Skip to main content
SpringerLink
Log in

A Consistency-Based Method to Decision Making with Triangular Fuzzy Multiplicative Preference Relations

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

Abstract

Triangular fuzzy multiplicative preference relations (TFMPRs) enable decision makers to apply triangular fuzzy numbers to denote their preferences, which can express their vagueness and fuzziness. Consistency analysis is an important research topic that can guarantee logical ranking order. Considering previous research regarding the consistency of TFMPRs, several issues and limitations exist with these relations. The researches cannot cope well with certain situations, such as unacceptably consistent TFMPRs and incomplete TFMPRs. In this paper, a new consistency definition for TFMPRs is defined that can overcome the issues in the previous concepts. Next, several desirable properties of the definition are discussed, and the relationship between the new definition and two existing ones is presented. A linear goal programming model is also built to judge the consistency of TFMPRs, and a method is introduced to obtain an acceptably consistent TFMPR from an inconsistent one. Furthermore, several goal programming models are constructed to estimate missing values in an incomplete TFMPR. Finally, a decision-making method with TFMPRs is developed. Illustrative examples are offered to demonstrate the concrete application of the developed procedure, and a comparison analysis is also made.

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

Fig. 1

Similar content being viewed by others

References

  1. Buckley, J.J.: Fuzzy hierarchical analysis. Fuzzy Sets Syst. 17(3), 233–247 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  2. Brunelli, M.: A note on the article “Inconsistency of pair-wise comparison matrix with fuzzy elements based on geometric mean” [Fuzzy Sets and Systems161 (2010) 1604–1613]. Fuzzy Sets Syst. 176(1), 76–78 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chou, C.C., Liu, L.J., Huang, S.F., Yih, J.M., Han, T.C.: An evaluation of airline service quality using the fuzzy weighted SERVQUAL method. Appl. Soft Comput. 11(2), 2117–2128 (2011)

    Article  Google Scholar 

  4. Dubois, D.: The role of fuzzy sets in decision sciences: old techniques and new directions. Fuzzy Sets Syst. 184(1), 3–28 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Gong, Z.W.: Least-square method to priority of the fuzzy preference relations with incomplete information. Int. J. Approx. Reason. 47(2), 258–264 (2008)

    Article  MATH  Google Scholar 

  6. Gong, Z.W., Zhang, L.F., Liu, S.F.: Study on group decision making based on the triangular fuzzy number preference relations under incomplete information. J. Syst. Eng. 23(3), 269–275 (2008)

    MATH  Google Scholar 

  7. Hu, J.H., Yang, Y., Chen, X.H.: A novel TODIM method based three-way decision model for medical treatment selection. Int. J. Fuzzy Syst (2017). doi:10.1007/s40815-017-0320-3

    Google Scholar 

  8. Kwiesielewicz, M.: A note on the fuzzy extension of Saaty’s priority theory. Fuzzy Sets Syst. 95(2), 161–172 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  9. Leung, L.C., Cao, D.: On consistency and ranking of alternatives in fuzzy AHP. Eur. J. Oper. Res. 124(1), 102–113 (2000)

    Article  MATH  Google Scholar 

  10. Liu, F., Zhang, W.G., Zhang, L.H.: Consistency analysis of triangular fuzzy reciprocal preference relations. Eur. J. Oper. Res. 235(3), 718–726 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Liu, F., Pedrycz, W., Wang, Z.X., Zhang, W.G.: An axiomatic approach to approximation-consistency of triangular fuzzy reciprocal preference relations”. Fuzzy Sets Syst. (2017). doi:10.1016/j.fss.2017.02.004

    MathSciNet  MATH  Google Scholar 

  12. Liu, F., Pedrycz, W., Zhang, W.G.: Limited rationality and its quantification through the interval number judgments with permutations”. IEEE Trans. Cybern. (2016). doi:10.1109/TCYB.2016.2594491

    Google Scholar 

  13. Liu, F., Zhang, W.G., Zhang, L.H.: A group decision-making model based on a generalized ordered weighted geometric average operator with interval-valued preference relations. Fuzzy Sets Syst. 246, 1–18 (2014)

    Article  MATH  Google Scholar 

  14. Liu, F., Zhang, W.G.: TOPSIS-based consensus model for group decision making with incomplete interval fuzzy preference relations. IEEE Trans. Cybern. 44(8), 1283–1294 (2014)

    Article  MathSciNet  Google Scholar 

  15. Liang, R.X., Wang, J.Q., Zhang, H.Y.: A multi-criteria decision-making method based on single-valued trapezoidal neutrosophic preference relations with complete weight information. Neural Comput. Appl. (2017). doi:10.1007/s00521-017-2925-8

    Google Scholar 

  16. Meng, F.Y., Chen, X.H.: A new method for triangular fuzzy compare wise judgment matrix process based on consistency analysis. Int. J. Fuzzy Syst. 19(1), 27–46 (2017)

    Article  MathSciNet  Google Scholar 

  17. Meng, F.Y., Tan, C.Q., Chen, X.H.: Multiplicative consistency analysis for interval reciprocal preference relations: a comparative study. Omega 68, 17–38 (2017)

    Article  Google Scholar 

  18. Meng, F.Y., Tang, J., An, Q.X., Chen, X.H.: Decision making with intuitionistic linguistic preference relations”. Int. Trans. Oper. Res. (2017). doi:10.1111/itor.12383

    Google Scholar 

  19. F.Y. Meng and X.H. Chen, “A robust additive consistency-based method for decision making with triangular fuzzy reciprocal preference relations,” Fuzzy Opt. Decis. Ma., doi:10.1007/s10700-016-9262-8

  20. Mikhailov, L.: Deriving priorities from fuzzy pairwise comparison judgments. Fuzzy Sets Syst. 134(3), 365–385 (2003)

    Article  MATH  Google Scholar 

  21. Ramik, J., Korviny, P.: Inconsistency of pair-wise comparison matrix with fuzzy elements based on geometric mean. Fuzzy Sets Syst. 161(11), 1604–1613 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  22. Rezaei, J., Ortt, R., Scholten, V.: An improved fuzzy preference programming to evaluate entrepreneurship orientation. Appl. Soft Comput. 13(5), 2749–2758 (2013)

    Article  Google Scholar 

  23. Saaty, T.L.: The Analytic Hierarchy Process. McGraw-Hill, New York (1980)

    MATH  Google Scholar 

  24. Saaty, T.L., Vargas, L.G.: Comparison of eigenvalue, logarithmic least squares and least squares methods in estimating ratios. Math. Model. 5(5), 309–324 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  25. Ureña, R., Chiclana, F., Morente-Molinera, J.A., Herrera-Viedma, E.: Managing incomplete preference relations in decision making: a review and future trends. Inf. Sci. 302, 14–32 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  26. Ureña, R., Chiclana, F., Fujita, H., Herrera-Viedma, E.: Confidence-consistency driven group decision making approach with incomplete reciprocal intuitionistic preference relations. Knowle. Based Syst 89, 86–96 (2015)

    Article  Google Scholar 

  27. van Laarhoven, P.J.M., Pedrycz, W.: A fuzzy extension of Saaty’s priority theory. Fuzzy Sets Syst. 11(1–3), 229–241 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  28. Wang, Y.M., Chin, K.S.: A linear goal programming priority method for fuzzy analytic hierarchy process and its applications in new product screening. Int. J. Approx. Reason. 49(2), 451–465 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  29. Wang, Y.M., Luo, Y., Hua, Z.S.: On the extent analysis method for fuzzy AHP and its applications. Eur. J. Oper. Res. 186(2), 735–747 (2008)

    Article  MATH  Google Scholar 

  30. Wang, Z.J.: Consistency analysis and priority derivation of triangular fuzzy preference relations based on modal value and geometric mean. Inf. Sci. 314, 169–183 (2015)

    Article  MathSciNet  Google Scholar 

  31. Wang, Z.J., Tong, X.Y.: Consistency analysis and group decision making based on triangular fuzzy additive reciprocal preference relations. Inf. Sci. 361, 29–47 (2016)

    Article  Google Scholar 

  32. Wang, Z.J., Lin, J.: Acceptability measurement and priority weight elicitation of triangular fuzzy multiplicative preference relations based on geometric consistency and uncertainty indices. Inf. Sci. 402, 105–123 (2017)

    Article  Google Scholar 

  33. Wang, Z.J., Li, K.W.: A multi-step goal programming approach for group decision making with incomplete interval additive reciprocal comparison matrices. Eur. J. Oper. Res. 242, 890–900 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  34. Wu, J., Chiclana, F.: Visual information feedback mechanism and attitudinal prioritisation method for group decision making with triangular fuzzy complementary preference relations. Inf. Sci. 279, 716–734 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  35. Xia, M.M., Xu, Z.S.: Methods for fuzzy complementary preference relations based on multiplicative consistency. Comput. Ind. Eng. 61(4), 930–935 (2011)

    Article  MathSciNet  Google Scholar 

  36. Xu, Y.J., Wang, H.M.: A comment on “Incomplete fuzzy linguistic preference relations under uncertain environments”. Inf. Fusion 20(1), 2–5 (2014)

    Article  MathSciNet  Google Scholar 

  37. Xu, Y.J., Herrera, F., Wang, H.M.: A distance-based framework to deal with ordinal and additive inconsistencies for fuzzy reciprocal preference relations. Inf. Sci. 328, 189–205 (2016)

    Article  Google Scholar 

  38. Xu, Y.J., Ma, F., Xu, W.J., Wang, H.M.: An incomplete multi-granular linguistic model and its application in emergency decision of unconventional outburst incidents. J. Intell. Fuzzy Syst. 29(2), 619–633 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  39. Xu, Y.J., Chen, L., Li, K.W., Wang, H.M.: A chi-square method for priority derivation in group decision making with incomplete reciprocal preference relations. Inf. Sci. 306, 166–179 (2015)

    Article  Google Scholar 

  40. Xu, R.: Fuzzy least-squares priority method in the analytic hierarchy process. Fuzzy Sets Syst. 112(3), 359–404 (2000)

    Article  MathSciNet  Google Scholar 

  41. Yager, R.R.: A procedure for ordering fuzzy subsets of the unit interval. Inf. Sci. 24(2), 143–161 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  42. Yuen, K.K.F., Lau, H.C.W.: A fuzzy group analytical hierarchy process approach for software quality assurance management: fuzzy logarithmic least squares method. Expert Syst. Appl. 38(8), 10292–10302 (2011)

    Article  Google Scholar 

  43. Zadeh, L.A.: Fuzzy sets. Inf. Con. 8(1), 338–353 (1965)

    Article  MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Nos. 71571192, 71671188, and 71501189), the Innovation-Driven Planning Foundation of Central South University (No. 2016CXS027), the State Key Program of National Natural Science of China (No. 71431006), the Projects of Major International Cooperation NSFC (No. 71210003), the Hunan Province Foundation for Distinguished Young Scholars of China (No. 2016JJ1024) and the China Postdoctoral Science Foundation (No. 2016M602170).

Author information

Authors and Affiliations

  1. Business School, Central South University, Changsha, 410083, China

    Jie Tang & Fanyong Meng

  2. School of International Audit, Nanjing Audit University, Nanjing, 211815, China

    Fanyong Meng

Authors
  1. Jie Tang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fanyong Meng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fanyong Meng.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tang, J., Meng, F. A Consistency-Based Method to Decision Making with Triangular Fuzzy Multiplicative Preference Relations. Int. J. Fuzzy Syst. 19, 1317–1332 (2017). https://doi.org/10.1007/s40815-017-0333-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40815-017-0333-y

Keywords

Search

Navigation