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Article dans une revue Fuzzy Sets and Systems Année : 2022

On the convex hull of k-additive 0-1 capacities and its application to model identification in decision making

Résumé

The Choquet integral w.r.t. a capacity is a versatile tool commonly used in decision making. Its practical identification requires, however, to solve an optimization problem with exponentially many variables and constraints. The introduction of k-additive capacities, through the use of the Möbius transform, permits to reduce the number of variables to a polynomial size, but leaves the number of constraints exponential. When k = 2, the use of vertices of the set of 2-additive capacities permits to solve the problem as the number of vertices is polynomial. When k > 2, this solution is no more applicable as the set of vertices of k-additive capacities is not known. We propose in this paper to use instead the set of vertices which are 0-1 valued. We show that the number of such vertices is polynomial, and we observe that the loss of generality is very small for n = 4, k = 3, and conjecture that this still holds for larger values of n. Also, we study the geometric properties of the convex hull of 0-1 valued k-additive capacities.
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Dates et versions

halshs-03881431, version 1 (01-12-2022)

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  • HAL Id : halshs-03881431 , version 1

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Michel Grabisch, Christophe Labreuche. On the convex hull of k-additive 0-1 capacities and its application to model identification in decision making. Fuzzy Sets and Systems, 2022, 451, pp.228-252. ⟨halshs-03881431⟩

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