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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Choquet, G.: Theory of Capacities. Annales de l’Institute Fourier (Grenoble), vol. 5, pp. 131–295 (1953)
De Cooman, G., Kerre, E.: Order norms on bounded partially ordered sets. J. Fuzzy Math. 2, 281–310 (1994)
Dempster, A.P.: Upper and lower probabilities induced by a multi-valued mapping. Ann. Math. Stat. 38, 325–339 (1967)
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)
Grabisch, M.: Fuzzy integral in multicriteria decision making. Fuzzy Sets Syst. 69(3), 279–298 (1995)
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)
Grabisch, M., Marichal, J.L., Mesiar, R., Pap, E.: Aggregation Functions (Encyclopedia of Mathematics and Its Applications). Cambridge University Press, New York (2009)
Jiroušek, R., Vejnarová, J., Daniel, M.: Compositional models for belief functions. In: Proceedings of ISIPTA 2007, Prague, 243–252 (2007)
Jin, L.: OWA monoid, submitted
Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer Academic Publisher, Dordrecht (2000)
Komorníková, M., Mesiar, R.: Aggregation functions on bounded partially ordered sets and their classification. Fuzzy Sets Syst. 175(1), 48–56 (2011)
Narukawa, Y., Torra, V.: Twofold integral and multi-step Choquet integral. Kybernetika 40(1), 39–50 (2004)
Schmeidler, D.: Subjective probability and expected utility without additivity. Econometrica 57, 571–587 (1989)
Schmeidler, D.: Integral representation without additivity. Proc. Am. Math. Soc. 2 97, 255–261 (1986)
Šipoš, J.: Integral with respect to a pre-measure. Math. Slov. 29, 141–155 (1979)
Yager, R.R.: On ordered weighted averaging aggregation operators in multicriteria decision-making. IEEE Trans. Syst. Man Cybern. 18(1), 183–190 (1988)
Acknowledgment
The work on this contribution was supported by the grant APVV-14-0013.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this paper
Cite this paper
Mesiar, R., Šipeky, L., Šipošová, A. (2016). Aggregation of Choquet Integrals. In: Carvalho, J., Lesot, MJ., Kaymak, U., Vieira, S., Bouchon-Meunier, B., Yager, R. (eds) Information Processing and Management of Uncertainty in Knowledge-Based Systems. IPMU 2016. Communications in Computer and Information Science, vol 610. Springer, Cham. https://doi.org/10.1007/978-3-319-40596-4_6
Download citation
DOI: https://doi.org/10.1007/978-3-319-40596-4_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-40595-7
Online ISBN: 978-3-319-40596-4
eBook Packages: Computer ScienceComputer Science (R0)