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. We show that, under rather natural conditions, the discrete Choquet integral is an adequate aggregation… (More)
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 ag-gregation functions. In this first part, the stress is put on means, i.e., averaging aggregation functions, both with fixed arity (n-ary… (More)
This paper introduces four alternative representations of a set function: the Möbius transformation, the co-Möbius 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.
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.
We present a model allowing to aggregate decision criteria when the available information is of a qualitative nature. The use of the Sugeno integral as an aggregation function is justified by an axiomatic approach. It is also shown that the mutual preferential independence of criteria reduces the Sugeno integral to a dictatorial aggregation.
The Sugeno integral, for a given fuzzy measure, is studied under the viewpoint of aggre-gation. 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.
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 present the ordered linkage property. The second class deals with (MN)-idempotent aggregators which are stable for positive linear… (More)
We define the concept of weighted lattice polynomial functions as lattice polynomial functions constructed from both variables and parameters. We provide equivalent forms of these functions in an arbitrary bounded distributive lattice. We also show that these functions include the class of discrete Sugeno integrals and that they are characterized by a… (More)
An axiomatization of the concept of entropy of a discrete Choquet capacity is given. It is based on three axioms: the symmetry property, a boundary condition for which the entropy reduces to the classical Shannon entropy, and a generalized version of the well-known recursivity property. This entropy, recently introduced to extend the Shannon entropy to… (More)
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… (More)