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 Maker's (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.     

Using ELECTRE TRI outranking method to sort MOMILP nondominated solutions

Several interactive methods exist to identify nondominated solutions in a Multiple Objective Mixed Integer Linear Program. But what if the Decision Maker is also interested in sorting those solutions (assigning them to pre-established ordinal categories)? We propose an interactive “branch-and-bound like” technique to progressively build the nondominated set, combined with ELECTRE TRI method (Pessimistic procedure) to sort identified nondominated solutions. A disaggregation approach is considered in order to avoid direct definition of all ELECTRE TRI preference parameters. Weight-importance coefficients are inferred and category reference profiles are determined based on assignment examples provided by the Decision Maker. A computation tool was developed with a twofold purpose: support the Decision Maker involved in a decision process and provide a test bed for research purposes.     

Eliciting Electre Tri category limits for a group of decision makers

Multiple criteria sorting aims at assigning alternatives evaluated on several criteria to predefined ordered categories. In this paper, we consider a well known multiple criteria sorting method, Electre Tri, which involves three types of preference parameters: (1) category limits defining the frontiers between consecutive categories, (2) weights and majority level specifying which coalitions form a majority, and (3) veto thresholds characterizing discordance effects. We propose an elicitation procedure to infer category limits from assignment examples provided by multiple decision makers. The procedure computes a set of category limits and vetoes common to all decision makers, with variable weights for each decision maker. Hence, the method helps reaching a consensus among decision makers on the category limits and veto thresholds, whereas finding a consensus on weights is left aside. The inference procedure is based on mixed integer linear programming and performs well even for datasets corresponding to real-world decision problems. We provide an illustrative example of the use of the method and analyze the performance of the proposed algorithms.     

An algorithm for ordinal sorting based on ELECTRE with categories defined by examples

This work proposes a Progressive Assisted Sorting Algorithm (PASA) based on a multicriteria evaluation ELECTRE-type method. The purpose of the PASA is to aid a decision maker to progressively sort a set of alternatives into a set of categories, which we considered are ordered (ordinal sorting), following a consistency principle. We consider the principle that if an alternative outranks (is as good as) a second one, then it must belong to the same category or to a better category. The set of alternatives already sorted by the decision maker will implicitly define the categories, and will constrain the range of categories where other alternatives may be sorted. We show how the same idea may be used in an aggregation/disaggregation approach, considering some parameters of ELECTRE are not fixed a priori, but are constrained only by the examples provided. In this context, we establish a “convex-shape property” stating that the range of possible categories for an alternative is always an interval of categories. A discussion contrasting this approach with ELECTRE TRI is included in the conclusions.     

An axiomatic approach to noncompensatory sorting methods in MCDM,I: The case of two categories

In the literature on MCDM, many methods have been proposed in order to sort alternatives evaluated on several attributes into ordered categories. Most of them were proposed on an ad hoc basis. The purpose of this paper is to contribute to a recent trend of research aiming at giving these methods sound theoretical foundations. Using tools from conjoint measurement, we provide an axiomatic analysis of the partitions of alternatives into two categories that can be obtained using what we call “noncompensatory sorting models”. These models have strong links with the pessimistic version of ELECTRE TRI. Our analysis allows to pinpoint what appears to be the main distinctive features of ELECTRE TRI when compared to other sorting methods. It also gives hints on the various methods that have been proposed to assess the parameters of ELECTRE TRI on the basis of assignment examples.     

A stochastic method for robustness analysis in sorting problems

ELECTRE TRI is a multiple criteria decision aiding sorting method with a history of successful real-life applications. In ELECTRE TRI, values for certain parameters have to be provided. We propose a new method, SMAA-TRI, that is based on stochastic multicriteria acceptability analysis (SMAA), for analyzing the stability of such parameters. The stability analysis can be used for deriving robust conclusions. SMAA-TRI allows ELECTRE TRI to be used with uncertain, arbitrarily distributed values for weights, the lambda cutting level, and profiles. The method consists of analyzing finite spaces of arbitrarily distributed parameter values. Monte Carlo simulation is applied in this in order to describe for each alternative the share of parameter values that have it assigned to different categories. We show the real-life applicability by re-analyzing a case study in the field of risk assessment.     

An evolutionary approach to construction of outranking models for multicriteria classification: The case of the ELECTRE TRI method

Outranking methods constitute an important class of multicriteria classification models. Often, however, their implementation is cumbersome, due to the large number of parameters that the decision maker must specify. Past studies tried to address this issue using linear and nonlinear programming, to elicit the necessary preferential information from assignment examples. In this study, an evolutionary approach, based on the differential evolution algorithm, is proposed in the context of the ELECTRE TRI method. Computational results are given to test the effectiveness of the methodology and the quality of the obtained models.     

Dealing with inconsistent judgments in multiple criteria sorting models

Sorting models consist in assigning alternatives evaluated on several criteria to ordered categories. To implement such models it is necessary to set the values of the preference parameters used in the model. Rather than fixing the values of these parameters directly, a usual approach is to infer these values from assignment examples provided by the decision maker (DM), i.e., alternatives for which (s)he specifies a required category. However, assignment examples provided by DMs can be inconsistent, i.e., may not match the sorting model. In such situations, it is necessary to support the DMs in the resolution of this inconsistency. In this paper, we extend algorithms from mous5ejor03 that calculate different ways to remove assignment examples so that the information can be represented in the sorting model. The extension concerns the possibility to relax (rather than to delete) assignment examples. These algorithms incorporate information about the confidence attached to each assignment example, hence providing inconsistency resolutions that the DMs are most likely to accept. Received: September 2004, Revised: June 2005 AMS classification: 90B50, 91B08, 90C05     

An axiomatic approach to noncompensatory sorting methods in MCDM,II: More than two categories

A number of recent papers have investigated the foundations of methods allowing to sort multi-attributed alternatives between several ordered categories. This paper has a similar objective. Our analysis uses a general conjoint measurement framework, encompassing most sorting models used in MDCM, that was proposed in the literature. Within this framework, we provide an axiomatic analysis of what we call noncompensatory sorting models, with or without veto effects. These noncompensatory sorting models contain the pessimistic version of ELECTRE TRI as a particular case. Our analysis can be seen as an attempt to give a firm axiomatic basis to ELECTRE TRI, while emphasizing its specific feature, i.e., the rather poor information that this model uses on each attribute.     

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.     

Multiple criteria sorting with a set of additive value functions

We present a new multiple criteria sorting method that aims at assigning actions evaluated on multiple criteria to p pre-defined and ordered classes. The preference information supplied by the decision maker (DM) is a set of assignment examples on a subset of actions relatively well known to the DM. These actions are called reference actions. Each assignment example specifies a desired assignment of a corresponding reference action to one or several contiguous classes. The set of assignment examples is used to build a preference model of the DM represented by a set of general additive value functions compatible with the assignment examples. For each action a, the method computes two kinds of assignments to classes, concordant with the DM’s preference model: the necessary assignment and the possible assignment. The necessary assignment specifies the range of classes to which the action can be assigned considering all compatible value functions simultaneously. The possible assignment specifies, in turn, the range of classes to which the action can be assigned considering any compatible value function individually. The compatible value functions and the necessary and possible assignments are computed through the resolution of linear programs.     

An interactive sorting approach based on the assignment examples of multiple decision makers with different priorities

Although group decision-making is often adopted by many organizations in today??s highly complicated business environment, the multiple criteria sorting (MCS) problem in the context of group decision-making has not been studied sufficiently. To this end, we propose a new interactive method to assist a group of decision makers (DMs) with different priorities. With the goal of relieving the cognitive effort exerted by DMs, this method uses the assignment examples provided by the DMs to draw the parameters for the group preference model. In the iterative MCS process that we employ, the DMs are supported from two perspectives. When the assignment examples provided by the DMs are inconsistent, a RINCON algorithm is developed to identify all the possible solutions that the DMs can use to resolve the conflicts. When the examples are consistent, the potential and the fittest assignments of each alternative are deduced using linear programming techniques. These are then presented to the DMs to help them provide more information for the decision-making process. Furthermore, the priority of each DM is objectively and subjectively evaluated, and then progressively updated to reflect the decision-making performance of a DM at each iteration. Meanwhile, the priorities are integrated into the linear programming model to deduce the fittest assignment, as well as into the RINCON algorithm. Hence, the assignment examples of the DMs with higher priorities are emphasized in the fittest assignment, and are less likely to be revised for inconsistency. A practical example featuring MBA programs is also presented to demonstrate the proposed method.     

Sensitivity analysis for the ELECTRE multicriteria method

In this paper, we develop sensitivity ranges for parameters used in the ELECTRE I multicriteria decision method. Parameters studied include criteria weights as well as significance thresholds. For these parameters, intervals are determined in which parameter changes will not affect the set of non-outranked alternatives. The approach developed is illustrated by a numerical example and possible applications are discussed.

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Zusammenfassung In dieser Arbeit wird ein Ansatz zur Sensitivitätsanalyse für das Multikriteria-Entscheidungsverfahren ELECTRE I vorgestellt. Für die Gewichte der Kriterien sowie die im Verfahren verwendeten Signifikanzschwellen werden jeweils Intervalle konstruiert, innerhalb derer Parameteränderungen keine Änderungen im Ergebnis des Verfahrens bewirken. Die entwickelte Sensitivitätsanalyse wird anhand eines Beispiels illustriert und mögliche Anwendungen des Konzeptes werden aufgezeigt.

     

DIS-CARD: a new method of multiple criteria sorting to classes with desired cardinality

In this paper, we present a new preference disaggregation method for multiple criteria sorting problems, called DIS-CARD. Real-life experience indicates the need of considering decision making situations in which a decision maker (DM) specifies a desired number of alternatives to be assigned to single classes or to unions of some classes. These situations require special methods for multiple criteria sorting subject to desired cardinalities of classes. DIS-CARD deals with such a problem, using the ordinal regression approach to construct a model of DM’s preferences from preference information provided in terms of exemplary assignments of some reference alternatives, together with the above desired cardinalities. We develop a mathematical model for incorporating such preference information via mixed integer linear programming (MILP). Then, we adapt the MILP model to two types of preference models: an additive value function and an outranking relation. Illustrative example is solved to illustrate the methodology.     

Multicriteria sorting using a valued indifference relation under a preference disaggregation paradigm

Most multicriteria decision methods need the definition of a significant amount of preferential information from a decision agent. The preference disaggregation analysis paradigm infers the model’s parameter values from holistic judgments provided by a decision agent. Here, a new method for inferring the parameters of a fuzzy outranking model for multicriteria sorting is proposed. This approach allows us to use most of the preferential information contained in a reference set. The central idea is to characterize the quality of the model by measuring discrepancies and concordances amongst (i) the preference relations derived from the outranking model, and (ii) the preferential information contained in the reference set. The model’s parameters are inferred from a multiobjective optimization problem, according to some additional preferential information from a decision agent. Once the model has been fitted, sorting decisions about new objects are performed by using a fuzzy indifference relation. This proposal performs very well in some examples.     

Interactive outranking approaches for multicriteria decision-making problems with imprecise information

PROMETHEE is a powerful method, which can solve many multiple criteria decision making (MCDM) problems. It involves sophisticated preference modelling techniques but requires too much a priori precise information about parameter values (such as criterion weights and thresholds). In this paper, we consider a MCDM problem where alternatives are evaluated on several conflicting criteria, and the criterion weights and/or thresholds are imprecise or unknown to the decision maker (DM). We build robust outranking relations among the alternatives in order to help the DM to rank the alternatives and select the best alternative. We propose interactive approaches based on PROMETHEE method. We develop a decision aid tool called INTOUR, which implements the developed approaches.     

Multicriteria decision making for contract research organisation choice in the pharmaceutical industry

The choice of a Contract Research Organisation (CRO-Sponsor partnership or CRO) in clinical development is fundamental for the medical department of a pharmaceutical company. A decision analysis tool can help in this choice. Multicriteria analysis (ELECTRE 1 and 2) has the capacity to integrate qualitative judgement criteria with weights in incorporating the concordance and discordance indexes. The objective of this study is to present the theory and one application in the selection of CROs. How can one CRO be selected from a group of nine pre-selections by using seven judgement criteria with different weights? The results in this paper are: with the ELECTRE 1 method six CROs were selected and only one CRO with the ELECTRE 2 method.     

Building a binary outranking relation in uncertain, imprecise and multi-experts contexts: The application of evidence theory

We consider multicriteria decision problems where the actions are evaluated on a set of ordinal criteria. The evaluation of each alternative with respect to each criterion may be uncertain and/or imprecise and is provided by one or several experts. We model this evaluation as a basic belief assignment (BBA). In order to compare the different pairs of alternatives according to each criterion, the concept of first belief dominance is proposed. Additionally, criteria weights are also expressed by means of a BBA. A model inspired by ELECTRE I is developed and illustrated by a pedagogical example.     

Multi-attribute decision making: A simulation comparison of select methods

Several methods have been proposed for solving multi-attribute decision making problems (MADM). A major criticism of MADM is that different techniques may yield different results when applied to the same problem. The problem considered in this study consists of a decision matrix input of N criteria weights and ratings of L alternatives on each criterion. The comparative performance of some methods has been investigated in a few, mostly field, studies. In this simulation experiment we investigate the performance of eight methods: ELECTRE, TOPSIS, Multiplicative Exponential Weighting (MEW), Simple Additive Weighting (SAW), and four versions of AHP (original vs. geometric scale and right eigenvector vs. mean transformation solution). Simulation parameters are the number of alternatives, criteria and their distribution. The solutions are analyzed using twelve measures of similarity of performance. Similarities and differences in the behavior of these methods are investigated. Dissimilarities in weights produced by these methods become stronger in problems with few alternatives; however, the corresponding final rankings of the alternatives vary across methods more in problems with many alternatives. Although less significant, the distribution of criterion weights affects the methods differently. In general, all AHP versions behave similarly and closer to SAW than the other methods. ELECTRE is the least similar to SAW (except for closer matching the top-ranked alternative), followed by MEW. TOPSIS behaves closer to AHP and differently from ELECTRE and MEW, except for problems with few criteria. A similar rank-reversal experiment produced the following performance order of methods: SAW and MEW (best), followed by TOPSIS, AHPs and ELECTRE. It should be noted that the ELECTRE version used was adapted to the common MADM problem and therefore it did not take advantage of the method's capabilities in handling problems with ordinal or imprecise information.     

Assessing the performance of biogas plants with multi-criteria and data envelopment analysis

This paper performs an assessment of 41 agricultural biogas plants located in Austria to determine their relative performance in terms of economic, environmental, and social criteria and corresponding indicators. The comparison of these renewable energy conversion plants is based on two complementary analyses. Data envelopment analysis (DEA) was conducted to provide measures of radial efficiency relative to the observed frontier of production possibilities. Multi-criteria decision analysis (MCDA), using the IRIS/ELECTRE TRI methodology, was conducted to obtain a different perspective on the results, and as a tool that would enable to incorporate managerial preferences easily. To be able to use IRIS while keeping the spirit behind DEA, the evaluation criteria were defined as different output/input efficiency ratios, and no information about criteria weights was introduced at the outset. The results suggest that MCDA, and the use of IRIS in particular, constitutes a useful approach that can be applied in a complementary way to DEA.