Elicitation of a 2-Additive Bi-capacity through Cardinal Information on Trinary Actions

  • Brice Mayag
  • Antoine Rolland
  • Julien Ah-Pine
Part of the Communications in Computer and Information Science book series (CCIS, volume 300)

Abstract

In the context of MultiCriteria Decision Aid, we present new properties of a 2-additive bi-capacity by using a bipolar Möbius transform. We use these properties in the identification of a 2-additive bi-capacity when we represent a cardinal information by a Choquet integral with respect to a 2-additive bi-capacity.

Keywords

MCDA Preference modeling bi-capacity Choquet integral 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Angilella, S., Greco, S., Matarazzo, B.: Non-additive robust ordinal regression: A multiple criteria decision model based on the Choquet integral. European Journal of Operational Research 41(1), 277–288 (2009)Google Scholar
  2. 2.
    Bana e Costa, C.A., De Corte, J.-M., Vansnick, J.-C.: On the mathematical foundations of MACBETH. In: Figueira, J., Greco, S., Ehrgott, M. (eds.) Multiple Criteria Decision Analysis: State of the Art Surveys, pp. 409–437. Springer, Heidelberg (2005)Google Scholar
  3. 3.
    Choquet, G.: Theory of capacities. Annales de l’Institut Fourier 5, 131–295 (1953)MathSciNetCrossRefGoogle Scholar
  4. 4.
    De Corte, J.M.: Un logiciel d’Exploitation d’Information Préférentielles pour l’Aide à la Décision. Bases Mathématiques et Algorithmiques. PhD thesis, University of Mons-Hainaut, Mons (2002)Google Scholar
  5. 5.
    Figueira, J.R., Greco, S., Slowinski, R.: Building a set of additive value functions representing a reference preorder and intensities of preference: Grip method. European Journal of Operational Research 195(2), 460–486 (2009)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Fujimoto, K.: New characterizations of k-additivity and k-monotonicity of bi-capacities. In: SCIS-ISIS 2004, 2nd Int. Conf. on Soft Computing and Intelligent Systems and 5th Int. Symp. on Advanced Intelligent Systems, Yokohama, Japan (September 2004)Google Scholar
  7. 7.
    Fujimoto, K., Murofushi, T.: Some characterizations of k-monotonicity through the bipolar möbius transform in bi-capacities. J. of Advanced Computational Intelligence and Intelligent Informatics 9(5), 484–495 (2005)Google Scholar
  8. 8.
    Fujimoto, K., Murofushi, T., Sugeno, M.: k-additivity and \(\mathcal{C}\)-decomposability of bi-capacities and its integral. Fuzzy Sets and Systems 158, 1698–1712 (2007)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Grabisch, M.: k-order additive discrete fuzzy measures and their representation. Fuzzy Sets and Systems 92, 167–189 (1997)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Grabisch, M., Labreuche, C.: Fuzzy measures and integrals in MCDA. In: Figueira, J., Greco, S., Ehrgott, M. (eds.) Multiple Criteria Decision Analysis: State of the Art Surveys, pp. 565–608. Springer (2005)Google Scholar
  11. 11.
    Grabisch, M., Labreuche, C.: Bi-capacities. Part I: definition, Möbius transform and interaction. Fuzzy Sets and Systems 151, 211–236 (2005)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Grabisch, M., Labreuche, C.: Bi-capacities. Part II: the Choquet integral. Fuzzy Sets and Systems 151, 237–259 (2005)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Grabisch, M., Labreuche, C.: A decade of application of the Choquet and Sugeno integrals in multi-criteria decision aid. 4OR 6, 1–44 (2008)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Mayag, B., Grabisch, M., Labreuche, C.: An Interactive Algorithm to Deal with Inconsistencies in the Representation of Cardinal Information. In: Hüllermeier, E., Kruse, R., Hoffmann, F. (eds.) IPMU 2010. CCIS, vol. 80, pp. 148–157. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  15. 15.
    Mayag, B., Grabisch, M., Labreuche, C.: A characterization of the 2-additive Choquet integral through cardinal information. Fuzzy Sets and Systems 184(1), 84–105 (2011)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Simon, H.: Rational choice and the structure of the environment. Psychological Review 63(2), 129–138 (1956)CrossRefGoogle Scholar
  17. 17.
    Slovic, P., Finucane, M., Peters, E., MacGregor, D.G.: The affect heuristic. In: Gilovitch, T., Griffin, D., Kahneman, D. (eds.) Heuristics and Biases: The Psychology of Intuitive Judgment, pp. 397–420. Cambridge University Press (2002)Google Scholar
  18. 18.
    Tversky, A., Kahneman, D.: Advances in prospect theory: cumulative representation of uncertainty. J. of Risk and Uncertainty 5, 297–323 (1992)MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Brice Mayag
    • 1
  • Antoine Rolland
    • 2
  • Julien Ah-Pine
    • 2
  1. 1.LAMSADEUniversity Paris DauphineParisFrance
  2. 2.ERICUniversity Lumière Lyon 2Bron CedexFrance

Personalised recommendations