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

https://doi.org/10.1016/j.ejor.2007.08.013Get rights and content

Abstract

We present a new method, called UTAGMS, 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 AR  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 UTAGMS method is intended to be used interactively, with an increasing subset AR 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, UTAGMS 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 UTAGMS software. Some extensions of the method are also presented.

Introduction

We are considering a decision situation in which a finite and non-empty set of alternatives (actions) A = {a, b, c, d, …} is evaluated on a family of n criteria g1,  , gi,  , gn, with gi:AR for all i  G = {1,  , n}. We assume, without loss of generality, that the greater gi(a), a  A, the better alternative a on criterion gi, for all i  G. A decision maker (DM) is willing to rank the alternatives in A from the best to the worst, according to his/her preferences. The ranking can be complete or partial, depending on the preference information supplied by the DM and on the way of using this information. The family of criteria G is supposed to satisfy the following consistency conditions (see [25]):

  • exhaustivity – any two alternatives having the same evaluations on all criteria from G should be considered indifferent,

  • monotonicity – when comparing two alternatives, an improvement of one of them on at least one criterion from G should not deteriorate its comparison to the other alternative,

  • non-redundancy – deletion of any criterion from G will contradict one of the two above conditions.

Such a decision problem is called multiple criteria ranking problem. It is known that the only information coming out from the formulation of this problem is the weak dominance relation. Let us recall that the weak dominance relation is a partial preorder, i.e. it is a reflexive and transitive binary relation. According to the weak dominance relation, alternative a  A is preferred to alternative b  A if and only if gi(a)  gi(b) for all i  G, with at least one strict inequality; moreover, a is indifferent to b if and only if gi(a) = gi(b) for all i  G; finally, a is incomparable with b otherwise, i.e. if gi(a) > gi(b) for at least one criterion i  G and gj(a) < gj(b) for at least another criterion j  G. Since incomparability is very often the most frequent situation, the weak dominance relation is usually very poor.

In order to enrich the weak dominance relation, multiple criteria decision aiding (MCDA) helps in construction of an aggregation model on the base of preference information provided by the DM. Such an aggregation model is called preference model – it induces a preference structure in set A whose proper exploitation permits to work out a ranking proposed to the DM.

The preference information may be either direct or indirect, depending if it specifies directly values of some parameters used in the preference model (e.g. trade-off weights, aspiration levels, discrimination thresholds, etc.), or if it specifies some examples of holistic judgments from which compatible values of the preference model parameters are induced. Direct preference information is used in the traditional aggregation paradigm, according to which the aggregation model is first constructed and then applied on set A to rank the alternatives.

Indirect preference information is used in the disaggregation (or regression) paradigm, according to which the holistic preferences on a subset of alternatives AR  A are known first, and then a consistent aggregation model is inferred from this information to be applied on set A in order to rank the alternatives.

Presently, MCDA methods based on indirect preference information and the disaggregation paradigm are of increasing interest for they require relatively less cognitive effort from the DM. Indeed, the disaggregation paradigm is consistent with the “posterior rationality” postulated by March [19] and with the inductive learning used in artificial intelligence approaches (see [20]). Typical applications of this paradigm in MCDA are presented in [30], [23], [10], [13], [1], [22], [7], [8], [9].

Let Xi denote the evaluation scale of criterion gi, i  G. Consequently, X=i=1nXi is the evaluation space, and x = [x1,  , xn]  X denotes a profile in the evaluation space.

From a pragmatic point of view, it is reasonable to assume that Xi = [αi, βi], i.e. the evaluation scale on each criterion gi is bounded, such that αi < βi are the worst and the best (finite) evaluations, respectively. Thus, gi : A  Xi, i  G, therefore, each alternative a  A is associated with the profile [g1(a),  , gn(a)] in the evaluation space X. In consequence, A is obviously associated with a finite subset of X.

In this paper, we are considering the aggregation model in form of an additive value function U:XR, such that, for each x  X,U(x)=i=1nui(xi),where ui are non-decreasing marginal value functions, ui:XiR,i=1,,n.

To simplify notation, when considering any alternative a  A, we shall write U(a) instead of U(g1(a),  , gn(a)), and ui(a) instead of ui(gi(a)), even if U:XR and gi : A  Xi, i  G.

While the additive value function involves compensation between criteria and requires a rather strong assumption about their independence in the sense of preference [11], it is often used for its intuitive interpretation and relatively easy computation. The weighted-sum aggregation model, which is a particular case of the additive value function, is used even more frequently, in spite of its simplistic form (see e.g. [1], [26]).

We are using the additive aggregation model in the settings of the disaggregation paradigm, as it has been proposed in the UTA method (see [10]). In fact, our method generalizes the UTA method in three aspects:

  • it takes into account all additive value functions (1) compatible with indirect preference information, while UTA is using only one such function,

  • the marginal value functions of (1) are general non-decreasing functions, and not piecewise linear, as in UTA,

  • the DM’s ranking of reference alternatives does not need to be complete.

The preference information used by our method is provided in the form of a set of pairwise comparisons of some alternatives from a subset AR  A, called reference alternatives. The method is producing two rankings in the set of alternatives A, such that for any pair of alternatives a, b  A:

  • in the necessary ranking, a is ranked at least as good as b if and only if, U(a)  U(b) for all value functions compatible with the preference information,

  • in the possible ranking, a is ranked at least as good as b if and only if, U(a)  U(b) for at least one value function compatible with the preference information.

The necessary ranking can be considered as robust with respect to the indirect preference information. Such robustness of the necessary ranking refers to the fact that any pair of alternatives compares in the same way whatever the additive value function compatible with the indirect preference information. Indeed, when no indirect preference information is given, the necessary ranking boils down to the weak dominance relation, and the possible ranking is a complete relation. Every new pairwise comparison of reference alternatives, for which the dominance relation does not hold, is enriching the necessary ranking and it is impoverishing the possible ranking, so that they converge with the growth of the preference information.

Another appeal of such an approach stems from the fact that it gives space for interactivity with the DM. Presentation of the necessary ranking, resulting from an indirect preference information provided by the DM, is a good support for generating reactions from the DM. Namely, (s)he could wish to enrich the ranking or to contradict a part of it. This reaction can be integrated in the indirect preference information in the next iteration.

The organization of the paper is the following. In the next section, we will outline the principle of the ordinal regression via linear programming, as proposed in the original UTA method (see [10]). In Section 3, we give a brief overview of existing approaches to multiple criteria ranking using a set of additive value functions, and we provide motivations for our approach. The new UTAGMS method is presented in Section 4. Some extensions are considered in Section 5. Section 6 provides an illustrative example showing how the method can be applied in practice. The last section includes conclusions.

Section snippets

Ordinal regression via linear programming – principle of the UTA method

In the following, we recall the principle of the UTA method as presented recently in [29]. The indirect preference information is given in the form of a complete preorder ≿ on a subset of reference alternatives AR  A, called reference preorder, such that, for all a, b  AR:abais at least as good asb.This weak preference relation can be decomposed into its asymmetric and symmetric parts, as follows:

  • ab[abandnot(ba)] “a is preferred to b”,

  • ab[abandba]“a is indifferent to b”.

The reference

Existing approaches and motivations for a new method

Our work aims at generalizing the UTA method in order to consider the set of all value functions compatible with the indirect preference information rather than choosing a single value function within the set of compatible ones. The literature concerning MCDA methods involving a set of additive value functions can be viewed from three points of view:

  • The methods are designed for different problem statements (problematics, see [24]):

    • choice of the best alternative (e.g. [2], [14], [17], [5], [27]),

Presentation of the method

The new UTAGMS method is an ordinal regression method using a set of additive value functions U(a)=i=1nui(a) as a preference model. One of its characteristic features is that it takes into account the set of all value functions compatible with the preference information provided by the DM. Moreover, it considers general non-decreasing marginal value functions instead of piecewise linear only.

We suppose that the DM provides preference information in form of pairwise comparisons of reference

Specification of pairwise comparisons with gradual confidence levels

The UTAGMS method presented in the previous section is intended to support the DM in an interactive process. Indeed, defining a large set of pairwise comparisons of reference alternatives at once can be difficult for the DM. Therefore, one way to reduce the difficulty of this task would be to permit the DM an incremental specification of pairwise comparisons. This way of proceeding allows the DM to control the evolution of the necessary and possible weak preference relations.

Another way of

Illustrative example

In this section, we illustrate how a decision aiding process can be supported by the UTAGMS method. We consider the following hypothetical decision problem. AGRITEC is a medium size firm (350 persons approx.) producing some tools for agriculture. The CEO, Mr Becault, intends to double the production and multiply exports by 4 within 5 years. Therefore, he wants to hire a new international sales manager. A recruitment agency has interviewed 17 potential candidates which have been evaluated on 3

Conclusion

The new UTAGMS method presented in this paper is an ordinal regression method supporting multiple criteria ranking of alternatives; it is distinguished from previous methods of this kind by the following new features:

  • the method considers general additive value functions rather than piecewise linear ones,

  • the final rankings are defined using all value functions compatible with the provided preference information,

  • the method provides two final rankings: the necessary ranking identifies “sure”

References (32)

  • L.C. Dias et al.

    Additive aggregation with variable interdependent parameters: The VIP analysis software

    Journal of the Operational Research Society

    (2000)
  • J. Fodor et al.

    Fuzzy Preference Modelling and Multicriteria Decision Support

    (1994)
  • S. Greco et al.

    The use of rough sets and fuzzy sets in MCDM

  • S. Greco et al.

    Decision rule approach

  • R.L. Keeney et al.

    Decisions with Multiple Objectives: Preferences and Value Tradeoffs

    (1976)
  • C.W. Kirkwood et al.

    Ranking with partial information: A method and an application

    Operations Research

    (1985)
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