Jean-Luc Marichal

An axiomatic approach of the discrete Choquet integral as a tool to aggregate interacting criteria

The most often used operator to aggregate criteria in decision making problems is the classical weighted arithmetic mean. However, in many problems the criteria considered interact and a substitute to the weighted arithmetic mean has to be adopted. We show that, under rather natural conditions, the discrete Choquet integral is an adequate aggregation operator that extends the weighted arithmetic mean by taking into consideration of the interaction among criteria. The axiomatic that supports the Choquet integral is presented and an intuitive approach is proposed as well.

Characterization of the ordered weighted averaging operators

This paper deals with the characterization of two classes of monotonic and neutral (MN) aggregation operators. The first class corresponds to (MN) aggregators which are stable for the same positive linear transformations and presents the ordered linkage property. The second class deals with (MN)-idempotent aggregators which are stable for positive linear transformations with the same unit, independent zeroes and ordered values. These two classes correspond to the weighted ordered averaging operator (OWA) introduced by Yager in 1988. It is also shown that the OWA aggregator can be expressed as a Choquet integral. >

Determination of weights of interacting criteria from a reference set

Abstract In this paper, we present a model allowing to determine the weights related to interacting criteria. This is done on the basis of the knowledge of a partial ranking over a reference set of alternatives (prototypes), a partial ranking over the set of criteria, and a partial ranking over the set of interactions between pairs of criteria.

Aggregation functions: Means

This two-part state-of-the-art overview on aggregation theory summarizes the essential information concerning aggregation issues. An overview of aggregation properties is given, including the basic classification on aggregation functions. In this first part, the stress is put on means, i.e., averaging aggregation functions, both with fixed arity (n-ary means) and with multiple arities (extended means).

On Sugeno integral as an aggregation function

The Sugeno integral, for a given fuzzy measure, is studied under the viewpoint of aggregation. In particular, we give some equivalent expressions of it. We also give an axiomatic characterization of the class of all the Sugeno integrals. Some particular subclasses, such as the weighted maximum and minimum functions are investigated as well.

Equivalent Representations of Set Functions

This paper introduces four alternative representations of a set function: the MA¶bius transformation, the co-MA¶bius transformation, and the interactions between elements of any subset of a given set as extensions of Shapley and Banzhaf values. The links between the five equivalent representations of a set function are emphasized in this presentation.

The use of the discrete Sugeno integral in decision making: a survey

An overview of the use of the discrete Sugeno integral as either an aggregation tool or a preference functional is presented in the qualitative framework of two decision paradigms: multi-criteria decision-making and decision-making under uncertainty. The parallelism between the representation theorems in both settings is stressed, even if a basic requirement like the idempotency of the aggregation scheme should be explicitely stated in multi-criteria decision-making, while its counterpart is implicit in decision under uncertainty by equating the utility of a constant act with the utility of its consequence. Important particular cases of Sugeno integrals such as prioritized minimum and maximum operators, their ordered versions, and Boolean max-min functions are studied.

Tolerant or intolerant character of interacting criteria in aggregation by the Choquet integral

Abstract In many multi-criteria decision-making problems the decision criteria present some interaction whose nature may vary from one situation to another. For example, some criteria may be statistically correlated, thus making them somewhat redundant or opposed. Some others may be somewhat substitutive or complementary depending on the behavior of the decision maker. Some others may be decisive in the sense that the global score (of any alternative) obtained by aggregation is bounded by the partial score along one of them. In this paper we analyze this latter form of interaction. When a criterion bounds the global score from above, it is called a blocker or a veto, due to its rather intolerant character. When it bounds the global score from below, it is then called a pusher or a favor. We thus investigate the tolerance of criteria, or equivalently, the tolerance of the weighted aggregation operator (here the Choquet integral) that is used to aggregate criteria. More specifically, we propose (axiomatically) indices to appraise the extent to which each criterion behaves like a veto or a favor in the aggregation by the Choquet integral. Previous to this, we also propose global tolerance degrees measuring the extent to which the Choquet integral is conjunctive or disjunctive.

Aggregation of interacting criteria by means of the discrete Choquet integral

The most often used operator to aggregate criteria in decision making problems is the classical weighted arithmetic mean. In many problems however, the criteria considered interact, and a substitute to the weighted arithmetic mean has to be adopted. Under rather natural conditions, the discrete Choquet integral is proved to be an adequate aggregation operator that extends the weighted arithmetic mean by the taking into consideration of the interaction among criteria. The axiomatic that supports the Choquet integral is presented and some subfamilies are studied.

Entropy of discrete Choquet capacities

We introduce a measure of entropy for any discrete Choquet capacity and we interpret it in the setting of aggregation by the Choquet integral.