Parametric Fuzzy Distribution Sets

Fuzzy distribution set (FDS) is a fuzzy set defined on a finite domain subject to a sum of membership values equal to 1. Such fuzzy sets can model subjective probability and weight distributions. The paper gives a short introduction to the operations of complement, union, and intersection of fuzzy distributive sets. As a complement of FDS we use recently introduced an involutive negation of probability distributions. The operations of complement and intersection of FDS are constructed as an extension of these operations defined for fuzzy sets and based on t-conorms and t-norms. The paper introduces parametric FDS as a generalization of the main types of parametric membership functions of fuzzy sets: triangular, trapezoidal, Gaussian, and bell membership functions. We depict examples of such FDS, and examples of their union and intersection. Examples of union and intersection of FDS show that, unlike for fuzzy sets, the union of FDS can decrease membership values of resulting distribution, but an intersection can increase some membership values of resulting distributions. Introduced parametric FDS can be used as parametric probability distributions and weight distributions in multi-criteria, multi-person, and multi-alternative decision-making and probabilistic reasoning models.