New Consistency Properties for Preference Relations

Consistency of preference relations is associated with the study of the transitivity property. This is clearly related to the scales used for associating preferences values to judgements, and the conditions used to represent transitivity. Both issues, the scale to use and the consistency property, have been studied in the multiplicative model as well as in the fuzzy one. However, we make note the existence of a conflict between the consistency property for multiplicative preference relations and the scale proposed to provide a such preference relation. Obviously, the same problem exists when dealing with fuzzy preference relations. In order to overcome this problem, in this paper we analyze the properties to be verified by a function, T , in order to obtain the value of preference of the alternative xi over the alternative xk when we already have the values of the preference of xi over xj, and of xj over xk. As a consequence, we define T -additive transitivity property as the consistency property for fuzzy preference relations and T -multiplicative transitivity property for the case of multiplicative preference relations.