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Eurofuse 2011 pp 243-255 | Cite as

Construction of Interval-Valued Fuzzy Preference Relations Using Ignorance Functions: Interval-Valued Non Dominance Criterion

  • Edurne Barrenechea
  • Alberto Fernández
  • Francisco Herrera
  • Humberto Bustince
Part of the Advances in Intelligent and Soft Computing book series (AINSC, volume 107)

Abstract

In this work we present a construction method for interval-valued fuzzy preference relations from a fuzzy preference relation and the representation of the lack of knowledge or ignorance that experts suffer when they define the membership values of the elements of that fuzzy preference relation.We also prove that, with this construction method, we obtain membership intervals for an element which length is equal to the ignorance associated with that element. We then propose a generalization of Orlovsky’s non dominance method to solve decision making problems using interval-valued fuzzy preference relations.

Keywords

Fuzzy Preference Relation Dominance Criterion Strict Preference Relation Reciprocity Property Fuzzy Binary Relation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    De Baets, B., Van de Walle, B., Kerre, E.: Fuzzy preference structures without incomparability. Fuzzy Sets and Systems 76 (3), 333–348 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Bilgiç, T.: Interval-valued preference structures. European Journal of Operational Research 105 (1), 162–183 (1998)zbMATHCrossRefGoogle Scholar
  3. 3.
    Chiclana, F., Herrera, F., Herrera-Viedma, E.: Integrating three representation models in fuzzy multipurpose decision making based on fuzzy preference relations. Fuzzy Sets and Systems 97, 33–48 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Bustince, H., Pagola, M., Barrenechea, E., Fernandez, J., Melo-Pinto, P., Couto, P., Tizhoosh, H.R., Montero, J.: Ignorance functions. An application to the calculation of the threshold in prostate ultrasound images. Fuzzy Sets and Systems 161 (1), 20–36 (2010)MathSciNetGoogle Scholar
  5. 5.
    Deschrijver, G., Cornelis, C., Kerre, E.E.: On the representation of intuitionistic fuzzy T-norms and T-conorms. IEEE Transactions on Fuzzy Systems 12(1), 45–61 (2004)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Fernández, A., Calderón, M., Barrenechea, E., Bustince, H., Herrera, F.: Solving multi-class problems with linguistic fuzzy rule based classification systems based on pairwise learning and preference relations. Fuzzy Sets and Systems 161(23), 3064–3080 (2010)zbMATHCrossRefGoogle Scholar
  7. 7.
    Fodor, J., Roubens, M.: Fuzzy Preference Modelling and Multicriteria Decision Support. Kluwer Academic Publishers, Dordrecht (1994)zbMATHGoogle Scholar
  8. 8.
    González-Pachón, J., Gómez, D., Montero, J., Yáñez, J.: Searching for the dimension of valued preference relations. International Journal of Approximate Reasoning 33 (2), 133–157 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Herrera, F., Martínez, L., Sánchez, P.J.: Managing non-homogeneous information in group decision making. European Journal of Operational Research 166 (1), 115–132 (2005)zbMATHCrossRefGoogle Scholar
  10. 10.
    Hüllermeier, E., Brinker, K.: Learning valued preference structures for solving classification problems. Fuzzy Sets and Systems 159(18), 2337–2352 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Kacprzyk, J.: Group decision making with a fuzzy linguistic majority. Fuzzy Sets and Systems 18(2), 105–118 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Montero, F.J., Tejada, J.: Some problems on the definition of fuzzy preference relations. Fuzzy Sets and Systems 20, 45–53 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Montero, F.J., Tejada, J.: A necessary and sufficient condition for existence of Orlovsky’s choice set. Fuzzy Sets and Systems 26, 121–125 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Orlovsky, S.A.: Decision-making with a fuzzy preference relation. Fuzzy Sets and Systems 1 (3), 155–167 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Ovchinnikov, S.V., Roubens, M.: On Strict Preference Relations. Fuzzy Sets and Systems 43 (3), 319–326 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Ovchinnikov, S.V., Ozernoy, V.M.: Using fuzzy binary relations for identifying noninferior decision alternatives. Fuzzy Sets and Systems 25 (1), 21–32 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Perny, P., Roy, B.: The use of fuzzy outranking relations in preference modelling. Fuzzy Sets and Systems 49, 33–53 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Roubens, M., Vincke, P.: Preference Modelling. Lecture Notes in Economics and Mathematical Systems, vol. 250. Springer, Berlin (1985)zbMATHGoogle Scholar
  19. 19.
    Sambuc, J.: Function Φ-Flous, Application a l’aide au Diagnostic en Pathologie Thyroidienne. These de Doctorat en Medicine, Marseille (1975)Google Scholar
  20. 20.
    Sanz, J., Fernández, A., Bustince, H., Herrera, F.: A genetic tuning to improve the performance of fuzzy rule-based classification systems with interval-valued fuzzy sets: degree of ignorance and lateral position. International Journal of Approximate Reasoning (2011); doi:10.1016/j.ijar.2011.01.011Google Scholar
  21. 21.
    Szmidt, E., Kacprzyk, J.: Using intuitionistic fuzzy sets in group decision making. Control and Cybernetics 31, 1037–1053 (2002)zbMATHGoogle Scholar
  22. 22.
    Szmidt, E., Kacprzyk, J.: A consensus-reaching process under intuitionistic fuzzy preference relations. International Journal of Intelligent Systems 18(7), 837–852 (2003)zbMATHCrossRefGoogle Scholar
  23. 23.
    Xu, Z.: On compatibility of interval fuzzy preference relations. Fuzzy Optimization and Decision Making 3 (3), 217–225 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Xu, Z., Yager, R.R.: Some geometric aggregation operators based on intuitionistic fuzzy sets. International Journal of General Systems 35, 417–433 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Xu, Z.: A method based on distance measure for interval-valued intuitionistic fuzzy group decision making. Information Sciences 180 (1), 181–190 (2010)zbMATHCrossRefGoogle Scholar
  26. 26.
    Zadeh, L.A.: Fuzzy sets. Information Control 8, 338–353 (1965)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Zadeh, L.A.: The concept of a linguistic variable and its application to approximate reasoning I. Information Sciences 8, 199–249 (1975)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Edurne Barrenechea
    • 1
  • Alberto Fernández
    • 2
  • Francisco Herrera
    • 3
  • Humberto Bustince
    • 1
  1. 1.Dept. Automática y ComputaciónUniversidad Pública de NavarraSpain
  2. 2.Dept. of Computer ScienceUniversity of JaénSpain
  3. 3.Dept. of Computer Science and Artificial IntelligenceUniversidad de GranadaSpain

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