Advertisement

Aggregation of Choquet Integrals

  • Radko Mesiar
  • Ladislav ŠipekyEmail author
  • Alexandra Šipošová
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 610)

Abstract

Aggregation functions acting on the lattice of all Choquet integrals on a fixed measurable space \((\mathrm {X},\mathcal {A})\) are discussed. The only direct aggregation of Choquet integrals resulting into a Choquet integral is linked to the convex sums, i.e., to the weighted arithmetic means. We introduce and discuss several other approaches, for example one based on compatible aggregation systems. For \(\mathrm {X}\) finite, the related aggregation of OWA operators is obtained as a corollary. The only exception, with richer structure of aggregation functions, is the case \(card \ \mathrm {X} = 2\), when the lattice of all OWA operators forms a chain.

Keywords

Aggregation function Capacity Choquet integral OWA operator 

Notes

Acknowledgment

The work on this contribution was supported by the grant APVV-14-0013.

References

  1. 1.
    Choquet, G.: Theory of Capacities. Annales de l’Institute Fourier (Grenoble), vol. 5, pp. 131–295 (1953)Google Scholar
  2. 2.
    De Cooman, G., Kerre, E.: Order norms on bounded partially ordered sets. J. Fuzzy Math. 2, 281–310 (1994)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Dempster, A.P.: Upper and lower probabilities induced by a multi-valued mapping. Ann. Math. Stat. 38, 325–339 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Denneberg, D.: Non-additive Measure and Integral. Theory and Decision Library (Series B Mathematical and Statistical Methods), vol. 27, p. ix–178. Kluwer Academic Publishers, Dordrecht (1994)Google Scholar
  5. 5.
    Grabisch, M.: Fuzzy integral in multicriteria decision making. Fuzzy Sets Syst. 69(3), 279–298 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Grabisch, M., Labreuche, C.: A decade of application of the Choquet and Sugeno integrals in multi-criteria decision aid. 4OR 6(1), 1–44 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Grabisch, M., Marichal, J.L., Mesiar, R., Pap, E.: Aggregation Functions (Encyclopedia of Mathematics and Its Applications). Cambridge University Press, New York (2009)CrossRefzbMATHGoogle Scholar
  8. 8.
    Jiroušek, R., Vejnarová, J., Daniel, M.: Compositional models for belief functions. In: Proceedings of ISIPTA 2007, Prague, 243–252 (2007)Google Scholar
  9. 9.
    Jin, L.: OWA monoid, submittedGoogle Scholar
  10. 10.
    Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer Academic Publisher, Dordrecht (2000)CrossRefzbMATHGoogle Scholar
  11. 11.
    Komorníková, M., Mesiar, R.: Aggregation functions on bounded partially ordered sets and their classification. Fuzzy Sets Syst. 175(1), 48–56 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Narukawa, Y., Torra, V.: Twofold integral and multi-step Choquet integral. Kybernetika 40(1), 39–50 (2004)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Schmeidler, D.: Subjective probability and expected utility without additivity. Econometrica 57, 571–587 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Schmeidler, D.: Integral representation without additivity. Proc. Am. Math. Soc. 2 97, 255–261 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Šipoš, J.: Integral with respect to a pre-measure. Math. Slov. 29, 141–155 (1979)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Yager, R.R.: On ordered weighted averaging aggregation operators in multicriteria decision-making. IEEE Trans. Syst. Man Cybern. 18(1), 183–190 (1988)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Radko Mesiar
    • 1
  • Ladislav Šipeky
    • 1
    Email author
  • Alexandra Šipošová
    • 1
  1. 1.Faculty of Civil Engineering, Department of Mathematics and Descriptive GeometrySlovak University of TechnologyBratislavaSlovakia

Personalised recommendations