ELECTRE: Robust ordinal regression for outranking methods

We present a new method, called ELECTRE^GKMS, which employs robust ordinal regression to construct a set of outranking models compatible with preference information. The preference information supplied by the decision maker (DM) is composed of pairwise comparisons stating the truth or falsity of the outranking relation for some real or fictitious reference alternatives. Moreover, the DM specifies some ranges of variation of comparison thresholds on considered pseudo-criteria. Using robust ordinal regression, the method builds a set of values of concordance indices, concordance thresholds, indifference, preference, and veto thresholds, for which all specified pairwise comparisons can be restored. Such sets are called compatible outranking models. Using these models, two outranking relations are defined, necessary and possible. Whether for an ordered pair of alternatives there is necessary or possible outranking depends on the truth of outranking relation for all or at least one compatible model, respectively. Distinguishing the most certain recommendation worked out by the necessary outranking, and a possible recommendation worked out by the possible outranking, ELECTRE^GKMS answers questions of robustness concern. The method is intended to be used interactively with incremental specification of pairwise comparisons, possibly with decreasing confidence levels. In this way, the necessary and possible outranking relations can be, respectively, enriched or impoverished with the growth of the number of pairwise comparisons. Furthermore, the method is able to identify troublesome pieces of preference information which are responsible for incompatibility. The necessary and possible outranking relations are to be exploited as usual outranking relations to work out recommendation in choice or ranking problems. The introduced approach is illustrated by a didactic example showing how ELECTRE^GKMS can support real-world decision problems.     

Weighted Overlap Dominance – A procedure for interactive selection on multidimensional interval data

We present an outranking procedure that supports selection of alternatives represented by multiple attributes with interval valued data. The procedure is interactive in the sense that the decision maker directs the search for preferred alternatives by providing weights of the different attributes as well as parameters related to risk attitude and weighted dominance. The outranking relation builds on pairwise comparisons between optimistic and pessimistic weighted values as well as weighted dominance relations supported by volume based measures. The suggested procedure is referred to as the Weighted Overlap Dominance procedure (WOD).     

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.     

Handling multicriteria preferences in cluster analysis

In the framework of multicriteria decision aid, a lot of interest has been devoted to sorting problems, in which the set of categories is pre-defined. Besides, preference oriented multicriteria clustering has received little attention. Usual geometric and related metrics are not well suited for this problem. Here, we propose a clustering method based on a valued indifference relation inspired by outranking methods. We suggest a method (based on comparing cluster centers and an average net flow score of clusters) to build a complete ranking of the set of clusters, that is, a way of defining a set of ordered categories for sorting purposes. The new approach performs very well in some examples.     

How to select and how to rank projects: The Promethee method

In this paper, we present the Promethee methods, a new class of outranking methods in multicriteria analysis. Their main features are simplicity, clearness and stability. The notion of generalized criterion is used to construct a valued outranking relation. All the parameters to be defined have an economic signification, so that the decision maker can easily fix them. Two ways of treatment are proposed: It is possible to obtain either a partial preorder (Promethee I) or a complete one (Promethee II), both on a finite set of feasible actions. A comparison is made with the Electre III method. The stability of the results given by the two methods is analysed. Numerical applications are given in order to illustrate the properties of the new methods and some further problems are discussed.     

<Emphasis Type="Bold">R</Emphasis><Emphasis Type="SmallCaps">UBIS</Emphasis>: a bipolar-valued outranking method for the choice problem

The main concern of this article is to present the R UBIS method for tackling the choice problem in the context of multiple criteria decision aiding. Its genuine purpose is to help a decision maker to determine a single best decision alternative. Methodologically we focus on pairwise comparisons of these alternatives which lead to the concept of bipolar-valued outranking digraph. The work is centred around a set of five pragmatic principles which are required in the context of a progressive decision aiding methodology. Their thorough study and implementation in the outranking digraph lead us to define a choice recommendation as an extension of the classical digraph kernel concept.      

Preference relations on actions and criteria in multicriteria decision making

Let A be a finite set of feasible actions which are judged following several criteria. An outranking relation is defined on A by considering preference of the decision maker as a weak order on each criterion and the relation among criteria as a semi-order on the given set of criteria.Several ways of constructing outranking relations have been proposed. One of the most popular, introduced by B. Roy, for instance ELECTRE(s), is based on the use of weights related to criteria. In our approach, the knowledge of weights is replaced by the existence of a semi-order.A case study is developed. It deals with a computer selection problem.     

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.     

The extended QUALIFLEX method for multiple criteria decision analysis based on interval type-2 fuzzy sets and applications to medical decision making

QUALIFLEX, a generalization of Jacquet-Lagreze’s permutation method, is a useful outranking method in decision analysis because of its flexibility with respect to cardinal and ordinal information. This paper develops an extended QUALIFLEX method for handling multiple criteria decision-making problems in the context of interval type-2 fuzzy sets. Interval type-2 fuzzy sets contain membership values that are crisp intervals, which are the most widely used of the higher order fuzzy sets because of their relative simplicity. Using the linguistic rating system converted into interval type-2 trapezoidal fuzzy numbers, the extended QUALIFLEX method investigates all possible permutations of the alternatives with respect to the level of concordance of the complete preference order. Based on a signed distance-based approach, this paper proposes the concordance/discordance index, the weighted concordance/discordance index, and the comprehensive concordance/discordance index as evaluative criteria of the chosen hypothesis for ranking the alternatives. The feasibility and applicability of the proposed methods are illustrated by a medical decision-making problem concerning acute inflammatory demyelinating disease, and a comparative analysis with another outranking approach is conducted to validate the effectiveness of the proposed methodology.     

A Graded Quadrivalent Logic for Ordinal Preference Modelling: Loyola–Like Approach

We extend a quadrivalent logic of Belnap to graded truth values in order to handle graded relevance of positive and negative arguments provided in preferential information concerning ranking of a finite set of alternatives. This logic is used to design the preference modelling and exploitation phases of decision aiding with respect to the ranking problem. The graded arguments are presented on an ordinal scale and their aggregation leads to preference model in form of four graded outranking relations (true, false, unknown and contradictory). The exploitation procedure combines the min-scoring procedure with the leximin rule. Aggregation of positive and negative arguments as well as exploitation of the resulting outranking relations is concordant with an advice given by St. Ignatius of Loyola (1548) how to make a good choice.     

A fuzzy outranking relation in multicriteria decision making

It frequently happens that a decision maker must establish a ranking within a finite set of alternatives with respect to multiple criteria. The subjective evaluation of each alternative according to each criterion is expressed in the form of a distributive evaluation. To capture the preferences of one alternative over another, a concept of fuzzy outranking relation can be used. This fuzzy outranking relation is characterized by a degree of credibility which is computed from two indices: a confidence index and a doubt index. Each of these indices is calculated from the distributive evaluations over the various criteria. In this paper, such a fuzzy outranking relation (fuzzy binary relation) is constructed and an application is presented.     

Uncertainty index based interval assignment by Interval AHP

In a multi-attribute decision making problem, indigenous values are assigned to attributes based on a decision maker’s subjective judgments. The given judgments are often uncertain, because of the uncertainty of situations and intuitiveness of human judgments. In order to reflect the uncertainty in the assigned values, they are denoted as intervals whose widths represent the possibilities of attributes. Since it is difficult for a decision maker to assign values directly to attributes in case of more than two attributes, he/she gives a pairwise comparison matrix by comparing two attributes at one occasion. The given matrix contains two kinds of uncertainty, one is inconsistency among comparisons and the other is incompleteness of comparisons. This paper proposes the models to obtain intervals of attributes from the given uncertain pairwise comparison matrix. At first, the uncertainty indexes of a set of intervals are defined from the viewpoints of entropy in probability, sum or maximum of widths, or ignorance. Then, considering that too uncertain information is not useful, the intervals of attributes are obtained by minimizing their uncertainty indexes.     

Some models for deriving the priority weights from interval fuzzy preference relations

Interval fuzzy preference relation is a useful tool to express decision maker’s uncertain preference information. How to derive the priority weights from an interval fuzzy preference relation is an interesting and important issue in decision making with interval fuzzy preference relation(s). In this paper, some new concepts such as additive consistent interval fuzzy preference relation, multiplicative consistent interval fuzzy preference relation, etc., are defined. Some simple and practical linear programming models for deriving the priority weights from various interval fuzzy preference relations are established, and two numerical examples are provided to illustrate the developed models.     

Outranking under uncertainty using scenarios

This paper considers the use of scenarios to treat uncertain attribute evaluations in the outranking methods. The scenario-based approach allows the decision maker to think deterministically about the problem by attaching causal links to a small number of potential outcomes, instead of using probability distributions. The scenario approach can be expressed as a simplified version of the comprehensive but practically complex “distributive” outranking method of d’Avignon and Vincke. Using a scenario approach has distinct practical advantages, but also presents the inherent danger that meaningful information is ignored. The extent of this danger is assessed using a simulation experiment, where it is found to be of a magnitude that is non-trivial but still potentially acceptable for certain decision contexts.     

Preference elicitation from inconsistent judgments using multi-objective optimization

Several decision-making techniques involve pairwise comparisons to elicit the preferences of a decision maker (DM). This paper proposes a new approach for prioritization from pairwise comparisons using the concept of indirect judgments. No method exists that simultaneously minimizes deviations from both direct and indirect judgments. In order to estimate preferences, it is sensible to consider both the acquired judgments and the other judgments latent in the DM’s mind. Hence, a technique is developed here to minimize the deviations from both types of judgments.     

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.     

A new data envelopment analysis method for priority determination and group decision making in the analytic hierarchy process

The DEAHP method for weight deviation and aggregation in the analytic hierarchy process (AHP) has been found flawed and sometimes produces counterintuitive priority vectors for inconsistent pairwise comparison matrices, which makes its application very restrictive. This paper proposes a new data envelopment analysis (DEA) method for priority determination in the AHP and extends it to the group AHP situation. In this new DEA methodology, two specially constructed DEA models that differ from the DEAHP model are used to derive the best local priorities from a pairwise comparison matrix or a group of pairwise comparison matrices no matter whether they are perfectly consistent or inconsistent. The new DEA method produces true weights for perfectly consistent pairwise comparison matrices and the best local priorities that are logical and consistent with decision makers (DMs)’ subjective judgments for inconsistent pairwise comparison matrices. In hierarchical structures, the new DEA method utilizes the simple additive weighting (SAW) method for aggregation of the best local priorities without the need of normalization. Numerical examples are examined throughout the paper to show the advantages of the new DEA methodology and its potential applications in both the AHP and group decision making.     

An outranking method under uncertainty

The uncertainty of consequences and the imprecision of data often imply, in multicriteria decision problems, the use of probability distributions to characterize the evaluation of each action with respect to eacg criterion. To keep as much information as possible, the analysis should treat directly these probability distributions instead of reducing them to single values such as mean or median. In this context, the paper proposes a multicriteria procedure which transforms these distributive evaluations of actions, according to decisionmaker's preferences, in order to progress to a ranking of these actions. The procedure consists, for each couple of actions, to construct a distributive preference degree with respect to each criterion and a distributive outranking degree over all criteria. These distributive outranking degrees are then explored in order to rank the actions, totally or partially.     

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.     

Performance evaluation: An integrated method using data envelopment analysis and fuzzy preference relations

In a multi-attribute decision-making (MADM) context, the decision maker needs to provide his preferences over a set of decision alternatives and constructs a preference relation and then use the derived priority vector of the preference to rank various alternatives. This paper proposes an integrated approach to rate decision alternatives using data envelopment analysis and preference relations. This proposed approach includes three stages. First, pairwise efficiency scores are computed using two DEA models: the CCR model and the proposed cross-evaluation DEA model. Second, the pairwise efficiency scores are then utilized to construct the fuzzy preference relation and the consistent fuzzy preference relation. Third, by use of the row wise summation technique, we yield a priority vector, which is used for ranking decision-making units (DMUs). For the case of a single output and a single input, the preference relation can be directly obtained from the original sample data. The proposed approach is validated by two numerical examples.