliakov, G., James, S.: On extending generalized Bonferroni means to Atanassov orthopairs in decision making contexts. Fuzzy Sets Syst. 211, 84–98 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. Liao, H.C., Xu, Z.S.: Intuitionistic fuzzy hybrid weighted aggregation operators. Int. J. Intell. Syst. 29, 971–993 (2014)

    Article  Google Scholar 

  25. Xu, Z.S., Yager, R.R.: Some geometric aggregation operators based on intuitionistic fuzzy sets. Int. J. Intell. Syst. 35, 417–433 (2006)

    MathSciNet  MATH  Google Scholar 

  26. Xu, Z.S., Yager, R.R.: Intuitionistic fuzzy Bonferroni means. IEEE Trans. Syst. Man Cybern. B 41, 568–578 (2011)

    Article  Google Scholar 

  27. Zhou, W., He, J.: Intuitionistic fuzzy normalized weighted Bonferroni mean and its application in multicriteria decision making. J. Appl. Math. 2012, 1–22 (2012)

    MathSciNet  MATH  Google Scholar 

  28. Muneeza, Abdullah S.: Multicriteria group decision-making for supplier selection based on intuitionistic cubic fuzzy aggregation operators. Int. J. Fuzzy Syst. 22, 810–823 (2020)

    Article  Google Scholar 

  29. Xu, C.Y., Ma, Z.M.: Symmetric intuitionistic multiplicative aggregation operator for group decision making in intuitionistic multiplicative environments. J. Intell. Fuzzy Syst. 36, 5909–5918 (2019)

    Article  Google Scholar 

  30. Rahman, K., Abdullah, S., Jamil, M., Khan, M.Y.: Some generalized intuitionistic fuzzy Einstein hybrid aggregation operators and their application to multiple attribute group decision making. Int. J. Fuzzy Syst. 20, 1567–1575 (2018)

    Article  MathSciNet  Google Scholar 

  31. Liu, P., Wang, P.: Multiple attribute group decision making method based on intuitionistic fuzzy Einstein interactive operations. Int. J. Fuzzy Syst. 22, 790–809 (2020)

    Article  Google Scholar 

  32. Xu, Z.S.: Priority weight intervals derived from intuitionistic multiplicative preference relations. IEEE Trans. Fuzzy Syst. 21, 642–654 (2013)

    Article  Google Scholar 

  33. Jin, F., Ni, Z., Chen, H., Li, Y.: Approaches to group decision making with intuitionistic fuzzy preference relations based on multiplicative consistency. Knowl. Based Syst. 97, 48–59 (2016)

    Article  Google Scholar 

  34. Liao, H.C., Xu, Z.S.: Consistency and consensus of intuitionistic fuzzy preference relations in group decision making, imprecision and uncertainty in information representation and processing. Stud. Fuzzin. Soft Comput. 332, 189–206 (2016)

    Article  MATH  Google Scholar 

  35. Zadeh, L.A.: Fuzzy sets. Inform. Contr. 8, 338–353 (1965)

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  37. Bonferroni, C.: Sulle medie multiple di potenze. Bolletino Matematica Italiana 5, 267–270 (1950)

    MathSciNet  MATH  Google Scholar 

  38. Beliakov, G., Pradera, A., Calvo, T.: Aggregation Functions: a Guide for Practitioners. Springer, Berlin (2007)

    MATH  Google Scholar 

  39. Yager, R.R.: On generalized Bonferroni mean operators for multi-criteria aggregation. Int. J. Approx. Reason. 50, 1279–1286 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  40. Xu, Z.S.: Intuitionistic preference relations and their application in group decision making. Inform. Sci. 177, 2363–2379 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  41. Liao, H.C., Xu, Z.S.: Priorities of intuitionistic fuzzy preference relation based on multiplicative consistency. IEEE Trans. Fuzzy Syst. 22, 1669–1681 (2014)

    Article  Google Scholar 

  42. Wang, Z.J.: Derivation of intuitionistic fuzzy weights based on intuitionistic fuzzy preference relations. Appl. Math. Model. 37, 6377–6388 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  43. Lehmer, D.H.: On the compounding of certain means. J. Math. Analy. Appl. 36, 183–200 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  44. Yang, Y., Wang, X., Xu, Z.S.: The multiplicative consistency threshold of intuitionistic fuzzy preference relation. Inform. Sci. 477, 349–368 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  45. Xu, Z.S., Liao, H.C.: Intuitionistic fuzzy analytic hierarchy process. IEEE Trans. Fuzzy Syst. 22, 749–761 (2014)

    Article  Google Scholar 

Download references

Acknowledgements

This research was supported by the Natural Science Foundation of Shandong Province (Grant No. ZR2017MG027).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Wei Yang.

Ethics declarations

Conflicts of interest

The authors declare that they have no conflict of interest.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yang, W., Jhang, S.T., Shi, S.G. et al. Aggregating Intuitionistic Fuzzy Preference Relations with Symmetrical Intuitionistic Fuzzy Bonferroni Mean Operators in Group Decision Making. Int. J. Fuzzy Syst. 23, 455–473 (2021). https://doi.org/10.1007/s40815-020-00960-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40815-020-00960-4

Keywords

Navigation