Variable consistency and variable precision models for dominance-based fuzzy rough set analysis of possibilistic information systems

The dominance-based fuzzy rough set approach (DFRSA) is a theoretical framework that can deal with multi-criteria decision analysis of possibilistic information systems. While a set of comprehensive decision rules can be induced from a possibilistic information system by using DFRSA, generation of several intuitively justified rules is sometimes blocked by objects that only partially satisfy the antecedents of the rules. In this paper, we use the variable consistency models and variable precision models of DFRSA to cope with the problem. The models admit rules that are not satisfied by all objects. It is only required that the proportion of objects satisfying the rules must be above a threshold called a consistency level or a precision level. In the presented models, the proportion of objects is represented as a relative cardinality of a fuzzy set with respect to another fuzzy set. We investigate three types of models based on different definitions of fuzzy cardinalities including-counts, possibilistic cardinalities, and probabilistic cardinalities; and the consistency levels or precision levels corresponding to the three types of models are, respectively, scalars, fuzzy numbers, and random variables.