The soft consensus model in the multidistance framework

In the context of the soft consensus model due to (Fedrizzi et al. in International Journal of Intelligent Systems 14:63–77, 1999) [27], (Fedrizzi et al. in New Mathematics and Natural Computation 3:219–237, 2007) [28], (Fedrizzi et al. in Preferences and Decisions: Models and Applications, Studies in Fuzziness and Soft Computing, Springer, Heidelberg, pp. 159–182, 2010) [30], we investigate the reformulation of the soft dissensus measure in relation with the notion of multidistance, recently introduced by Martín and Mayor (Information processing and management of uncertainty in knowledge-based systems. Theory and methods, Communications in Computer and Information Science, Springer, Heidelberg, pp. 703–711 2010) [43], Martín and Mayor (Fuzzy Sets and Systems 167:92–100, 2011) [44]. The concept of multidistance is an extension of the classical concept of binary distance, obtained by means of a generalization of the triangular inequality. The new soft dissensus measure introduced in this paper is a particular form of sum-based multidistance. This multidistance is constructed on the basis of a binary distance defined by means of a subadditive scaling function, whose role is that of emphasizing small distances and attenuating large distances in preferences. We present a detailed study of the subadditive scaling function, which is analogous but not equivalent to the one used in the traditional form of the soft consensus model.