Short CommunicationA new distance measure including the weak preference relation: Application to the multiple criteria aggregation procedure for mixed evaluations
Introduction
An abundant literature exists addressing the problem of appropriate distance measures between two preorders. The context of much of this literature is group decision making where individual preorders have to be reconciled into a collective or compromised preorder. Kemeny and Snell (1962) were the first to use a distance-based model for this purpose, presenting a set of conditions that a distance measure must satisfy. Many authors, such as Cook and Seiford, 1978, Cook and Kress, 1985 amongst others, have proposed similar conditions to those of Kemeny and Snell (1962) to construct different distance measures between two total preorders (including the indifference and the preference relations). However, few studies have examined the case of partial preorders (including the incomparability relation). Bogart (1975) generalized the model of Kemeny and Snell (1962) to accommodate partial preorders but excluded the indifference relation. Cook et al., 1986a, Cook et al., 1986b included the indifference relation presenting a set of axioms showing the existence and uniqueness of a distance measure between two individual preorders. Yet they did not address the aggregation problem for determining a collective preorder. In a multicriteria decision-aiding context, the aggregation problem with partial preorders was addressed by constructing convex cones to partially order the set of alternatives (Dehnokhalaji, Korhonen, Köksalan, Nasrabadi, & Wallenius, 2011). In a multicriteria analysis on water supply systems, Roy and Slowinski (1993) were the first to introduce the idea of a distance measure between pairs of binary relations. Their approach was adapted by Ben Khélifa and Martel (2001) to tackle the aggregation problem and an algorithm proposed to determine a total collective preorder from partial individual preorders. Jabeur, Martel, and Ben Khélifa (2004) presented a new distance measure considered to improve these previous models. Indeed, they proposed a minimal set of conditions to construct a metric. Recently, they used the same set of conditions to assign new values to their distance measure (Jabeur & Martel, 2010). Other approaches to measure the distance between preference relations have been proposed but they do not include the incomparability relation (Meskanen and Nurmi, 2006, Nitzan, 1981).
We introduce a new distance measure between preorders, including the weak preference relation, extending the work of Jabeur et al. (2004). The weak preference relation (Q) is an intermediate relation between preference and indifference first introduced by Roy (1978) with the ELECTRE III method. It is meant to capture situations where distinguishing between preference and indifference is problematic because of ambiguity and/or uncertainty.
Section 2 presents the new distance measure D including the weak preference relation. It is used in the Multiple Criteria Aggregation Procedure (MCAP) for mixed evaluations (Ben Amor, Jabeur, & Martel, 2007) to aggregate n unicriterion (local) preorders into a multicriterion (global) preorder. The MCAP is extended here to include the weak preference relation to afford the decision maker greater modeling power to elucidate preferences. Several changes required for this purpose and an illustrative numerical example, are presented in Section 3. Conclusions and future work follow in Section 4.
Section snippets
Extended distance measure D
Let A be a finite set of objects (alternatives, options or actions in this context) and (ai, ak) an ordered pair of objects belonging to A. In order to set a preference between ai and aj, four binary relations are considered: the strict preference (P), the weak preference (Q), the indifference (I) and the incomparability (?). For the sake of convenience, we use ai P−1 ak for ak P ai and ai Q−1 ak for ak Q ai, P−1 and Q−1 stand for the inverse strict preference and the inverse weak preference
Multiple Criteria Aggregation Procedure (MCAP) for mixed evaluations including the weak preference relation
The MCAP for mixed evaluations (Ben Amor et al., 2007) is designed to deal with a large spectrum of mixed evaluation types including deterministic, fuzzy (linguistic), stochastic, possibilistic, evidential and missing evaluations. It leads to non-valued global (multicriterion) preference relations where the p.r.s. consists of the strict preference, the indifference and the incomparability relations. In fact, numerical values associated with global preference relations can be difficult to
Conclusion
We propose a new distance measure between two preorders where the weak preference relation is taken into account. The MCAP designed to handle mixed evaluations (stochastic, fuzzy, possibilistic and others) is adapted to lead to an enriched preference relational system including the weak preference relation. For this purpose, the new measure of distance is used to aggregate different unicriterion preorders into one multicriterion preorder. An adapted exploitation procedure is also proposed.
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