Uncertainty and reduction of variable precision multigranulation fuzzy rough sets based on three-way decisions

This paper studies reduction of a multigranulation fuzzy information system by using uncertainty measures based on variable precision multigranulation decision-theoretic fuzzy rough sets, avoiding the positive region and negative region changing to small ones. Firstly we review variable precision multigranulation fuzzy rough sets and the decision method based on three-way decisions. Then, a double parameter rough membership degree of a fuzzy set is constructed based on variable precision multigranulation decision-theoretic fuzzy rough sets. A double parameter uncertainty measure of a multigranulation decision system is also proposed. In order to keep the decision-makings of certain elements, which are in the positive region or negative region, unchanged, we use the double parameter uncertainty measure of a multigranulation decision system to reduce the consistent decision system. Finally, we propose a conditional uncertainty measure and discuss the decision-theoretic reduction of granulation set and conditional attribute set in inconsistent views.