Searching for the dimension of valued preference relations

European Journal of Operational Research, 1992

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.

MICAI 2004: Advances in …, 2004

Fuzzy (valued) preference relations (FPR) give possibility to take into account the intensity of preference between alternatives. The refinement of crisp (non-valued) preference relations by replacing them with valued preference relations often transforms crisp preference relations with cycles into acyclic FPR. It gives possibility to make decisions in situations when crisp models do not work. Different models of rationality of strict FPR defined by the levels of transitivity or acyclicity of these relations are considered. The choice of the best alternatives based on given strict FPR is defined by a fuzzy choice function (FCF) ordering alternatives in given subset of alternatives. The relationships between rationality of strict FPR and rationality of FCF are studied. Several valued generalizations of crisp group decision-making procedures are proposed. As shown on examples of group decision-making in multiagent systems, taking into account the preference values gives possibility to avoid some problems typical for crisp procedures.

Journal of logic and …, 2002

The paper is a theoretical study of a generalization of the lexicographic rule for combining ordering relations. We define the concept of priority operator: a priority operator maps a family of relations to a single relation which represents their lexicographic combination according to a certain priority on the family of relations. We present four kinds of results.

Mathematical Social Sciences, 1995

We consider a class of relations which includes irreflexive preference relations and interdependent preferences. For this class, we obtain necessary and sufficient conditions for representation of the relation by two numerical functions in the sense of a ~ x if and only if u(a) < vex).

2007

We introduce two criteria for judging "goodness" of the result when combining preference relations in information systems: completeness and consistency. Completeness requires that the result must be the union of all preference relations, while consistency requires that the result must be an acyclic relation. In other words, completeness requires that the result contain all pairs appearing in the preference relations, and only those pairs; while consistency requires that for every pair (x, y) in the result, it must be able to decide which of x and y is preferred to the other. Obviously, when combining preference relations, there is little hope for the result to satisfy both requirements. In this paper, we classify the various methods for combining preference relations, based on the degree to which the result satisfies completeness and consistency. Our results hold independently of the nature of preference relations (quantitative or qualitative); and also independently of the preference elicitation method (i.e. whether the preference relations are obtained by the system using query-log analysis or whether the user states preferences explicitly). Moreover, we assume no constraints whatsoever on the preference relations themselves (such as transitivity, strict ordering and the like).

2015

We consider the problem of normalizing the priority vectors associated with fuzzy preference relations and we show that a widely used normalization procedure may lead to unsatisfactory results whenever additive consistency is involved. We give some examples from the literature and we propose an alternative normalization procedure which is compatible with additive consistency and leads to better results.

2015

summary:In decision processes some objects may not be comparable with respect to a preference relation, especially if several criteria are considered. To provide a model for such cases a poset valued preference relation is introduced as a fuzzy relation on a set of alternatives with membership values in a partially ordered set. We analyze its properties and prove the representation theorem in terms of particular order reversing involution on the co-domain poset. We prove that for every set of alternatives there is a poset valued preference whose cut relations are all relations on this domain. We also deal with particular transitivity of such preferences

Journal of Mathematical Economics, 2013

A classical approach to model a preference on a set A of alternatives uses a reflexive, transitive and complete binary relation, i.e. a total preorder. Since the axioms of a total preorder do not usually hold in many applications, preferences are often modeled by means of weaker binary relations, dropping either completeness (e.g. partial preorders) or transitivity (e.g. interval orders and semiorders). We introduce an alternative approach to preference modeling, which uses two binary relations -the necessary preference N and the possible preference P -to fulfill completeness and transitivity in a mixed form. Formally, a NaP-preference (necessary and possible preference) on A is a pair

In practical decision-making, it seems clear that if we hope to make an optimal or at least defensible decision, we must weigh our alternatives against each other and come to a principled judgment between them. In the formal literature of classical decision theory, it is taken as an indispensable axiom that cardinal rankings of alternatives be defined for all possible alternatives over which we might have to decide. Whether there are any items " beyond compare " is thus a crucial question for decision theorists to consider when constructing a formal framework. At the very least, it seems problematic to presuppose that no such incommensurability is possible on the grounds that it would make formalizing axioms for decision-making more difficult, or even intractable. With this in mind, I plan to argue in this paper that a formal notion of comparability can be introduced to the classical understanding of preference relations such that the question of comparability between alternatives can be taken non-trivially. Building on the work of Richard Bradley and Ruth Chang, I argue that the comparability relation should be understood to be transitive but not complete. I contend that this understanding of comparability within decision theory can explain both why we believe that some alternatives may be incommensurable, yet we are still able to make justified decisions despite incomplete preference relations. In Section I, I lay the groundwork for understanding the conceptual relationship between comparability and commensurability with respect to decision-making. In Section II, I will argue that Bradley's definition of the preference relation with comparability leads to absurdity and contradiction due to a small oversight, which I propose to remedy. Then,

In decision making, consistency in fuzzy preference relations is associated with the study of transitivity property. While using additive consistency property to complete incomplete preference relations, the preference values found may lie outside the interval [0, 1] or the resultant relation may itself be inconsistent. This paper proposes a method that avoids inconsistency and completes an incomplete preference relation using an upper bound condition. Additionally, the paper extends the upper bound condition for multiplicative reciprocal preference relations. The proposed methods ensure that if (n − 1) preference values are provided by an expert, such that they satisfy the upper bound condition, then the preference relation is completed such that the estimated values lie inside the unit interval [0, 1] in the case of preference relations and [1/9, 9] in the case of multiplicative preference relation. Moreover, the resultant preference relation obtained using the proposed method is transitive.