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.Choquet integral, Sugeno integral, capacity, bipolarity, preferences

Bipolar and bivariate models in multi-criteria decision analysis: descriptive and constructive approaches

Multi-criteria decision analysis studies decision problems in which the alternatives are evaluated on several dimensions or viewpoints. In the problems we consider in this paper, the scales used for assessing the alternatives with respect to a viewpoint are bipolar and univariate or unipolar and bivariate. In the former case, the scale is divided in two zones by a neutral point; a positive feeling is associated to the zone above the neutral point and a negative feeling to the zone below this point. On unipolar bivariate scales, an alternative can receive both a positive and a negative evaluation, reflecting contradictory feelings or stimuli. The paper discusses procedures and models that have been proposed to aggregate multi-criteria evaluations when the scale of each criterion is of one of the two types above. We present both a constructive and a descriptive view on this question; the descriptive approach is concerned with characterizations of models of preference, while the constructive approach aims at building preferences by questioning the decision maker. We show that these views are complementary.Multiple criteria, Decision analysis, Preference, Bipolarmodels, Choquet integral

A Representation of Preferences by the Choquet Integral with Respect to a 2-Additive Capacity

In the context of Multiple criteria decision analysis, we present the necessary and sufficient conditions allowing to represent an ordinal preferential information provided by the decision maker by a Choquet integral w.r.t a 2-additive capacity. We provide also a characterization of this type of preferential information by a belief function which can be viewed as a capacity. These characterizations are based on three axioms, namely strict cycle-free preferences and some monotonicity conditions called MOPI and 2-MOPI.multicriteria decision making; Choquet integral; 2-additive capacity; MACBETH

On the Choquet multiple criteria preference aggregation model: theoretical and practical insights from a real-world application

We consider the use of the Choquet integral for evaluating projects or actions in a real-world application starting from the case of the re-qualification of an abandoned quarry. Despite the Choquet integral being a very well-known preference model for which there is a rich and well developed theory, its application in a multiple criteria decision aiding perspective requires some specific methodological developments. This led us to work out and implement, in practice, two new procedures: A first procedure to build interval scales with the objective of assigning utility values on a common scale to the criteria performances, and a second one to construct a ratio scale for assigning numerical values to the capacities of the Choquet integral. This article discusses the strengths and weaknesses of the Choquet integral as appearing in the case study, proposing as well insights related to the interaction of the experts within a focus grou

Learning nonlinear monotone classifiers using the Choquet Integral

In der jüngeren Vergangenheit hat das Lernen von Vorhersagemodellen, die eine monotone Beziehung zwischen Ein- und Ausgabevariablen garantieren, wachsende Aufmerksamkeit im Bereich des maschinellen Lernens erlangt. Besonders für flexible nichtlineare Modelle stellt die Gewährleistung der Monotonie eine große Herausforderung für die Umsetzung dar. Die vorgelegte Arbeit nutzt das Choquet Integral als mathematische Grundlage für die Entwicklung neuer Modelle für nichtlineare Klassifikationsaufgaben. Neben den bekannten Einsatzgebieten des Choquet-Integrals als flexible Aggregationsfunktion in multi-kriteriellen Entscheidungsverfahren, findet der Formalismus damit Eingang als wichtiges Werkzeug für Modelle des maschinellen Lernens. Neben dem Vorteil, Monotonie und Flexibilität auf elegante Weise mathematisch vereinbar zu machen, bietet das Choquet-Integral Möglichkeiten zur Quantifizierung von Wechselwirkungen zwischen Gruppen von Attributen der Eingabedaten, wodurch interpretierbare Modelle gewonnen werden können. In der Arbeit werden konkrete Methoden für das Lernen mit dem Choquet Integral entwickelt, welche zwei unterschiedliche Ansätze nutzen, die Maximum-Likelihood-Schätzung und die strukturelle Risikominimierung. Während der erste Ansatz zu einer Verallgemeinerung der logistischen Regression führt, wird der zweite mit Hilfe von Support-Vektor-Maschinen realisiert. In beiden Fällen wird das Lernproblem imWesentlichen auf die Parameter-Identifikation von Fuzzy-Maßen für das Choquet Integral zurückgeführt. Die exponentielle Anzahl von Freiheitsgraden zur Modellierung aller Attribut-Teilmengen stellt dabei besondere Herausforderungen im Hinblick auf Laufzeitkomplexität und Generalisierungsleistung. Vor deren Hintergrund werden die beiden Ansätze praktisch bewertet und auch theoretisch analysiert. Zudem werden auch geeignete Verfahren zur Komplexitätsreduktion und Modellregularisierung vorgeschlagen und untersucht. Die experimentellen Ergebnisse sind auch für anspruchsvolle Referenzprobleme im Vergleich mit aktuellen Verfahren sehr gut und heben die Nützlichkeit der Kombination aus Monotonie und Flexibilität des Choquet Integrals in verschiedenen Ansätzen des maschinellen Lernens hervor

Modelling multicriteria value interactions with Reasoning Maps

Idiographic causal maps are extensively employed in Operational Research to support problem structuring and complex decision making processes. They model means-end or causal discourses as a network of concepts connected by links denoting influence, thus enabling the representation of chains of arguments made by decision-makers. There have been proposals to employ such structures to support the structuring of multicriteria evaluation models, within an additive value measurement framework. However, a drawback of this multi-methodological modelling is the loss of richness of interactions along the means-end chains when evaluating options. This has led to the development of methods that make use of the structure of the map itself to evaluate options, such as the Reasoning Maps method, which employs ordinal scales and ordinal operators for such evaluation. However, despite their potential, Reasoning Maps cannot model explicitly value interactions nor perform a quantitative ranking of options, limiting their applicability and usefulness. In this article we propose extending the Reasoning Maps approach through a multilinear evaluation model structure, built with the MACBETH multicriteria method. The model explicitly captures the value interactions between concepts along the map and employs the MACBETH protocol of questioning to assess the strength of influence for each means-end link. The feasibility of the proposed approach to evaluate options and to deal with multicriteria interactions is tested in a real-world application to support the construction of a population health index

New axiomatizations of the Shapley interaction index for bi-capacities

International audienceBi-capacities are 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. After a short presentation of the basis structure, we introduce the Shapley value and the interaction index for capacities. Afterwards, the case of bi-capacities is studied with new axiomatizations of the interaction index