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Fuzzy Measures and Integrals in MCDA

In: Multiple Criteria Decision Analysis

Author

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  • Michel Grabisch

    (Université Paris I - Panthéon-Sorbonne)

  • Christophe Labreuche

    (Thales Research & Technology)

Abstract

This chapter aims at a unified presentation of various methods of MCDA based on fuzzy measures (capacity) and fuzzy integrals, essentially the Choquet and Sugeno integral. A first section sets the position of the problem of multicriteria decision making, and describes the various possible scales of measurement (cardinal unipolar and bipolar, and ordinal). Then a whole section is devoted to each case in detail: after introducing necessary concepts, the methodology is described, and the problem of the practical identification of fuzzy measures is given. The important concept of interaction between criteria, central in this chapter, is explained in detail. It is shown how it leads to k-additive fuzzy measures. The case of bipolar scales leads to the general model based on bi-capacities, encompassing usual models based on capacities. A general definition of interaction for bipolar scales is introduced. The case of ordinal scales leads to the use of Sugeno integral, and its symmetrized version when one considers symmetric ordinal scales. A practical methodology for the identification of fuzzy measures in this context is given.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:isochp:978-1-4939-3094-4_14
    DOI: 10.1007/978-1-4939-3094-4_14
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    1. Mustapha Ridaoui & Michel Grabisch & Christophe Labreuche, 2019. "Interaction indices for multichoice games," Documents de travail du Centre d'Economie de la Sorbonne 19019, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    2. Grabisch, Michel & Labreuche, Christophe & Vansnick, Jean-Claude, 2003. "On the extension of pseudo-Boolean functions for the aggregation of interacting criteria," European Journal of Operational Research, Elsevier, vol. 148(1), pages 28-47, July.
    3. Tversky, Amos & Kahneman, Daniel, 1992. "Advances in Prospect Theory: Cumulative Representation of Uncertainty," Journal of Risk and Uncertainty, Springer, vol. 5(4), pages 297-323, October.
    4. Michel Grabisch & Christophe Labreuche, 2002. "The symmetric and asymmetric Choquet integrals on finite spaces for decision making," Statistical Papers, Springer, vol. 43(1), pages 37-52, January.
    5. Michel Grabisch, 2009. "Subjective Evaluation," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-00671303, HAL.
    6. Marc Roubens & Michel Grabisch, 1999. "An axiomatic approach to the concept of interaction among players in cooperative games," International Journal of Game Theory, Springer;Game Theory Society, vol. 28(4), pages 547-565.
    7. Michel Grabisch, 2003. "The Symmetric Sugeno Integral," Post-Print hal-00272084, HAL.
    8. Christophe Labreuche & Michel Grabisch, 2003. "The Choquet integral for the aggregation of interval scales in multicriteria decision making," Post-Print hal-00272090, HAL.
    9. Pedro Miranda & Michel Grabisch, 2004. "p-symmetric bi-capacities," Post-Print halshs-00188173, HAL.
    10. Dubois, Didier & Prade, Henri & Sabbadin, Regis, 2001. "Decision-theoretic foundations of qualitative possibility theory," European Journal of Operational Research, Elsevier, vol. 128(3), pages 459-478, February.
    11. Pedro Miranda & Michel Grabisch & Pedro Gil, 2002. "p-symmetric fuzzy measures," Post-Print hal-00273960, HAL.
    12. Dieter Denneberg & Michel Grabisch, 2004. "Measure and integral with purely ordinal scales," Post-Print hal-00272078, HAL.
    13. MoshÊ Machover & Dan S. Felsenthal, 1997. "Ternary Voting Games," International Journal of Game Theory, Springer;Game Theory Society, vol. 26(3), pages 335-351.
    14. Michel Grabisch & Jacques Duchene & Frédéric Lino & Patrice Perny, 2002. "Subjective Evaluation of Discomfort in Sitting Positions," Post-Print halshs-00273179, HAL.
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