Selection of a representative value function in robust multiple criteria sorting
Introduction
Real-world decision problems usually require analysis of pertinent factors that affect a final recommendation. Goals to be attained, impacts of considered actions, and their performances on multiple criteria representing conflicting viewpoints are these factors. Complexity of actual decision situations has led to the development of the field of multiple criteria decision aiding (MCDA), which offers a diversity of methods supporting the decision makers (DMs).
Generally, decision problems considered by decision analysts concern situations in which a finite set of actions is evaluated on a consistent family of m criteria. Taking into account the way actions are perceived during analysis, and the type of expected results, Roy distinguishes four major problematics: description, choice, ranking, and sorting [24]. In the first case, actions shall be described in terms of their performances on the criteria, and some major distinguishing features should be identified. In the following, two actions are compared and the results express relative judgments with the use of comparative notions. In the multiple criteria choice problem, the aim is to select a subset of the best actions, while in the multiple criteria ranking problem, actions are to be ranked from the best to the worst. Finally, in the multiple criteria sorting problem (MCSP, also called ordinal classification), a DM is willing to assign actions to homogeneous classes which are given in a preference order. The classes considered here have a semantic definition, and are pre-defined, which means that they, unlike clusters, do not result from the analysis. Sorting is among the most frequent real-world decision problems in such various fields as finance, marketing, project management, environmental protection, tourism or medicine [13], [32]. Banks and international lending institutions sort countries, companies, and individuals into different groups based on their reliability to repay debts. Firms classify portfolios with respect to inherent risk, assign personnel into appropriate teams according to their qualifications, and sort employees into classes associated with incentive packages. In tourism, one has star-based categories for classification of hotels, beaches, and restaurants.
Among many approaches that have been developed to deal with the decision problems, multi-attribute utility theory (MAUT, see [19]) seems to prevail. The simplest and most straightforward way to use the theory is based on indirect preference information and on the disaggregation–aggregation paradigm implemented for the first time in the UTA method [18]. Here, the DM provides some holistic judgments on a subset of actions which are called reference actions and play the role of a training set composed of decision examples. On the basis of this information, the parameters of a preference model are induced using a methodology called ordinal regression. Then, a consistent preference model is applied on the set of actions A to give a recommendation.
Usually, among many sets of parameters of a preference model representing the provided preference information, only one specific set is used to give a recommendation. As far as value functions are concerned as a preference model, the feasible polyhedron of all functions compatible with the stated indirect preference information can be, in general, quite large. This means that many value functions lead to accurate representation of the exemplary decisions. Usually, in such a case, the choice of a single representative value function was either arbitrary or left to the DM. The problem of choosing the “best” value function in the ordinal regression approach to ranking problems is an exception, however. In the UTA method [18], Jacquet-Lagrèze and Siskos suggested several algorithms for exploration of the set of admissible value functions and for testing the stability of ranking using post-optimality analysis. In result of exploration of the polyhedron of value functions one can observe marginal value functions attaining maximum or minimum weight. Moreover, branch and bound methods [29] and techniques based on the notion of the labyrinth in a graph, such as Tarry's method [5], are said to be useful for the analysis of the polyhedron around the optimum obtained with UTA. On the other hand, UTA STAR method [28] finds the “mean” additive value function through maximization of the sum of differences between marginal values of consecutive evaluations. Furthermore, Beuthe and Scanella proposed the UTAMP1 model which maximizes the minimum difference between the values of two consecutive actions in the reference ranking given by the DM [2]. They extended this approach in the UTAMP2 model, stressing not only the difference of values between actions, but also the difference between values at successive bounds [3]. In [27], Siskos introduced the notion of a system of additive utilities. Such a system contains 2m utilities obtained by solving the linear programs consisting in minimization and maximization of the weight of each of the m criteria in the admissible polyhedron. A recent meta-UTA technique, called ACUTA [4], computes the analytic center of a polyhedron of value functions compatible with the preference information. It takes advantage of non-linear programming, and selects a value function which is central by definition and uniquely defined. With respect to multiple criteria sorting problems, Doumpos and Zopounidis proposed three variants of the UTADIS method in order to achieve higher discriminating and predicting ability [30], [31]. The original objective of UTADIS was to estimate an additive value function and the value thresholds delimiting the classes that would classify the actions in their original classes with the minimum classification error. Its first variant (UTADIS I), apart from the classification error also incorporates the distances of the correctly classified actions from the value thresholds, which have to be maximized in order to achieve as sharp discrimination as possible. The second variant (UTADIS II) is based on a mixed-integer programming formulation in order to minimize the number of misclassifications instead of their magnitude. The third variant (UTADIS III) combines UTADIS I and II.
Recently, Greco, Mousseau, and SŁowiński proposed in [17] the so-called robust ordinal regression with the aim of taking into account complete sets of instances of a preference model compatible with the information given by the DM. This extension has been implemented originally in a method for multiple criteria ranking problems, called UTAGMS [14], further generalized in another method, called GRIP [9], and then developed by analogy in UTADISGMS [16] to deal with multiple criteria sorting problems. Those methods use the whole set of compatible general additive value functions as DM's preference model, and supply the DM with two kinds of results (respectively rankings or assignments): necessary and possible. The necessary results specify the most certain recommendations worked out on the basis of all compatible value functions considered simultaneously, while the possible results identify all possible recommendations which are made by at least one compatible value function. In [20], Köksalan and Özpeynirci proposed an interactive procedure to sort the actions. It is based on consideration of all additive value functions compatible with the assignment examples and determination of the range of possible classes to which the actions can be assigned. The considered marginal value functions are piecewise linear functions, and not just monotonic as in UTADISGMS. Then, during the solution process, the DM is occasionally required to place some additional reference actions into classes, that affects categorization of the remaining actions as well. Let us also mention a parent methodology proposed for multiple criteria sorting, based on consideration of a set of compatible outranking models instead of a set of compatible value functions [7], [22]. Results provided by all those approaches answer the robustness concerns, since they are in general “more robust” than a recommendation made by an arbitrarily chosen compatible preference model. However, in some decision-making situations, the DM or the decision analyst would like to see a representative preference model among all the compatible ones.
As far as the set of compatible value functions was considered as a preference model, the initial idea of selection of a representative value function has been presented in [8] for robust ordinal regression in ranking problems. In this paper, we introduce the concept of a representative value function in robust ordinal regression applied to multiple criteria sorting problems. Within an interactive procedure for selection of a representative value function, the DM is allowed to choose out of five pre-defined targets that could be attained by this function. The targets concern enhancement of differences between possible assignments of two actions, which is conditioned by the fulfillment of a corresponding binary relation. Those relations enable comparison of a pair of actions on the basis of comparison of subsets of classes to which they can be assigned by any compatible value function considered individually, or all consistent value functions considered simultaneously. In particular, the DM may wish to differentiate values of actions such that one action is always in a better class than the other, or it is always in a class not worse and for at least one value function in a class strictly better, and/or to equalize values of actions assigned by all compatible value functions to either the same class or to different classes, but without the clear advantage of one of the actions. (S)he is also allowed to specify the order of optimizing different targets, to indicate whether in a given iteration one should optimize a single target or a few of them, and to allow worsening of an already optimized target up to a small proportion in the following iterations. In this way, the identification of a representative value function consists in an iterative process during which one deals with incremented set of constraints which account for results from the previous iterations, and uses successive optimizations for potential tie-breaking in case there is more than one value function for which the primary targets are attained.
Consequently, the proposed concept of a representative value function builds on results of the robust ordinal regression, i.e. it makes use of necessary and possible results stemming from robust ordinal regression applied to multiple criteria sorting. The representativeness of a selected value function is understood in the sense of the robustness preoccupation. This function highlights the possible assignments which correspond to the most stable part of the robust sorting. Therefore, it is an original proposal, which is different from other concepts of representative value functions like “mean”, “central”, or “the most discriminant” value function.
Introducing the concept of a representative value function, we extend UTADISGMS in its capacity of explaining the final output in terms of a sorting model which can be displayed to the DM. This enables a synthetic representation of the result of the robust ordinal regression. For any user, the analysis of a single, representative value function is surely less abstract than that of the whole set of compatible value functions. In such a way, the DM can see a score of each action and easily assess relative importance of the criteria understood as a share of a given criterion in the comprehensive value.
Furthermore, we use a representative value function to drive an example-based sorting procedure. In the proposed methodology, outcomes of all compatible sorting models are represented by a single value function, which has a positive impact on the accuracy of the sorting decisions taken through this model, as they are not affected by non-uniqueness of sorting models. In this way, our approach can be viewed as an alternative and improvement to other MCDA methods which choose randomly one sorting model out of the compatible ones, or define the “best” model on the basis of a limited set of value functions compatible with the provided preference information. The presented proposal refers to “one for all, all for one” motto, i.e. the representative value function represents all compatible value functions, which also do contribute to its definition.
Moreover, results obtained with the use of a representative value function can be analyzed in the context of final outcomes of UTADISGMS. This is useful because, in general, they are more precise than possible assignments which can be very wide (even equal to the whole range of classes), and at the same time they are more general than necessary assignments which, on the other hand, can be empty.
The organization of the paper is the following. In the next section, we introduce definitions and notation that will be used along the paper. Section 3 recalls methodologies to multiple criteria sorting which have been used and extended in the proposed approach. The concept of a representative value function in robust multiple criteria sorting is introduced in Section 4. Further, we present alternative procedures for selection of a single value function in the context of multiple criteria sorting problems. In Section 5, we discuss results of numerical experiments with three case studies—they show how the presented method can be applied in practical decision support. The last section concludes the paper.
Section snippets
Concepts: definitions and notation
We shall use the following notation:
- •
A={a1,a2,…,ai,…,an}—a finite set of n actions to be assigned to classes; actions are described over a consistent family of m evaluation criteria.
- •
g1,g2,…,gj,…,gm—m evaluation criteria, for all .
The family of criteria G is supposed to satisfy the consistency conditions [25]:
- (i)
exhaustivity (any two actions described by the same evaluation vectors should be considered indifferent),
- (ii)
monotonicity (when comparing two actions, an improvement of one
- (i)
A reminder on example-based value driven sorting
In this paper, we are interested in sorting procedures that aim at assigning each action to one class or a set of contiguous classes. We focus on example-based value driven sorting procedures. They use a value function U to decide the assignments in such a way that if then a is assigned to a class not worse than b, and assume that classes are implicitly delimited by some assignment examples. This section recalls example-based value driven sorting procedures based on a single value
Selection of a representative value function
In this section, we introduce the concept of the representative value function in robust multiple criteria sorting. The presented approach adopts features of UTADISGMS as it 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 principle of a representative value function approach is “one for all, all for one”, i.e. on the one hand, one value function is
Numerical experiments
In this section, we report results of numerical experiments with three case studies. The first case study concerns an assessment of sales managers of a trading company into five preference ordered classes. We extend a basic analysis with a presentation of representative functions resulting from different priorities given to the targets by the DM, as well as single functions stemming from alternative procedures discussed in Section 4.5. The second case study deals with assignment of MBA programs
Conclusions
In this paper, we have introduced the concept of a representative value function in robust ordinal regression applied to multiple criteria sorting problems. In order to select a single value function representing the whole set of value functions compatible with the provided preference information, we proposed to explore results of the robust sorting method based on ordinal regression. The selected value function emphasizes the relations between actions that stem from all possible consequences
Acknowledgments
The second and the third authors wish to acknowledge financial support from the Polish Ministry of Science and Higher Education, Grant no. N519 314435.
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