Theory and Methodology
A multiple criteria ranking procedure based on distance between partial preorders

https://doi.org/10.1016/S0377-2217(00)00184-3Get rights and content

Abstract

In this paper, a multiple criteria ranking procedure based on distance between partial preorders is proposed. This method consists of two phases. In the first phase, the decision maker is asked to rank alternatives with a preorder (complete or partial) for each criterion and provide complete or partial linear information about the relative importance (weights) of the criteria. In the second phase, we introduce a distance procedure to aggregate the above individual rankings into a global ranking (a partial preorder). An algorithm for the aggregation procedure is proposed, followed by a numerical illustration.

Introduction

A multiple criteria decision making (MCDM) method usually consists of two main phases: (1) construction and information input, and (2) aggregation and exploitation (Guitouni and Martel, 1998; Pirlot, 1994). In the first phase, the decision maker is asked to determine the alternatives and criteria and provide the performance information of alternatives with respect to each criterion according to the specific situation. The types of performance information depend on the decision situation and the decision maker's capability and willingness. They can be cardinal, ordinal or mixed. Some multiple criteria decision methods such as Weighted Sum, TOPSIS (Hwang and Yoon, 1981), SMART (Edwards, 1977), and MAUT (Keeney and Raiffa, 1976) need cardinal performance information and they start with a decision matrix (Hwang and Yoon, 1981). Other multiple criteria decision methods such as QUALIFLEX (Paelinck, 1978), ORESTE (Roubens, 1982), the Bernardo model (Bernardo, 1976), and the Cook and Kress model (Cook and Kress, 1991) deal with ordinal performance information. It is generally thought that ordinal performance information is less demanding on the decision maker than cardinal information, because the latter require accurate evaluation of the performances of the alternatives on the given criteria, which is usually inaccurate, unreliable or even unavailable, especially under an uncertain environment (Roy, 1991).

The relative importance (weights) of criteria is another important inter-criteria information in the decision process. The weights can be provided by the decision maker based on some previous experience or obtained by some elaborate approach such as Saaty's method (Saaty, 1980), the centroid method (Solymosi and Dombi, 1986) and Simos' method (Simos, 1990). When there is insufficient information about the weights from the decision maker, some decision techniques with partial weighting information can be incorporated in the decision analysis. See, for example, Carrizosa et al. (1995) and Athanassopoulos and Podinovski (1997). In the second phase (aggregation and exploitation), the decision analyst will employ some procedure to aggregate the individual performances of the alternatives into an overall (global) performance and make recommendations to the decision maker. The results of aggregation usually fall into two main categories: cardinal and ordinal results. The procedures using a utility (value) function in aggregation always assign a numerical value to each alternative and the ranking of alternatives is straightforward: it is always a complete preorder of alternatives without incomparability. These methods are the so-called single synthesizing criterion approach (Guitouni and Martel, 1998). On the other hand, some MCDM methods based on outranking relations (Bouyssou, 1996) such as some of the ELECTRE methods (Roy, 1978) and the PROMETHEE methods (Brans et al., 1986) dedicate to producing ordinal results (complete or partial ranking of the alternatives). See Guitouni and Martel (1998) for a complete description of relevant features.

In this paper, we propose a multiple criteria ranking procedure based on distance between partial preorders. The method consists of two phases. In the first phase, the decision maker provides: (1) his ordinal preference information about alternatives on each criterion, and (2) complete or partial linear information about the relative importance (weights) of the criteria. The ordinal information can be in the form of complete or partial preorders. Partial preorder (accepting incomparability) is an important requirement in our model since in some circumstances this is the only form of input information available from the decision maker. In the second phase, an aggregation procedure based on distance between two preorders is proposed. In this procedure, we calculate the dominated index and the dominating index of each alternative, respectively, yielding two complete preorders P and P of the alternatives. The two complete preorders are then combined (by their intersection) into a final partial preorder P of the alternatives. This final ranking can be further analyzed and exploited by the decision maker before a decision is made.

The paper is organized as follows. In Section 2, we briefly review the basic concepts of the partial preference structure introduced, the distance between preorders (Roy and Slowinski, 1993; Ben Khélifa and Martel, 1998). Then the aggregation procedure is proposed. An algorithm for the aggregation is presented in Section 3 followed by a numerical example in Section 4. The algorithm is extended in Section 5 to handle partial linear information about the weights and the algorithm is illustrated by a numerical example.

Section snippets

Distance between preorders

Distance as a measure between two preorders has long been used in some works in group (collective) decision making models (Bogart, 1973, Bogart, 1975, Cook et al., 1986, Kemeny and Snell, 1962) and was used as a measure of closeness between two preorders (Roy and Slowinski, 1993; Roy et al., 1992). However, there has been no attempt to use distance between preorders in the MCDM aggregation approaches. In this paper, we use the distance measure introduced by Roy and Slowinski (1993) and improved

The aggregation algorithm for given weights

Now suppose relation R holds between the alternatives Ai and Ak for a certain criterion. As indicated before, the performance of alternative Ai can be measured by two types of distances: (1) the distance d(≻,R), between ≻ and AiRAk (for which less is better), and (2) the distance d(≺,R) between ≺ and AiRAk (for which more is better). The overall performance of alternative Ai can then be assessed by summing these distances over all alternatives. We now propose an algorithm for computing the

Numerical illustration

Let X={a1,a2,a3,a4,a5,a6} be the set of alternatives and C={C1,C2,C3} be the set of criteria with equal weights. For each criterion, the decision maker provides a partial ranking of the alternatives as shown in Fig. 1. For example, in criterion C2,a3a1 means a3a1,{a2,a5} means a2a5, while a1 and a2 are incomparable. Without loss of generality, we can assume a in Table 1 to be any positive number. For the sake of convenience, we assume a=3. Thus, in the following computation, d(≻,≺)=6,d

Extended aggregation algorithm

In the previous sections, it was assumed the decision maker was able to provide exact numerical values for the weights wj describing the relative importance of the criteria. It may very well be the case, in practice, that the decision maker only has partial information about the relative importance of the criteria. We now extend the aggregation algorithm of Section 3 to the case when the available information is in the form of linear constraints on the relative weights of the criteria. Without

Conclusion

In this paper, we propose a multiple criteria ranking procedure to aggregate the individual partial preorders of the criteria. The aggregation is based on the distance measure between order relations and is carried out from two different points of view, which produces two types of indexes: the dominated index and the dominating index. The former measures the intensity of one alternative being dominated by others while the latter measures the intensity of one alternative dominating others. For

Acknowledgements

The authors wish to thank the anonymous referees for their useful comments on an earlier version of the manuscript.

References (25)

  • K.P. Bogart

    Preference structures I: Distance between transitive preference relations

    Journal of Mathematical Sociology

    (1973)
  • K.P. Bogart

    Preference structures II: Distance between asymmetric relations

    SIAM Journal of Applied Mathematics

    (1975)
  • Cited by (0)

    This work was supported in part by China Scholarship Council and by the Fonds pour la Formation de Chercheurs et l'Aide à la Recherche du Québec under Grant 99-ER-1570.

    View full text