A New Uncertainty Measure of Covering-Based Rough Interval-Valued Intuitionistic Fuzzy Sets

Since some existing uncertainty measurement of covering-based rough intuitionistic fuzzy sets (CRIFSs) are unreasonable in some cases, an extended uncertainty measure criterion of CRIFSs is proposed. The discussion of the monotonicity of the criterion is further refined and more in line with the actual problem description. Taken into account this a modified roughness method is introduced, which is an uncertainty measure of CRIFSs satisfying the extended criterion. Furthermore, the criterion is extended to covering-based rough interval-valued intuitionistic fuzzy sets (CRIVIFSs), and the modified roughness method is proposed to measure the uncertainty of CRIVIFSs. Finally, the example is presented to illustrate the application of the modified roughness to attribute reductions. These conclusions provide a theoretical basis for the rationality (or irrationality) of existing uncertainty measures and also promote the application of CRIVIFSs.