7 research outputs found
A Discrete Choquet Integral for Ordered Systems
A model for a Choquet integral for arbitrary finite set systems is presented.
The model includes in particular the classical model on the system of all
subsets of a finite set. The general model associates canonical non-negative
and positively homogeneous superadditive functionals with generalized belief
functions relative to an ordered system, which are then extended to arbitrary
valuations on the set system. It is shown that the general Choquet integral can
be computed by a simple Monge-type algorithm for so-called intersection
systems, which include as a special case weakly union-closed families.
Generalizing Lov\'asz' classical characterization, we give a characterization
of the superadditivity of the Choquet integral relative to a capacity on a
union-closed system in terms of an appropriate model of supermodularity of such
capacities
A Discrete Choquet Integral for Ordered Systems
A model for a Choquet integral for arbitrary finite set systems is presented. The model includes in particular the classical model on the system of all subsets of a finite set. The general model associates canonical non-negative and positively homogeneous superadditive functionals with generalized belief functions relative to an ordered system, which are then extended to arbitrary valuations on the set system. It is shown that the general Choquet integral can be computed by a simple Monge-type algorithm for so-called intersection systems, which include as a special case weakly union-closed families. Generalizing LovĂĄsz' classical characterization, we give a characterization of the superadditivity of the Choquet integral relative to a capacity on a union-closed system in terms of an appropriate model of supermodularity of such capacities.Choquet integral, belief function, measurability, set systems, Monge algorithm, supermodularity
Robust Integrals
In decision analysis and especially in multiple criteria decision analysis,
several non additive integrals have been introduced in the last years. Among
them, we remember the Choquet integral, the Shilkret integral and the Sugeno
integral. In the context of multiple criteria decision analysis, these
integrals are used to aggregate the evaluations of possible choice
alternatives, with respect to several criteria, into a single overall
evaluation. These integrals request the starting evaluations to be expressed in
terms of exact-evaluations. In this paper we present the robust Choquet,
Shilkret and Sugeno integrals, computed with respect to an interval capacity.
These are quite natural generalizations of the Choquet, Shilkret and Sugeno
integrals, useful to aggregate interval-evaluations of choice alternatives into
a single overall evaluation. We show that, when the interval-evaluations
collapse into exact-evaluations, our definitions of robust integrals collapse
into the previous definitions. We also provide an axiomatic characterization of
the robust Choquet integral.Comment: 24 page