Vincent Mousseau

Ordinal regression revisited: Multiple criteria ranking using a set of additive value functions

We present a new method, called UTAGMS, for multiple criteria ranking of alternatives from set A using a set of additive value functions which result from an ordinal regression. The preference information provided by the decision maker is a set of pairwise comparisons on a subset of alternatives ARÂ [subset, double equals]Â A, called reference alternatives. The preference model built via ordinal regression is the set of all additive value functions compatible with the preference information. Using this model, one can define two relations in the set A: the necessary weak preference relation which holds for any two alternatives a, b from set A if and only if for all compatible value functions a is preferred to b, and the possible weak preference relation which holds for this pair if and only if for at least one compatible value function a is preferred to b. These relations establish a necessary and a possible ranking of alternatives from A, being, respectively, a partial preorder and a strongly complete relation. The UTAGMS method is intended to be used interactively, with an increasing subset AR and a progressive statement of pairwise comparisons. When no preference information is provided, the necessary weak preference relation is a weak dominance relation, and the possible weak preference relation is a complete relation. Every new pairwise comparison of reference alternatives, for which the dominance relation does not hold, is enriching the necessary relation and it is impoverishing the possible relation, so that they converge with the growth of the preference information. Distinguishing necessary and possible consequences of preference information on the complete set of actions, UTAGMS answers questions of robustness analysis. Moreover, the method can support the decision maker when his/her preference statements cannot be represented in terms of an additive value function. The method is illustrated by an example solved using the UTAGMS software. Some extensions of the method are also presented.

Inferring an ELECTRE TRI Model from Assignment Examples

Given a finite set of alternatives, the sorting problem consists in the assignment of each alternative to one of the pre-defined categories. In this paper, we are interested in multiple criteria sorting problems and, more precisely, in the existing method ELECTRE TRI. This method requires the elicitation of parameters (weights, thresholds, category limits,...) in order to construct the Decision Makers (DM) preference model. A direct elicitation of these parameters being rather difficult, we proceed to solve this problem in a way that requires from the DM much less cognitive effort. We elicit these parameters indirectly using holistic information given by the DM through assignment examples. We propose an interactive approach that infers the parameters of an ELECTRE TRI model from assignment examples. The determination of an ELECTRE TRI model that best restitutes the assignment examples is formulated through an optimization problem. The interactive aspect of this approach lies in the possibility given to the DM to revise his/her assignment examples and/or to give additional information before the optimization phase restarts.

A user-oriented implementation of the ELECTRE-TRI method integrating preference elicitation support

Multiple Criteria Sorting Problem consists in assigning a set of alternatives A={a1,a2,…,al} evaluated on n criteria g1,g2,…,gn to one of the categories which are pre-defined by some norms corresponding to vectors of scores on particular criteria, called profiles, either separating the categories or playing the role of central reference objects in the categories. The assignment of an alternative ak to a specific category results from a comparison of its evaluation on all criteria with the profiles defining the categories. This paper presents a new implementation of an existing method called ELECTRE TRI. It integrates specific functionalities supporting the decision maker (DM) in the preference elicitation process. These functionalities grouped in ELECTRE TRI Assistant aim at reducing the cognitive effort required from the DM in the phase of calibration of the preference model. The main characteristic feature of ELECTRE TRI Assistant is the inference of the ELECTRE TRI preferential parameters from assignment examples supplied by the DM. The software is presented through an illustrative example.

An aggregation/disaggregation approach to obtain robust conclusions with ELECTRE TRI

ELECTRE TRI is a well-known method to assign a set of alternatives to a set of predefined categories, considering multiple criteria. Using this method requires setting many parameters, which is often a difficult task. We consider the case where the decision makers (DMs) in the decision process are unsure of which values should each parameter take, which may result from uncertain, imprecise or inaccurately determined information, as well as from lack of consensus among them. This paper discusses the synergy between two approaches developed independently to deal with this difficulty. The first approach infers the value of parameters from assignment examples provided by the DMs, as an elicitation aid. Each assignment example originates mathematical constraints that the parameter values should satisfy. The second approach considers a set of constraints on the parameter values reflecting the imprecise information that the DMs are able to provide. Then, it computes the best and worst categories for each alternative compatible with constraints, in order to present robust conclusions. Both approaches avoid asking for precise values for the parameters. Rather, they proceed to solve the problem in a way that requires from the DMs much less effort. By integrating these two approaches, this paper proposes a new interactive approach, where the insight obtained during robustness analyses guides the DMs during the elicitation phase.

Using assignment examples to infer weights for ELECTRE TRI method: Some experimental results

Given a finite set of alternatives A, the sorting (or assignment) problem consists in the assignment of each alternative to one of the pre-defined categories. In this paper, we are interested in multiple criteria sorting problems and, more precisely, in the existing method ELECTRE TRI. This method requires the elicitation of preferential parameters (weights, thresholds, category limits,…) in order to construct a preference model which the decision maker (DM) accepts as a working hypothesis in the decision aid study. A direct elicitation of these parameters requiring a high cognitive effort from the DM (V. Mosseau, R. Slowinski, Journal of Global Optimization 12 (2) (1998) 174), proposed an interactive aggregation–disaggregation approach that infers ELECTRE TRI parameters indirectly from holistic information, i.e., assignment examples. In this approach, the determination of ELECTRE TRI parameters that best restore the assignment examples is formulated through a nonlinear optimization program. In this paper, we consider the subproblem of the determination of the weights only (the thresholds and category limits being fixed). This subproblem leads to solve a linear program (rather than nonlinear in the global inference model). Numerical experiments were conducted so as to check the behaviour of this disaggregation tool. Results showed that this tool is able to infer weights that restores in a stable way the assignment examples and that it is able to identify “inconsistencies” in the assignment examples.

Resolving inconsistencies among constraints on the parameters of an MCDA model

We consider a framework where decision makers (DMs) interactively define a multicriteria evaluation model by providing imprecise information (i.e., a linear system of constraints to the models parameters) and by analyzing the consequences of the information provided. DMs may introduce new constraints explicitly or implicitly (results that the model should yield). If a new constraint is incompatible with the previous ones, then the system becomes inconsistent and the DMs must choose between removing the new constraint or removing some of the older ones. We address the problem of identifying subsets of constraints which, when removed, lead to a consistent system. Identifying such subsets would indicate the reason for the inconsistent information given by DMs. There may exist several possibilities for the DMs to resolve the inconsistency. We present two algorithms to identify such possibilities, one using {0,1} mixed integer linear programming and the other one using linear programming. Both approaches are based on the knowledge that the system was consistent prior to introducing the last constraint. The output of these algorithms helps the DM to identify the conflicting pieces of information in a set of statements he/she asserted. The relevance of these algorithms for MCDA is illustrated by an application to an aggregation/disaggregation procedure for the Electre Tri method. � 2002 Elsevier Science B.V. All rights reserved.

A Theoretical Framework for Analysing the Notion of Relative Importance of Criteria

Multiple-criteria decision aid almost always requires the use of weights, importance coefficients or even a hierarchy of criteria, veto thresholds, etc. These are importance parameters that are used to differentiate the role devoted to each criterion in the construction of comprehensive preferences. Many researchers have studied the problem of how to assign values to such parameters, but few of them have tried to analyse in detail what underlies the notion of importance of criteria and to give a clear formal definition of it. In this paper our purpose is to define a theoretical framework so as to analyse the notion of the importance of criteria under very general conditions. Within this framework it clearly appears that the importance of criteria is taken into account in very different ways in various aggregation procedures. This framework also allows us to shed new light on fundamental questions such as: Under what conditions is it possible to state that one criterion is more important than another? Are importance parameters of the various aggregation procedures dependent on or independent of the encoding of criteria? What are the links between the two concepts of the importance of criteria and the compensatoriness of preferences? This theoretical framework seems to us sufficiently general to ground further research in order to define theoretically valid elicitation methods for importance parameters.

Valued outranking relations in ELECTRE providing manageable disaggregation procedures

In ELECTRE methods, the construction of an outranking relation amounts at validating or invalidating, for any pair of alternatives (a; b), the assertion ‘‘a is at least as good as b’’. This comparison is grounded on the evaluation vectors of both alternatives, and on additional information concerning the decision makers preferences, accounting for two conditions: concordance and non-discordance. In decision processes using these methods, the analyst should interact with the decision maker in order to elicit values for preferential parameters. This can be done either directly or through a disaggregation procedure that infers the parameters values from holistic judgements provided by the decision maker. Inference is usually performed through an optimization program that accounts for the aggregation model and minimizes an ‘‘error function’’. Although disaggregation approaches have been largely used in additive models, only few advances have been made towards a disaggregation approach for outranking methods. Indeed, outranking methods may lead to computationally difficult inference problems. In this paper we are concerned with a slight adaptation of the valued outranking relation used in the ELECTRE III and ELECTRE TRI. Such modification is shown to preserve the original discordance concept. We show that the modified outranking relation makes it easier to solve inference programs. � 2003 Elsevier B.V. All rights reserved.

Supporting groups in sorting decisions: Methodology and use of a multi-criteria aggregation/disaggregation DSS

This paper addresses the situation where a group wishes to cooperatively develop a common multicriteria evaluation model to sort actions (projects, candidates) into classes. It is based on an aggregation/disaggregation approach for the ELECTRE TRI method, implemented on the Decision Support System IRIS. We provide a methodology in which the group discusses how to sort some exemplary actions (possibly fictitious ones), instead of discussing what values the model parameters should take. This paper shows how IRIS may be used to help the group to iteratively reach an agreement on how to sort one or a few actions at a time, preserving the consistency of these sorting examples both at the individual level and at the collective level. The computation of information that may guide the discussion among the group members is also suggested. We provide an illustrative example and discuss some paths for future research motivated by this work.

Inferring Electre's veto-related parameters from outranking examples

When considering Electre’s valued outranking relations, aggregation/disaggregation methodologies have difficulties in taking discordance (veto) into account. We present a partial inference procedure to compute the value of the veto-related parameters that best restore a set of outranking statements (i.e., examples that an Electre model should restore) provided by a decision maker, given fixed values for the remaining parameters of the model. This paper complements previous work on the inference of other preference-related parameters (weights, cutting level, category limits, …), advancing toward an integrated framework of inference problems in Electre III and Tri methods. We propose mathematical programs to infer veto-related parameters, first considering only one criterion, then all criteria simultaneously, using the original version of Electre outranking relation and two variants. This paper shows that these inference procedures lead to linear programming, 0–1 linear programming, or separable programming problems, depending on the case.