Ranking methods for valued preference relations: A characterization of a method based on leaving and entering flows1

In this paper we study a particular method that builds a partial ranking on the basis of a valued preference relation. This method which is used in the MCDM method PROMETHEE I, is based on "leaving" and "entering" flows. We show that this method is characterized by a system of three independent axioms. IIntroduction Suppose that a number of decision alternatives are to be compared taking into account different points of view, e.g. several criteria or the opinion of several voters. As argued in Barrett et al. (1990) and Bouyssou (1990), a common practice in such situations is to associate with each ordered pair (a, b) of alternatives, a number indicating the strength or the credibility of the proposition "a is at least as good as b", e.g. the sum of the weights of the criteria favoring a or the percentage of voters declaring that a is preferred or indifferent to b. In this paper we study a particular method allowing to build a partial ranking, i.e. a reflexive and transitive binary (crisp) relation2, on A given such information. Since a partial ranking is not necessarily complete, the method considered in this paper will allow two alternatives to be declared incomparable. Though this may seem strange, it must not be forgotten that the available information may be very poor or conflictual. Declaring that a and b are incomparable thus means that it seems difficult to take, at least at this stage of the study, a definite position on the comparison of a and b. Let A be a finite set of objects called "alternatives" with at least three elements. We define a valued (binary) relation3 on A as a function R associating with each ordered pair of alternatives (a, b) ∈ A2 with a ≠ b an element of [0, 1]. A method ≥ building a partial ranking, or, for short, a partial ranking method, is a function assigning a partial ranking ≥(R) on A to any valued relation R on A. In this paper, we study a partial ranking method used in PROMETHEE I (see, e.g., Brans et 1 We wish to thank Marc Pirlot and Philippe Vincke for their helpful comments on earlier drafts of this text. 2 A (crisp) binary relation S on A is reflexive if a S a, for all a ∈ A. It is transitive if for all a, b, c ∈ A, a S b and b S c imply a S c. It is complete if for all a, b ∈ A, a S b or b S a. 3 From a technical point of view, the condition a ≠ b could be omitted from this definition at the cost of a minor modification of our axioms. However, since it is clear that the values R(a, a) are immaterial in order to rank the alternatives, we will use this definition throughout the paper.