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

We present a new method, called UTA^GMS, 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 A^R ⊆ 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 UTA^GMS method is intended to be used interactively, with an increasing subset A^R 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, UTA^GMS 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 UTA^GMS software. Some extensions of the method are also presented.     

Selection of a representative value function in robust multiple criteria ranking and choice

We introduce the concept of a representative value function in robust ordinal regression applied to multiple criteria ranking and choice problems. The proposed method can be seen as a new interactive UTA-like procedure, which extends the UTA^GMS and GRIP methods. The preference information supplied by the decision maker (DM) is composed of a partial preorder and intensities of preference on a subset of reference alternatives. Robust ordinal regression builds a set of general additive value functions which are compatible with the preference information, and returns two binary preference relations: necessary and possible. They identify recommendations which are compatible with all or at least one compatible value function, respectively. In this paper, we propose a general framework for selection of a representative value function from among the set of compatibles ones. There are a few targets which build on results of robust ordinal regression, and could be attained by a representative value function. In general, according to the interactively elicited preferences of the DM, the representative value function may emphasize the advantage of some alternatives over the others when all compatible value functions acknowledge this advantage, or reduce the ambiguity in the advantage of some alternatives over the others when some compatible value functions acknowledge an advantage and other ones acknowledge a disadvantage. The basic procedure is refined by few extensions. They enable emphasizing the advantage of alternatives that could be considered as potential best options, accounting for intensities of preference, or obtaining a desired type of the marginal value functions.     

Non-additive robust ordinal regression: A multiple criteria decision model based on the Choquet integral

Within the multicriteria aggregation–disaggregation framework, ordinal regression aims at inducing the parameters of a decision model, for example those of a utility function, which have to represent some holistic preference comparisons of a Decision Maker (DM). Usually, among the many utility functions representing the DM’s preference information, only one utility function is selected. Since such a choice is arbitrary to some extent, recently robust ordinal regression has been proposed with the purpose of taking into account all the sets of parameters compatible with the DM’s preference information. Until now, robust ordinal regression has been implemented to additive utility functions under the assumption of criteria independence. In this paper we propose a non-additive robust ordinal regression on a set of alternatives A, whose utility is evaluated in terms of the Choquet integral which permits to represent the interaction among criteria, modelled by the fuzzy measures, parameterizing our approach.     

Building a set of additive value functions representing a reference preorder and intensities of preference: GRIP method

We present a method called Generalized Regression with Intensities of Preference (GRIP) for ranking a finite set of actions evaluated on multiple criteria. GRIP builds a set of additive value functions compatible with preference information composed of a partial preorder and required intensities of preference on a subset of actions, called reference actions. It constructs not only the preference relation in the considered set of actions, but it also gives information about intensities of preference for pairs of actions from this set for a given decision maker (DM). Distinguishing necessary and possible consequences of preference information on the considered set of actions, GRIP answers questions of robustness analysis. The proposed methodology can be seen as an extension of the UTA method based on ordinal regression. GRIP can also be compared to the AHP method, which requires pairwise comparison of all actions and criteria, and yields a priority ranking of actions. As for the preference information being used, GRIP can be compared, moreover, to the MACBETH method which also takes into account a preference order of actions and intensity of preference for pairs of actions. The preference information used in GRIP does not need, however, to be complete: the DM is asked to provide comparisons of only those pairs of reference actions on particular criteria for which his/her judgment is sufficiently certain. This is an important advantage comparing to methods which, instead, require comparison of all possible pairs of actions on all the considered criteria. Moreover, GRIP works with a set of general additive value functions compatible with the preference information, while other methods use a single and less general value function, such as the weighted-sum.     

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.     

Robust multi-criteria ranking with additive value models and holistic pair-wise preference statements

We consider a problem of ranking alternatives based on their deterministic performance evaluations on multiple criteria. We apply additive value theory and assume the Decision Maker’s (DM) preferences to be representable with general additive monotone value functions. The DM provides indirect preference information in form of pair-wise comparisons of reference alternatives, and we use this to derive the set of compatible value functions. Then, this set is analyzed to describe (1) the possible and necessary preference relations, (2) probabilities of the possible relations, (3) ranges of ranks the alternatives may obtain, and (4) the distributions of these ranks. Our work combines previous results from Robust Ordinal Regression, Extreme Ranking Analysis and Stochastic Multicriteria Acceptability Analysis under a unified decision support framework. We show how the four different results complement each other, discuss extensions of the main proposal, and demonstrate practical use of the approach by considering a problem of ranking 20 European countries in terms of 4 criteria reflecting the quality of their universities.     

As simple as possible but not simpler in Multiple Criteria Decision Aiding: the robust-stochastic level dependent Choquet integral approach

The level dependent Choquet integral has been proposed to handle decision making problems in which the importance and the interaction of criteria may depend on the level of the alternatives’ evaluations. This integral is based on a level dependent capacity, which is a family of single capacities associated to each level of evaluation for the considered criteria. We present two possible formulations of the level dependent capacity where importance and interaction of criteria are constant inside each one of the subintervals in which the interval of evaluations for considered criteria is split or vary with continuity inside the whole interval of evaluations. Since, in general, there is not only one but many level dependent capacities compatible with the preference information provided by the Decision Maker, we propose to take into account all of them by using the Robust Ordinal Regression (ROR) and the Stochastic Multicriteria Acceptability Analysis (SMAA). On one hand, ROR defines a necessary preference relation (if an alternative a is at least as good as an alternative b for all compatible level dependent capacities), and a possible preference relation (if a is at least as good as b for at least one compatible level dependent capacity). On the other hand, considering a random sampling of compatible level dependent capacities, SMAA gives the probability that each alternative reaches a certain ranking position as well as the probability that an alternative is preferred to another. A real-world decision problem on rankings of universities is provided to illustrate the proposed methodology.     

A visual interactive approach for multiple criteria decision making with monotone utility functions

We consider the problem of choosing the best of a set of alternatives where each alternative is evaluated on multiple criteria. We develop a visual interactive approach assuming that the decision maker (DM) has a general monotone utility function. The approach partitions the criteria space into nonoverlapping cells. The DM uses various graphical aids to move between cells and to further manipulate selected cells with the goal of creating cells that have ideal points less preferred than an alternative. When the DM identifies such cells, all alternatives in those cells are eliminated from further consideration. The DM may also compare pairs of alternatives. The approach terminates with the most preferred alternative of the DM.     

Families of relations modelling preferences under incomplete information

Let us consider a preferential information of type preference–indifference–incomparability (P, I, J), with additional information about differences in attractiveness between pairs of alternatives. The present paper offers a theoretical framework for the study of the “level of constraint” of this kind of partial preferential information. It suggests a number of structures as potential models being less demanding than the classical one in which differences in utilities can be used to represent the comparison of differences in attractiveness. The models are characterized in the more general context of families of non-complete preference structures, according to two different perspectives (called “semantico-numerical” and “matrix”). Both perspectives open the door to further practical applications connected with elicitation of the preferences of a decision maker.     

Dealing with interaction between bipolar multiple criteria preferences in PROMETHEE methods

In this paper we extend the PROMETHEE methods to the case of interacting criteria on a bipolar scale, introducing the bipolar PROMETHEE method based on the bipolar Choquet integral. In order to elicit parameters compatible with preference information provided by the Decision Maker (DM), we propose to apply the Robust Ordinal Regression (ROR). ROR takes into account simultaneously all the sets of parameters compatible with the preference information provided by the DM considering a necessary and a possible preference relation.     

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.     

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.     

Kernel-based learning methods for preference aggregation

The mathematical representation of human preferences has been a subject of study for researchers in different fields. In multi-criteria decision making (MCDM) and fuzzy modeling, preference models are typically constructed by interacting with the human decision maker (DM). However, it is known that a DM often has difficulties to specify precise values for certain parameters of the model. He/she instead feels more comfortable to give holistic judgements for some of the alternatives. Inference and elicitation procedures then assist the DM to find a satisfactory model and to assess unjudged alternatives. In a related but more statistical way, machine learning algorithms can also infer preference models with similar setups and purposes, but here less interaction with the DM is required/allowed. In this article we discuss the main differences between both types of inference and, in particular, we present a hybrid approach that combines the best of both worlds. This approach consists of a very general kernel-based framework for constructing and inferring preference models. Additive models, for which interpretability is preserved, and utility models can be considered as special cases. Besides generality, important benefits of this approach are its robustness to noise and good scalability. We show in detail how this framework can be utilized to aggregate single-criterion outranking relations, resulting in a flexible class of preference models for which domain knowledge can be specified by a DM.      

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.     

Solving multiple criteria problems by interactive decomposition

An interactive decomposition method is developed for solving the multiple criteria (MC) problem. Based on nonlinear programming duality theory, the MC problem is decomposed into a series of subproblems and relaxed master problems. Each subproblem is a bicriterion problem, and each relaxed master problem is a standard linear program. The prime objective of the decomposition is to simplify and facilitate the process of making preference assessments and tradeoffs across many conflicting objectives. Therefore, the decision-maker's preference function is not assumed to be known explicitly; rather, the decision maker is required to make only limited local preference assessments in the context of feasible and nondominated alternatives. Also, the preference assessments are of the form of ordinal paired comparisons, and in most of them only two criteria are allowed to change their values simultaneously, while the remaining (l–2) criteria are held fixed at certain levels.     

Ranking alternatives using fuzzy numbers

We investigate the problem of employing expert opinion to rank alternatives across a set of criteria. The experts use fuzzy numbers to express their preferences and we employ fuzzy arithmetic to compute an issue's fuzzy ranking. This leads to a partition of the alternatives into sets H₁, H₂,… where H₁ contains the highest ranked issues, H₂ has all the second highest ranked alternatives, etc. The total ranking process is shown to possess a number of important properties. An example is presented to illustrate the method.     

Interactive Polyhedral Outer Approximation (IPOA) strategy for general multiobjective optimization problems

We propose an interactive polyhedral outer approximation (IPOA) method to solve a broad class of multiobjective optimization problems (MOP) with, possibly, nonlinear and nondifferentiable objective and constraint functions, and with continuous or discrete decision variables. During the interactive optimization phase, the method progressively constructs a polyhedral approximation of the decision-maker’s (DM’s) unknown preference structure and a polyhedral outer-approximation of the feasible set of MOP. The piecewise linear approximation of the DM’s preferences also provides a mechanism for testing the consistency of the DM’s assessments and removing inconsistencies; it also allows post-optimality analysis. All the feasible trial solutions are non-dominated (efficient, or Pareto-optimal) so preference assessments are made in the context of non-dominated alternatives only. Upper and lower bounds on the yet unknown optimal value are produced at every iteration, allowing terminating the search prematurely at a good-enough solution and providing information about the closeness of this solution to the optimal solution. The IPOA method includes a preliminary phase in which a limited probe of the efficient set is conducted in order to find a good initial trial solution for the interactive phase. The computational requirements of the algorithm are relatively simple. The results of an extensive computational study are reported.     

A lexicographical compromise method for multiple criteria group decision problems with imprecise information

This paper addresses multiple criteria group decision making problems where each group member offers imprecise information on his/her preferences about the criteria. In particular we study the inclusion of this partial information in the decision problem when the individuals’ preferences do not provide a vector of common criteria weights and a compromise preference vector of weights has to be determined as part of the decision process in order to evaluate a finite set of alternatives. We present a method where the compromise is defined by the lexicographical minimization of the maximum disagreement between the value assigned to the alternatives by the group members and the evaluation induced by the compromise weights.     

Comparing influence theories in voting games under locally generated measures of dissatisfaction

The Isbell desirability relation (I), the Shapley?CShubik index (SS) and the Banzhaf?CColeman index (BC) are power theories that grasp the notion of individual influence in a yes?Cno voting rule. Also, a yes?Cno voting rule is often used as a tool for aggregating individual preferences over any given finite set of alternatives into a collective preference. In this second context, Diffo Lambo and Moulen (DM) have introduced a power relation which ranks the voters with respect to how ably they influence the collective preference. However, DM relies on the metric d that measures closeness between preference relations. Our concern in this work is: do I, SS, BC and DM agree when the same yes?Cno voting rule is the basis for collective decision making? We provide a concrete and intuitive class of metrics called locally generated (LG). We give a characterization of the LG metrics d for which I, SS, BC and DM agree on ranking the voters.