Brice Mayag

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.

Identification of a 2-Additive Bi-Capacity by Using Mathematical Programming

In some multi-criteria decision making problems, it is more convenient to express the decision maker preferences in bipolar scales. In such cases, the bipolar Choquet integral with respect to bi-capacities was introduced. In this paper, we address the problem of eliciting a bipolar Choquet integral with respect to a 2-additive bi-capacity. We assume that we are given a set of examples with i their scores distribution in regard to several criteria and ii their overall scores. We propose two types of optimization problems that allow identifying the parameters of a 2-additive bi-capacity such that the inferred bipolar Choquet integral is consistent with the given examples as much as possible. Furthermore, since the elicitation process we study has many relationships with problems in statistical machine learning, we also present the links between our models and concepts developed in the latter field.

An Interactive Algorithm to Deal with Inconsistencies in the Representation of Cardinal Information

We present a new interactive algorithm allowing to solve the inconsistencies problem, when the preferences of a decision maker cannot be representable by a numerical function. This algorithm is based on technics of linear programming and the type of preferences we use are cardinal information.

Decision Support for Disaster Risk Management: Integrating Vulnerabilities into Early-Warning Systems

Despite the potential of new technologies and the improvements of early-warning systems since the 2004 Tsunami, damage and harm caused by disasters do not stop to increase. There is a clear need for better integrating the fragmented landscape of researchers and practitioners working on different aspects of decision support for disaster risk reduction and response. To demonstrate and discuss the advantages of integrated systems, we will focus in this paper on vulnerabilities and early-warning systems. While vulnerabilities are mostly used to allocate risk management resources (preparedness), early-warning systems are designed to initiate the response phase. Indicator models have been used as a part of disaster risk reduction frameworks, and in the design of early-warning systems. In this paper we analyse the commonalities and differences between both, and outline how an integrated system that understands vulnerability assessments as part of both risk reduction programs and early-warning shall be designed in future.

Elicitation of a 2-additive bi-capacity through cardinal information on trinary actions

In the context of MultiCriteria Decision Aid, we present new properties of a 2-additive bi-capacity by using a bipolar Mobius transform. We use these properties in the identification of a 2-additive bi-capacity when we represent a cardinal information by a Choquet integral with respect to a 2-additive bi-capacity.

A Heuristic Approach to Test the Compatibility of a Preference Information with a Choquet Integral Model

This work deals with the problem of the existence of a Multicriteria Decision Aiding model, based on the Choquet integral, that represents the preferences of a decision maker. Given some preferences on a set of actions, our aim is to determine if those preferences are compatible with a Choquet integral model, where the utility function associated to each criterion and the capacity on the subsets of criteria are to be defined. Computing simultaneously the utility functions and the capacity leads to solving a mixed integer program with some quadratic constraints, which can not be performed efficiently. We propose here to solve this problem by using a linear approximation of the quadratic terms given by the Taylor’s formula, and then apply a standard mixed integer programming solver. We illustrate and analyze our approach with some numerical experiments.

A 2-Additive Choquet Integral Model for French Hospitals Rankings in Weight Loss Surgery

In a context of Multiple Criteria Decision Aid, we present a decision model explaining some French hospitals rankings in weight loss surgery. To take into account interactions between medical indicators, we elaborated a model based on the 2-additive Choquet integral. The reference subset, defined during the elicitation process of this model, is composed by some specific alternatives called binary alternatives. To validate our approach, we showed that the proposed 2-additive Choquet integral model is able to approximate the hospitals ranking, in weight loss surgery, published by the French magazine “Le Point” in August 2013.

Elicitation of a Utility from Uncertainty Equivalent Without Standard Gambles

In the context of decision under uncertainty, standard gambles are classically used to elicit a utility function on a set X of consequences. The utility of an element x in X is derived from the probability p for which a gamble giving the best outcome in X with probability p and the worst outcome in X otherwise, is indifferent to getting x for sure. In many situations, uncertainty that can be observed on the true value of X concerns only neighbour values. Uncertainty is then represented by a probability distribution whose support is an interval. In this case, standard gambles are unrealistic for the decision maker. We consider uncertainty represented by an equi-probability over an interval of X. This paper addresses the elicitation of a utility function on X by obtaining the certainty equivalent of an equi-probability over an interval of X. We show that not all utility models are suitable to accomplish this task.

MOPIC Properties in the Representation of Preferences by a 2-Additive Choquet Integral

In the context of Multiple Criteria Decision aiding MCDA, we present necessary conditions to obtain a representation of a cardinal information by a Choquet integral w.r.t a 2-additive capacity. A cardinal information is a preferential information provided by a Decision Maker DM containing a strict preference, a quaternary and indifference relations. Our work is focused on the representation of a cardinal information by a particular Choquet integral defined by a 2-additive capacity. Used as an aggregation function, it arises as a generalization of the arithmetic mean, taking into account the interaction between two criteria. Then, it is a good compromise between simple models like arithmetic mean and complex models like general Choquet integral. We consider also the set of fictitious alternatives called binary alternatives or binary actions from which the Choquet integral w.r.t a 2-additive capacity can be entirely specified. The proposed MOPIC MOnotonicity of Preferential Information for Cardinal conditions can be viewed as an alternative to balanced cyclones which are complex necessary and sufficient conditions, used in the characterization of a 2-additive Choquet integral through a cardinal information.

A Characterization of the 2-Additive Symmetric Choquet Integral Using Trinary Alternatives

In a context of Multiple Criteria Decision Aid, we present some necessary and sufficient conditions to obtain a symmetric Choquet integral compatible with some preferences on a particular set of alternatives. These axioms are based on the notion of strict cycle and the MOPI conditions.