We present variable precision generalized rough set approach to characterize incomplete decision tables. We show how to determine the discernibility threshold for a reflexive relational decision system in the variable precision generalized rough set model. We also point out some properties of positive regions and prove a statement of the necessary condition for weak consistency of an incomplete decision table. We present two examples to illustrate the results obtained in this paper.

In this paper, we provide a definition of α-fuzzified lower and upper approximations for fuzzy sets based on the α-cut of fuzzy binary relations. We show that the definition is a proper generalization of the previous one for approximations of crisp sets and compare it with an existing definition in the context of fuzzy tolerance relation.

The notions of approximation and definability in classical rough set theory and their generalizations have received much attention. In this paper, we study such generalizations from the perspective of neighborhood systems. We introduce four different types of definability, called interior definability, closure definability, interior-closure (IC) definability, and weak IC definability respectively. We also point out the relationship between IC definability and other types of definability for some special kinds of neighborhood systems. Several examples are presented to illustrate the concepts introduced in this paper.

In this paper, we present the connection between the concepts of Variable Precision Generalized Rough Set model (VPGRS-model) and Neighborhood Systems through binary relations. We provide characterizations of lower and upper approximations for VPGRS-model by introducing minimal neighborhood systems. Furthermore, we explore generalizations by investigating variable parameters which are limited by variable precision. We also prove some properties of lower and upper approximations for VPGRS-model.

Almost all practically encountered decision making problems can be treated as fuzzy decision making problems. In Bellman and Zadeh's seminal approach, fuzzy decision making problems are essentially reduced to the maximization of the aggregated objectives and constraints. Thus, the study of the properties of concavity before and after aggregation should be the a very useful approach for solving fuzzy decision making problems. In this paper, the concept of E-convex, which covers a wider class of sets and functions, is extended to fuzzy sets. Supp-E-concave and supp-E-quasiconcave fuzzy sets are first introduced. Then some aggregation rules for supp-E-concave and supp-E-quasiconcave fuzzy sets are given. Finally, some useful composition rules are developed. These aggregation and composition rules should form the basis for further exploration of fuzzy E convex sets in fuzzy decision-making