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An axiomatization of the Choquet integral in the context of multiple criteria decision making without any commensurability assumption

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  • Christophe Labreuche

    (Thales Research and Technology France)

Abstract

An axiomatization of the Choquet integral is proposed in the context of multiple criteria decision making without any commensurability assumption. The most essential axiom—named Commensurability Through Interaction—states that the importance of an attribute i takes only one or two values when a second attribute k varies. When the importance takes two values, the point of discontinuity is exactly the value on the attribute k that is commensurate to the fixed value on attribute i. If the weight of criterion i does not depend on criterion k, for any value of the other criteria than i and k, then criteria i and k are independent. Applying this construction to any pair i, k of criteria, one obtains a partition of the set of criteria. In each block, the criteria interact one with another, and it is thus possible to construct vectors of values on the attributes that are commensurate. There is complete independence between the criteria of any two blocks in this partition. Hence one cannot ensure commensurability between two blocks in the partition. But this is not a problem since the Choquet integral is additive between subsets of criteria that are independent.

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  • Christophe Labreuche, 2018. "An axiomatization of the Choquet integral in the context of multiple criteria decision making without any commensurability assumption," Annals of Operations Research, Springer, vol. 271(2), pages 701-735, December.
    • Handle: RePEc:spr:annopr:v:271:y:2018:i:2:d:10.1007_s10479-018-3046-1
    DOI: 10.1007/s10479-018-3046-1
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    References listed on IDEAS

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    1. Christophe Labreuche & M. Grabisch, 2007. "The representation of conditional relative importance between criteria," Annals of Operations Research, Springer, vol. 154(1), pages 93-122, October.
    2. Angilella, Silvia & Corrente, Salvatore & Greco, Salvatore, 2015. "Stochastic multiobjective acceptability analysis for the Choquet integral preference model and the scale construction problem," European Journal of Operational Research, Elsevier, vol. 240(1), pages 172-182.
    3. Michel Grabisch & Christophe Labreuche, 2010. "A decade of application of the Choquet and Sugeno integrals in multi-criteria decision aid," Annals of Operations Research, Springer, vol. 175(1), pages 247-286, March.
    4. Labreuche, Christophe & Grabisch, Michel, 2006. "Generalized Choquet-like aggregation functions for handling bipolar scales," European Journal of Operational Research, Elsevier, vol. 172(3), pages 931-955, August.
    5. Schmeidler, David, 1989. "Subjective Probability and Expected Utility without Additivity," Econometrica, Econometric Society, vol. 57(3), pages 571-587, May.
    6. Michel Grabisch & Christophe Labreuche, 2016. "Fuzzy Measures and Integrals in MCDA," International Series in Operations Research & Management Science, in: Salvatore Greco & Matthias Ehrgott & José Rui Figueira (ed.), Multiple Criteria Decision Analysis, edition 2, chapter 0, pages 553-603, Springer.
    7. Christophe Labreuche & Michel Grabisch, 2003. "The Choquet integral for the aggregation of interval scales in multicriteria decision making," Post-Print hal-00272090, HAL.
    8. Angilella, Silvia & Greco, Salvatore & Lamantia, Fabio & Matarazzo, Benedetto, 2004. "Assessing non-additive utility for multicriteria decision aid," European Journal of Operational Research, Elsevier, vol. 158(3), pages 734-744, November.
    9. Grabisch, Michel, 1996. "The application of fuzzy integrals in multicriteria decision making," European Journal of Operational Research, Elsevier, vol. 89(3), pages 445-456, March.
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