Fuzzy Optimization and Decision Making

, Volume 17, Issue 1, pp 49–73 | Cite as

A robust additive consistency-based method for decision making with triangular fuzzy reciprocal preference relations

  • Fanyong Meng
  • Xiaohong Chen


To express uncertain information in decision making, triangular fuzzy reciprocal preference relations (TFRPRs) might be adopted by decision makers. Considering consistency of this type of preference relations, this paper defines a new additive consistency concept, which can be seen as an extension of that for reciprocal preference relations. Then, a simple method to calculate the triangular fuzzy priority weight vector is introduced. When TFRPRs are inconsistent, a linear goal programming framework to derive the completely additive consistent TFRPRs is provided. Meanwhile, an improved linear goal programming model is constructed to estimate the missing values in an incomplete TFRPR that can address the situation where ignored objects exist. After that, an algorithm for decision making with TFRPRs is presented. Finally, numerical examples and comparison analysis are offered.


Decision making Triangular fuzzy reciprocal preference relation Additive consistency Goal programming model 



The authors first gratefully thank the Associate Editor and the anonymous referees for their valuable and constructive comments which have much improved the paper. This work was supported by the State Key Program of National Natural Science of China (No. 71431006), the Projects of Major International Cooperation NSFC (No. 71210003), the National Natural Science Foundation of China (Nos. 71571192, 71501189, 71201089, 71271217, and 51204100), the Hunan Province Foundation for Distinguished Young Scholars of China (2016JJ1024), and the Postdoctoral Science Special Foundation of China (2015T80901), the Innovation-Driven Planning Foundation of Central South University (No. 2016CXS027).


  1. Brunelli, M. (2011). 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 and Systems, 176, 76–78.MathSciNetCrossRefMATHGoogle Scholar
  2. Buckley, J. J. (1985). Fuzzy hierarchical analysis. Fuzzy Sets and Systems, 17, 233–247.MathSciNetCrossRefMATHGoogle Scholar
  3. Chang, D. Y. (1996). Application of the extent analysis method on fuzzy AHP. European Journal of Operational Research, 95, 649–655.CrossRefMATHGoogle Scholar
  4. Dong, Y. C., & Herrera-Viedma, E. (2015). Consistency-driven automatic methodology to set interval numerical scales of 2-tuple linguistic term sets and its use in the linguistic GDM with preference relations. IEEE Transactions on Cybernetics, 45, 780–792.CrossRefGoogle Scholar
  5. Kwiesielewicz, M. (1998). A note on the fuzzy extension of Saaty’s priority theory. Fuzzy Sets and Systems, 95, 161–172.MathSciNetCrossRefMATHGoogle Scholar
  6. Leung, L. C., & Cao, D. (2000). On consistency and ranking of alternatives in fuzzy AHP. European Journal of Operational Research, 124, 102–113.CrossRefMATHGoogle Scholar
  7. Liu, F., Zhang, W. G., & Zhang, L. H. (2014). Consistency analysis of triangular fuzzy reciprocal preference relations. European Journal of Operational Research, 235, 718–726.MathSciNetCrossRefMATHGoogle Scholar
  8. Meng, F. Y., & Chen, X. H. (2016). A new method for triangular fuzzy compare wise judgment matrix process based on consistency analysis. International Journal of Fuzzy Systems,. doi: 10.1007/s40815-016-0150-8.Google Scholar
  9. Meng, F. Y., Tan, C. Q., & Chen, X. H. (2016). Multiplicative consistency analysis for interval reciprocal preference relations: A comparative study. Omega,. doi: 10.1016/ Scholar
  10. Ramik, J., & Korviny, P. (2010). Inconsistency of pair-wise comparison matrix with fuzzy elements based on geometric mean. Fuzzy Sets and Systems, 161, 1604–1613.MathSciNetCrossRefMATHGoogle Scholar
  11. Saaty, T. L., & Vargas, L. G. (1987). Uncertainty and rank order in the analytic hierarchy process. European Journal of Operational Research, 32, 107–117.MathSciNetCrossRefMATHGoogle Scholar
  12. Ureña, R., Chiclana, F., Morente, J. A., & Herrera-Viedma, E. (2015). Managing incomplete preference relations in decision making: A review and future trends. Information Sciences, 302, 14–32.MathSciNetCrossRefMATHGoogle Scholar
  13. van Laarhoven, P. J. M., & Pedrycz, W. (1983). A fuzzy extension of Saaty’s priority theory. Fuzzy Sets and Systems, 11, 229–241.MathSciNetCrossRefMATHGoogle Scholar
  14. Wang, Z. J. (2015). Geometric consistency based interval weight elicitation from intuitionistic preference relations using logarithmic least square optimization. Fuzzy Optimization and Decision Making, 14, 289–310.MathSciNetCrossRefGoogle Scholar
  15. Wang, T. C., & Chen, Y. H. (2008). Applying fuzzy linguistic preference relations to the improvement of consistency of fuzzy AHP. Information Sciences, 178, 3755–3765.MathSciNetCrossRefMATHGoogle Scholar
  16. Wang, Y. M., & Chin, K. S. (2008). A linear goal programming priority method for fuzzy analytic hierarchy process and its applications in new product screening. International Journal of Approximate Reasoning, 49, 451–465.CrossRefMATHGoogle Scholar
  17. Wang, Y. M., Elhag, T. M. S., & Hua, Z. S. (2006). A modified fuzzy logarithmic least squares method for fuzzy analytic hierarchy process. Fuzzy Sets and Systems, 157, 3055–3071.MathSciNetCrossRefMATHGoogle Scholar
  18. Wu, J., & Chiclana, F. (2014a). Visual information feedback mechanism and attitudinal prioritisation method for group decision making with triangular fuzzy complementary preference relations. Information Sciences, 279, 716–734.MathSciNetCrossRefMATHGoogle Scholar
  19. Wu, J., & Chiclana, F. (2014b). Multiplicative consistency of intuitionistic reciprocal preference relations and its application to missing values estimation and consensus building. Knowledge-Based Systems, 71, 187–200.CrossRefGoogle Scholar
  20. Xia, M. M., & Xu, Z. S. (2011). Methods for fuzzy complementary preference relations based on multiplicative consistency. Computers and Industrial Engineering, 61, 930–935.CrossRefGoogle Scholar
  21. Xu, R. (2000). Fuzzy least-squares priority method in the analytic hierarchy process. Fuzzy Sets and Systems, 112, 359–404.MathSciNetCrossRefGoogle Scholar
  22. Xu, Z. S. (2001). A practical method for priority of interval number complementary judgment matrix. Operational Research and Management Sciences, 10, 16–19.Google Scholar
  23. Xu, Z. S. (2002). A method for priorities of triangular fuzzy number complementary judgement matrices. Fuzzy Systems and Mathematics, 16, 47–50.MathSciNetMATHGoogle Scholar
  24. Xu, Z. S. (2004). On compatibility of interval fuzzy preference relations. Fuzzy Optimization and Decision Making, 3, 217–225.MathSciNetCrossRefMATHGoogle Scholar
  25. Xu, Z. S. (2005). A procedure for decision making based on incomplete fuzzy preference relation. Fuzzy Optimization and Decision Making, 4, 175–189.MathSciNetCrossRefMATHGoogle Scholar
  26. Xu, Z. S. (2010). An integrated model-based interactive approach to FMAGDM with incomplete preference information. Fuzzy Optimization and Decision Making, 9, 333–357.MathSciNetCrossRefMATHGoogle Scholar
  27. Xu, Z. S., & Da, Q. L. (2003). An approach to improving consistency of fuzzy preference matrix. Fuzzy Optimization and Decision Making, 2, 3–12.MathSciNetCrossRefGoogle Scholar
  28. Xu, Y. J., Li, K. W., & Wang, H. M. (2014). Incomplete interval fuzzy preference relations and their applications. Computers and Industrial Engineering, 67, 93–103.CrossRefGoogle Scholar
  29. Yager, R. R. (1980). A procedure for ordering fuzzy subsets of the unit interval. Information Sciences, 24, 143–161.MathSciNetCrossRefMATHGoogle Scholar
  30. Zhang, G. Q., Dong, Y. C., & Xu, Y. F. (2012). Linear optimization modeling of consistency issues in fuzzy group decision making. Expert Systems with Applications, 39, 2415–2420.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.School of International AuditNanjing Audit UniversityNanjingChina
  2. 2.School of BusinessCentral South UniversityChangshaChina
  3. 3.School of AccountingHunan University of CommerceChangshaChina

Personalised recommendations