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Aggregation on bipolar scales

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  • Michel Grabisch

    (CES - Centre d'économie de la Sorbonne - UP1 - Université Paris 1 Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique)

Abstract

The paper addresses the problem of extending aggregation operators typically defined on $[0,1]$ to the symmetric interval $[-1,1]$, where the ``0'' value plays a particular role (neutral value). We distinguish the cases where aggregation operators are associative or not. In the former case, the ``0'' value may play the role of neutral or absorbant element, leading to pseudo-addition and pseudo-multiplication. We address also in this category the special case of minimum and maximum defined on some finite ordinal scale. In the latter case, we find that a general class of extended operators can be defined using an interpolation approach, supposing the value of the aggregation to be known for ternary vectors.

Suggested Citation

  • Michel Grabisch, 2006. "Aggregation on bipolar scales," Post-Print halshs-00187155, HAL.
    • Handle: RePEc:hal:journl:halshs-00187155
    Note: View the original document on HAL open archive server: https://shs.hal.science/halshs-00187155
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    References listed on IDEAS

    as
    1. Michel Grabisch & Bernard de Baets & Janos Fodor, 2004. "The quest for rings on bipolar scales," Post-Print hal-00271217, HAL.
    2. 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.
    3. Michel Grabisch, 2003. "The Symmetric Sugeno Integral," Post-Print hal-00272084, HAL.
    4. Dieter Denneberg & Michel Grabisch, 2004. "Measure and integral with purely ordinal scales," Post-Print hal-00272078, HAL.
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    Cited by:

    1. Merad, Myriam & Dechy, Nicolas & Serir, Lisa & Grabisch, Michel & Marcel, Frédéric, 2013. "Using a multi-criteria decision aid methodology to implement sustainable development principles within an organization," European Journal of Operational Research, Elsevier, vol. 224(3), pages 603-613.
    2. Christophe Labreuche & Michel Grabisch, 2008. "A value for bi-cooperative games," Post-Print halshs-00308738, HAL.
    3. Miguel Couceiro & Michel Grabisch, 2013. "On the poset of computation rules for nonassociative calculus," PSE-Ecole d'économie de Paris (Postprint) hal-00787750, HAL.
    4. Kojadinovic, Ivan, 2007. "A weight-based approach to the measurement of the interaction among criteria in the framework of aggregation by the bipolar Choquet integral," European Journal of Operational Research, Elsevier, vol. 179(2), pages 498-517, June.
    5. Kojadinovic, Ivan & Marichal, Jean-Luc, 2007. "Entropy of bi-capacities," European Journal of Operational Research, Elsevier, vol. 178(1), pages 168-184, April.
    6. Greco, Salvatore & Mousseau, Vincent & Słowiński, Roman, 2014. "Robust ordinal regression for value functions handling interacting criteria," European Journal of Operational Research, Elsevier, vol. 239(3), pages 711-730.

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