Christophe Labreuche

Bi-capacities—I: definition, Möbius transform and interaction

Bi-capacities arise as a natural generalization of capacities (or fuzzy measures) in a context of decision making where underlying scales are bipolar. They are able to capture a wide variety of decision behaviours, encompassing models such as cumulative prospect theory (CPT). The aim of this paper in two parts is to present the machinery behind bi-capacities, and thus remains on a rather theoretical level, although some parts are firmly rooted in decision theory, notably cooperative game theory. The present first part is devoted to the introduction of bi-capacities and the structure on which they are defined. We define the Mobius transform of bi-capacities, by just applying the well-known theory of Mobius functions as established by Rota to the particular case of bi-capacities. Then, we introduce derivatives of bi-capacities, by analogy with what was done for pseudo-Boolean functions (another view of capacities and set functions), and this is the key point to introduce the Shapley value and the interaction index for bi-capacities. This is done in a cooperative game theoretic perspective. In summary, all familiar notions used for fuzzy measures are available in this more general framework.

A decade of application of the Choquet and Sugeno integrals in multi-criteria decision aid

The main advances regarding the use of the Choquet and Sugeno integrals in multi-criteria decision aid over the last decade are reviewed. They concern mainly a bipolar extension of both the Choquet integral and the Sugeno integral, interesting particular submodels, new learning techniques, a better interpretation of the models and a better use of the Choquet integral in multi-criteria decision aid. Parallel to these theoretical works, the Choquet integral has been applied to many new fields, and several softwares and libraries dedicated to this model have been developed.

Bi-capacities—II: the Choquet integral

Bi-capacities arise as a natural generalization of capacities (or fuzzy measures) in a context of decision making where underlying scales are bipolar. They are able to capture a wide variety of decision behaviours, encompassing models such as cumulative prospect theory (CPT). The aim of this paper in two parts is to present the machinery behind bi-capacities, and thus remains on a rather theoretical level, although some parts are firmly rooted in decision theory, notably cooperative game theory. The present second part focuses on the definition of Choquet integral. We give several expressions of it, including an expression w.r.t. the Mobius transform. This permits to express the Choquet integral for 2-additive bi-capacities w.r.t. the interaction index.

A characterization of the 2-additive Choquet integral through cardinal information

In the context of multiple criteria decision analysis, we present the necessary and sufficient conditions to represent a cardinal preferential information by the Choquet integral w.r.t. a 2-additive capacity. These conditions are based on some complex cycles called cyclones.

Preference modeling on totally ordered sets by the Sugeno integral

We present in this paper necessary and sufficient conditions for the representation of preferences in a decision-making problem, by the Sugeno integral, in a purely ordinal framework. We distinguish between strong representation (exact) and weak representation (no contradiction on strict preferences).

A multilateral multi-issue negotiation protocol

In this paper, we present a new protocol to address multilateral multi-issue negotiation in a cooperative context. We consider complex dependencies between multiple issues by modelling the preferences of the agents with a multi-criteria decision aid tool, also enabling us to extract relevant information on a proposal assessment. This information is used in the protocol to help in accelerating the search for a consensus between the cooperative agents. In addition, the negotiation procedure is defined in a crisis management context where the common objective of our agents is also considered in the preferences of a mediator agent.

Integration and propagation of a multi-criteria decision making model in constraint programming

In this paper we propose a general integration scheme for a Multi-Criteria Decision Making model of the Multi-Attribute Utility Theory in Constraint Programming. We introduce the Choquet integral as a general aggregation function for multi-criteria optimization problems and define the Choquet global constraint that propagates this function during the Branch-and-Bound search. Finally the benefits of the propagation of the Choquet constraint are evaluated on the examination timetabling problem.

How to improve ACTS: an alternative representation of the importance of criteria in MCDM

The intepretation of aggregation functions in multicriteria decision making is often based on indices such as importance indices that measure the importance of one criterion. For instance, for the Choquet integral, the importance index is the so-called Shapley value. The use of an index must always be limited to the specific context it has been designed for. Here, we are interested in determining on which criteria acts should be improved if we want their global evaluation to increase as much as possible. The Shapley value is not suited for describing this. So, we introduce a new index of importance which represents the mean worth for acts to reach higher scores in a set of criteria. This index is defined for general aggregation functions with the help of several axioms. This importance index is then applied to the Choquet integral. In particular, we computer the worth to reach higher levels in one attribute, and in a couple of attributes. Interestingly, this leads to quantities that are closely related to the Shapley and interaction indices.

The representation of conditional relative importance between criteria

Abstract We study in this paper the notion of conditional relative importance in a quantitative framework. Bi-capacities are shown to be suitable to represent such a notion. We restrict ourself to the case when the relative importance of two criteria are conditional on a third being attractive or repulsive. We give two algorithms that enable to construct the neutral level from the preference relation.

MCS—A new algorithm for multicriteria optimisation in constraint programming

In this paper we propose a new algorithm called MCS for the search for solutions to multicriteria combinatorial optimisation problems. To quickly produce a solution that offers a good trade-off between criteria, the MCS algorithm alternates several Branch & Bound searches following diversified search strategies. It is implemented in CP in a dedicated framework and can be specialised for either complete or partial search.