Modeling Decisions for Artificial Intelligence

MDAI 2015: Modeling Decisions for Artificial Intelligence pp 66-77 | Cite as

Estimating Unknown Values in Reciprocal Intuitionistic Preference Relations via Asymmetric Fuzzy Preference Relations

  • Francisco Chiclana
  • Raquel Ureña
  • Hamido Fujita
  • Enrique Herrera-Viedma
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9321)


Intuitionistic preference relations are becoming increasingly important in the field of group decision making since they present a flexible and simple way to the experts to provide their preference relations, while at the same time allowing them to accommodate a certain degree of hesitation inherent to all decision making processes. In this contribution, we prove the mathematical equivalence between the set of asymmetric fuzzy preference relations and the set of reciprocal intuitionistic fuzzy preference relations. This result is exploited to tackle the presence of incomplete reciprocal intuitionistic fuzzy preference relation in decision making by developing a consistency driven estimation procedure via the corresponding equivalent incomplete asymmetric fuzzy preference relation.


Intuitionistic preference relation Asymmetric fuzzy preference relation Consistency Uninorm Incomplete information 



This work has been developed with the financing of the Andalusian Excellence research project TIC-5991 and FEDER funds in the Spanish National research project TIN2013-40658-P. Raquel Ureña would like to acknowledge the support received by the mobility grant program awarded by the University of Granada’s International Office. Prof. Francisco Chiclana and Prof. Hamido Fujita would like to acknowledge the support provided by the University of Granada ‘Strengthening through Short-Visits’ (Ref. GENIL-SSV 2015) programme.


  1. 1.
    Alonso, S., Cabrerizo, F., Chiclana, F., Herrera, F., Herrera-Viedma, E.: Group decision making with incomplete fuzzy linguistic preference relations. Int. J. Intell. Syst. 24(2), 201–222 (2009a)Google Scholar
  2. 2.
    Alonso, S., Chiclana, F., Herrera, F., Herrera-Viedma, E.: A Learning Procedure to Estimate Missing Values in Fuzzy Preference Relations Based on Additive Consistency. In: Torra, V., Narukawa, Y. (eds.) MDAI 2004. LNCS (LNAI), vol. 3131, pp. 227–238. Springer, Heidelberg (2004) CrossRefGoogle Scholar
  3. 3.
    Alonso, S., Chiclana, F., Herrera, F., Herrera-Viedma, E., Alcalá-Fdez, J., Porcel, C.: A consistency-based procedure to estimate missing pairwise preference values. Int. J. Intell. Syst. 23(2), 155–175 (2008)CrossRefMATHGoogle Scholar
  4. 4.
    Alonso, S., Herrera-Viedma, E., Chiclana, F., Herrera, F.: Individual and social strategies to deal with ignorance situations in multi-person decision making. Int. J. Inf. Technol. Decis. Making 8(2), 313–333 (2009b)Google Scholar
  5. 5.
    Alonso, S., Herrera-Viedma, E., Chiclana, F., Herrera, F.: A web based consensus support system for group decision making problems and incomplete preferences. Inf. Sci. 180(23), 4477–4495 (2010)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Atanassov, K.T.: Intuitionistic fuzzy sets. Fuzzy Sets Syst. 20(1), 87–96 (1986)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Bezdek, J., Spillman, B., Spillman, R.: A fuzzy relation space for group decisiontheory. Fuzzy Sets Syst. 1(4), 255–268 (1978)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Chiclana, F., Herrera-Viedma, E., Alonso, S., Herrera, F.: Cardinal consistency of reciprocal preference relations: a characterization of multiplicative transitivity. IEEE Trans. Fuzzy Syst. 17(1), 14–23 (2009)CrossRefGoogle Scholar
  9. 9.
    Ebenbach, D.H., Moore, C.: Incomplete information, inferences, and individual differences: the case of environmental judgments. Organ. Behav. Hum. Decis. Process. 81(1), 1–27 (2000)CrossRefGoogle Scholar
  10. 10.
    Fedrizzi, M., Giove, S.: Incomplete pairwise comparison and consistency optimization. Eur. J. Oper. Res. 183(1), 303–313 (2007)CrossRefMATHGoogle Scholar
  11. 11.
    Fishburn, P.: Utility theory for decision making. Krieger, Melbourne (1979)Google Scholar
  12. 12.
    Genc, S., Boran, F.E., Akay, D., Xu, Z.: Interval multiplicative transitivity for consistency, missing values and priority weights of interval fuzzy preference relations. Inf. Sci. 180(24), 4877–4891 (2010)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Herrera-Viedma, E., Alonso, S., Chiclana, F., Herrera, F.: A consensus model for group decision making with incomplete fuzzy preference relations. IEEE Trans. Fuzzy Syst. 15(5), 863–877 (2007a)Google Scholar
  14. 14.
    Herrera-Viedma, E., Chiclana, F., Herrera, F., Alonso, S.: Group decision-making model with incomplete fuzzy preference relations based on additive consistency. IEEE Trans. Syst. Man Cybern. Part B Cybern. 37(1), 176–189 (2007b)Google Scholar
  15. 15.
    Herrera-Viedma, E., Herrera, F., Chiclana, F., Luque, M.: Some issues on consistency of fuzzy preference relations. Eur. J. Oper. Res. 154(1), 98–109 (2004)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Lee, L.-W.: Group decision making with incomplete fuzzy preference relations based on the additive consistency and the order consistency. Expert Syst. Appl. 39(14), 11666–11676 (2012)CrossRefGoogle Scholar
  17. 17.
    Liu, X., Pan, Y., Xu, Y., Yu, S.: Least square completion and inconsistency repair methods for additively consistent fuzzy preference relations. Fuzzy Sets Syst. 198(1), 1–19 (2012)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Luce, R.D., Suppes, P.: Preferences, utility and subject probability. In: Handbook of Mathematical Psychology, New York, vol. 3 (1965)Google Scholar
  19. 19.
    Pérez-Asurmendi, P., Chiclana, F.: Linguistic majorities with difference in support. Appl. Soft Comput. 18, 196–208 (2014)CrossRefGoogle Scholar
  20. 20.
    Roubens, M., Vincke, P.: Preference modeling. Springer, Berlin (1985)CrossRefGoogle Scholar
  21. 21.
    Saaty, T.L.: The Analytic Hierarchy Process. McGraw-Hill, New York (1980)MATHGoogle Scholar
  22. 22.
    Szmidt, E., Kacprzyk, J.: Using intuitionistic fuzzy sets in group decision making. Control and Cybern. 31(4), 1037–1053 (2002)MATHGoogle Scholar
  23. 23.
    Tanino, T.: Fuzzy preference orderings in group decision making. Fuzzy Sets Syst. 12, 117–131 (1984)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Ureña, R., Chiclana, F., Morente-Molinera, J., Herrera-Viedma, E.: Managing incomplete preference relations in decision making: a review and future trends. Inf. Sci. 302, 14–32 (2015)CrossRefGoogle Scholar
  25. 25.
    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)CrossRefGoogle Scholar
  26. 26.
    Xu, Z., Cai, X., Szmidt, E.: Algorithms for estimating missing elements of incomplete intuitionistic preference relations. Int. J. Intell. Syst. 26(9), 787–813 (2011)CrossRefGoogle Scholar
  27. 27.
    Xu, Z., Liao, H.: A survey of approaches to decision making with intuitionistic fuzzy preference relations. Knowledge-Based Systems, 2014, doi: 10.1016/j.knosys.2014.12.034 (2015 in press)Google Scholar
  28. 28.
    Zadeh, L.A.: Fuzzy sets. Inf. Control 8(3), 338–357 (1965)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Zadeh, L.A.: The concept of a linguistic variable and its application to approximate reasoning-i. Inf. Sci. 8, 199–249 (1975)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Francisco Chiclana
    • 1
  • Raquel Ureña
    • 2
  • Hamido Fujita
    • 3
  • Enrique Herrera-Viedma
    • 2
  1. 1.Centre for Computational Intelligence, Faculty of TechnologyDe Montfort UniversityLeicesterUK
  2. 2.Department of Computer Science and Artificial IntelligenceUniversity of GranadaGranadaSpain
  3. 3.Iwate Prefectural UniversityTakizawaJapan

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