Decision SupportSelection of a representative value function in robust multiple criteria ranking and choice
Highlights
► Reverse engineering for selection of a single representative value function. ► Exploitation of the necessary, possible and extreme results of robust ordinal regression. ► Involvement of the decision maker in the interactive procedure. ► Comparison with other UTA-like procedures.
Introduction
The main purpose of Multiple Criteria Decision Aiding (MCDA) is to answer two basic questions. The first one is about providing an explanation of decisions in terms of the circumstances in which they were made. The other concerns giving a recommendation how to make a new decision under specific circumstances. MCDA is of essential help in structuring the decision problem, analyzing the preferences of the decision maker (DM), developing a comprehensive model of a decision situation, and its subsequent exploitation leading to a recommendation.
Roy distinguishes four major multiple criteria decision problems: choice, ranking, sorting, and description [20]. In this paper, we focus on the first two. Here, alternatives in set A are compared pairwise, and the results express relative judgments with the use of comparative notions. In ranking problems, alternatives are to be ordered from the best to the worst, whereas in choice problems, the aim is to select a small subset of the best alternatives, or from another perspective, to eliminate the greatest number of relatively bad alternatives. Ranking and choice are among the most frequent real-world decision problems. To provide some examples, let us recall power plant siting, highway planning, subway renovation planning, job evaluation, personnel selection, design of market penetration strategies, rankings of universities, study programs, or cities.
The most intuitive approach to deal with ranking and choice problems consists in assigning a score to each alternative. Such a score should reflect a comprehensive performance of the alternative on all criteria considered jointly. This idea underlies principles of multi-attribute utility theory (MAUT) [18]. Within MAUT the most important decision model is the additive value function. One of the methods designed for ranking problems that assumes such a model, is UTA [15], [23]. It adopts the preference disaggregation principle, and uses linear programming (LP) technique to infer additive value functions which make a faithful representation of the complete preorder on the set of so-called reference alternatives AR. However, when the preferences of the DM match the additive model, there is usually more than one compatible value function. Obviously, the rankings of alternatives from set A that can be obtained for any of these value functions, can vary up to a non-negligible degree. Therefore, the way of dealing with this indetermination is essential for the final recommendation, and different approaches have been proposed in the literature to address this problem.
Traditionally, if the set of value functions compatible with the stated preference information is not empty, the choice of a single value function is either left to the DM, or it is arbitrary. In the first case, the DM may benefit from the software which allows visual interactive modification of graphically presented marginal value functions within limits following from the solution of appropriate ordinal regression problems. Such user controlled modification of marginal value functions has been provided for the first time in UTA+ [19]. In the second case, one uses some predefined rules for selecting a single compatible value function, or for significantly reducing the set of compatible value functions. There are four main UTA-like techniques which employ such predefined rules in case of a non-empty set of compatible value functions. The post-optimality analysis of the traditional UTA method and one of its variants, UTASTAR [24], consists in averaging the extreme compatible solutions obtained from a sensitivity analysis conducted separately for the characteristic point corresponding to the best evaluation on each criterion. In this way, one selects a “mean” value function. Beuthe and Scannella proposed two algorithms, which implement the notion of centrality in the polyhedron of compatible value functions in a different manner [4]. In particular, UTAMP1 aims at accentuation of the difference between values of two consecutive alternatives in the reference ranking. Further, UTAMP2 apart from better identification of the preference relation between reference alternatives, emphasizes also the difference between values at successive characteristic points. The desired effect of the maximization of the minimal slope of each linear segment consists in selecting piecewise-linear marginal value functions which are as linear as possible. A very recent extension of this idea, called ACUTA [5], is based on the computation of the analytic center (AC) of a polyhedron. Precisely, it uses a nonlinear objective function which aims at maximizing the product of the slack variables, each corresponding to a single preference or monotonicity constraint (or equivalently, the sum of their logarithms). As a result, the selected value function is “central” and univocal.
Since any solution belonging to the convex set of solutions admitted by the constraints of the linear program provides a ranking identical with that given by the DM on reference alternatives, another reasonable way of dealing with this indetermination consists in applying all these functions, without selecting anyone of them. Even if these functions may differ significantly from one to another, it might be interesting to assess a preference relation on set A with respect to all of them. In this context, one may seek for robust conclusions that would be in agreement with all compatible instances of the preference model, and identify all possible recommendations which are consistent with at least one compatible value function. This idea underlies the principle of robust ordinal regression [14], which has been originally implemented in a method for multiple criteria ranking problems, called UTAGMS [13], and further generalized in another method, called GRIP [9]. These methods use the whole set of compatible general (not only piecewise-linear) additive value functions as DM’s preference model, and supply her/him with two kinds of results: necessary and possible. Whether for an ordered pair of alternatives there is necessary or possible weak preference depends on the truth of the weak preference relation for all or at least one compatible preference model, respectively. Both UTAGMS and GRIP allow the DM to provide incrementally the preference information by small pieces. This results in a progressive restriction of the set of compatible value functions, enrichment of the necessary relation, and impoverishment of the possible relation. In [16], one has extended the principle of robust ordinal regression for multiple criteria ranking by an analysis of extreme ranking results. Having considered complete rankings that follow the use of all compatible value functions, one can determine the best and the worst ranks taken by each alternative a ∈ A. In this way, one can assess the performance of a relative to all the remaining alternatives considered jointly.
In this paper, we introduce the concept of a representative value function for multiple criteria ranking and choice problems. We propose a new interactive UTA-like procedure, which aims at selecting a single value function representing the whole set of compatible value functions. This value function builds on results of the robust ordinal regression, i.e. it makes use of the necessary and possible preference relations stemming from the robust ordinal regression. Representativeness of the selected value function is understood in the sense of the robustness concern. The representative value function is expected to produce a robust recommendation with respect to the non-univocal preference model stemming from the input preference information. We are leaving the DM the freedom of assigning priorities to five targets which are supposed to be attained by the representative value function. They concern enhancement of differences between comprehensive values of two alternatives, which is conditioned by satisfying a specific binary relation. In particular, the DM may wish to emphasize the apparent advantage of some alternatives over the others, which is acknowledged by all compatible value functions, or reduce the ambiguity in the statement of such an advantage, if in the context of all rankings determined by the set of compatible value functions, the result of the comparison of a pair of alternatives is not univocal. Within an interactive procedure for selection of the representative value function, the DM may either wish that the targets are attained one after another, according to a given priority order, or that a compromise between the targets is attained according to some aggregation formula. Additionally, the DM may take advantage of a few extensions of the main procedure, like accounting for intensity of preference, indication of the desired character of marginal value functions, or the preface designed for multiple criteria choice problems. The latter requires involvement of the DM in specification of the conditions referring to the best and the worst ranks attained by each alternative, that the selected alternatives should satisfy. In this way, they could be perceived as the best in the context of the outcomes provided by the set of compatible value functions. Then, we emphasize their advantage with respect to all the remaining alternatives. Consequently, we are able to select a single value function which ranks all distinguished alternatives at the top positions in the ranking, in front of all remaining alternatives which are not considered by the DM as potential best options.
The comprehensive values assigned to considered alternatives by the representative value function can be used to rank the alternatives. From this perspective, such ranking can be considered as an output of an autonomous multiple criteria procedure. Moreover, the representative value function can play a significant role in helping the DM to understand the necessary and possible preference relations. Note that the first very general idea for selection of the representative function (called “the most representative” one) according to a pre-defined rule, i.e. without cooperation with the DM, has been presented in [8]. Furthermore, in [12], we considered selection of a representative value function for robust ordinal regression adapted to multiple criteria sorting. However, the selection procedure did not take into account any of the targets considered in this paper.
The organization of the paper is the following. In the next section, we introduce notation that will be used along the paper. In Section 3, we remind the principle of robust ordinal regression applied to multiple ranking problems. The concept of the representative value function in robust multiple criteria ranking and choice is introduced in Section 4. We also discuss alternative procedures for selection of a single value function, admitting the use of general value functions, and we compare them with the introduced concept of representativeness. Section 5 provides an illustrative example showing how the introduced approach can be applied in practice. This section is extended in the e-Appendix with presentation of results of numerical experiments with another two case studies. The last section concludes the paper, outlining also some possible ways of future development of the presented method.
Section snippets
Concepts: definitions and notation
We shall use the following notation:
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A = {a1, a2, … , ai, … , an} – a finite set of n alternatives described over m evaluation criteria.
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AR = {a∗, b∗, …} – a finite set of reference alternatives, on which the DM accepts to express preferences; usually, AR ⊆ A.
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G = {g1, g2, … , gj, … , gm} – a family of m evaluation criteria, for all j ∈ J = {1, 2, …, m}, which is supposed to satisfy the consistency conditions [20].
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Xj = {gj(ai), ai ∈ A} – the set of all different evaluations on gj, j ∈ J. We assume, without loss of generality,
Reminder on multiple criteria ranking with a set of value functions: overview of the UTAGMS method
The UTAGMS method [13] is a robust ordinal regression method that takes into account the set of all general additive value functions composed of monotonic non-linear marginal value functions compatible with the preference information provided by the DM. The procedure consists of three steps, which will be briefly recalled in this section. It starts with the preference elicitation process, followed by the statement of appropriate ordinal regression problems, and concludes with the calculation of
Selection of a representative value function
In this section, we introduce the concept of a representative value function in robust multiple criteria ranking and choice. We aim at selecting a single compatible value function which constitutes a synthetic representation of the outcomes of robust ordinal regression and extreme ranking analysis. To make this selection, we exploit the necessary, possible, and extreme results of applying all compatible value functions, and subsequently we enhance differences between comprehensive values of
Illustrative case study
In this section, we illustrate how a decision aiding process can be supported by our method of selection of the representative value function. We report results of numerical experiments with a case study concerning ranking countries according to their IT industry competitiveness. The data set concerning Asia–Pacific region for 2009 comes from the Economist [1].
This Section is extended in the e-Appendix. In particular, in the e-Appendix B, for the same case study we present single value
Conclusions
In this paper, we introduced the concept of a representative value function in robust ordinal regression applied to multiple criteria ranking and choice problems. Our proposal for selection of a representative value function refers to robust ranking results, i.e. to necessary and possible consequences of the provided preference information, as well as to extreme ranking results. This function emphasizes to the greatest possible extent the advantage of some alternatives over the others when all
Acknowledgments
The first and the third authors wish to acknowledge financial support from the Polish Ministry of Science and Higher Education, Grant No. N N519 441939. The authors thank two anonymous referees whose comments permitted to improve the previous version of the paper.
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