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Approximation of Fuzzy Measures Using Second Order Measures: Estimation of Andness Bounds

  • Marta Cardin
  • Silvio Giove
Conference paper
  • 1.3k Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8256)

Abstract

In this paper we analyze the compensation property of a second order fuzzy measure in the context of a multi-attribute problem. In particular, we show that the disjunction/conjunction behavior (andness/orness) changes with the number of criteria to be aggregated. Interpreting the spread between the maximum and the minimum orness as a measure of the representation capability, we obtain two bounds in function of which asymptotically converge to a limit interval.

Keywords

Fuzzy measures second order measures Choquet integral andness index OWA 

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References

  1. 1.
    Bortot, S., Marques Pereira, R.A.: The Generalized Gini Welfare Function in the Framework of Symmetric Choquet Integration. In: Ventre, A.G.S., Maturo, A., Hošková-Mayerová, Š., Kacprzyk, J. (eds.) Multicriteria & Multiagent Decision Making. STUDFUZZ, vol. 305, pp. 15–26. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  2. 2.
    Calvo, T., De Baets, B.: Aggregation operators defined by k-order additive/maxitive fuzzy measures. International Journal of Uncertainty, Fuzzyness and Knowledge-Based Systems 6(6), 533–550 (1998)CrossRefzbMATHGoogle Scholar
  3. 3.
    Chateauneuf, A., Jaffray, J.Y.: Some characterizations of lower probabilities and other monotone capacities throught the use of Möbius inversion. Mathematical Social Sciences 17(3), 263–283 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Choquet, G.: Theory of capacities. Annales de l’Institut Fourier 5, 131–295 (1953)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Denneberg, D.: Non-additive measure and integral. Kluwer Academic Publisher, Dordrecht (1994)CrossRefzbMATHGoogle Scholar
  6. 6.
    Gilboa, I., Schmeidler, D.: Additive representations of non-additive measures and the Choquet integral. Annals of Operations Research 52(1), 43–65 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Gilboa, I., Schmeidler, D.: Canonical representation of set functions. Mathematics of Operations Research 20(1), 197–212 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Grabish, M.: k-order additive discrete fuzzy measures and their representation. Fuzzy Sets and Systems 92, 167–189 (1997)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Grabish, M., Kojadinovic, I., Meyer, P.: A review of capacity identification methods for Choquet integral based multi-attribute theory– Applications of the Kappalab R package. European Journal of Operations Research 186(2), 766–785 (1995)CrossRefGoogle Scholar
  10. 10.
    Grabisch, M., Marichal, J.L., Mesiar, R., Pap, E.: Aggregation Functions, Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (2009)CrossRefGoogle Scholar
  11. 11.
    Kojadinovic, I.: Minimum variance capacity identification. European Journal of Operations Research 177(1), 498–514 (2007)CrossRefzbMATHGoogle Scholar
  12. 12.
    Marichal, J.L.: Aggregation operators for multicriteria decision aid. Ph.D. Thesis, University of Liège, Liège, Belgium (1998)Google Scholar
  13. 13.
    Mayag, B., Grabisch, M., Labreuche, C.: A representation of preferences by the Choquet integral with respect to a 2-additive capacity. Theory and Decision 71(3), 297–324 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Meyer, P., Ponthière, G.: Eliciting preferences on Multiattribute Societies with a Choquet Integral. Computational Economics 37(2), 133–168 (2011)CrossRefzbMATHGoogle Scholar
  16. 16.
    Pinar, M., Cruciani, C., Giove, S., Sostero, M.: Constructing the FEEM Sustainability Index: a Choquet integral application (submitted)Google Scholar
  17. 17.
    Yager, R.R.: On ordered weighted averaging aggregation operators in multicriteria decision making. IEEE Trans. on Systems, Man and Cybernetics 18(1), 183–190 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Yager, R.R.: Prioritized aggregations operators. International Journal of Approximate Reasoning 48, 263–274 (2008)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  • Marta Cardin
    • 1
  • Silvio Giove
    • 1
  1. 1.Department of EconomicsUniversity Cá Foscari of VeniceItaly

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