Rough Intuitionistic Fuzzy Sets And Its Uncertainty Measure

Rough set theory and intuitionistic fuzzy set theory are two effective methods for dealing with uncertainty. This paper combines the ideas of the two theories, and studies the uncertainty measures of rough intuitionistic fuzzy set and rough interval-valued intuitionistic fuzzy set under an equivalent relation and a covering, respectively, which will further enriches the rough intuitionistic fuzzy set theory. The main work is presented as follows:(1) By introducing the concept of boundary areas in rough intuitionistic fuzzy set under equivalent relation, we solve the problems that the intersection(or union) of the upper(or lower) approximations of two sets is unequal to the upper(or lower) approximation of the intersection(or union) of two sets. Then, the concepts of roughness and rough entropy are introduced into the two models and their uncertainly measure is discussed. Furthermore, we prove that the roughness(rough entropy)of a rough intuitionistic fuzzy set model and rough interval-valued intuitionistic fuzzy set are monotonously decreasing with the increasing of its uncertainty.(2) The uncertainty measure of two kinds of covering rough intuitionistic fuzzy sets is studied. For the first type of covering based rough intuitionistic fuzzy sets, the lower and upper approximations are described through the intersection of minimal description. Based on the analysis the definition of roughness in this model, we give some improvements for the existing roughness in reference [16]. Moreover, we point out that the third type of covering based rough intuitionistic fuzzy sets does not rely on the parameter set manually, which is just described through a confidence. The concepts of roughness and rough entropy are introduced into the two model and their uncertainly measure is discussed. Then, the uncertainly measure of the covering rough interval-valued intuitionistic fuzzy set is discussed and an example illustrate the efficiency of the proposed roughness entropy.