Constrained three-way approximations of fuzzy sets: From the perspective of minimal distance

Abstract

Three-way approximations of fuzzy sets aim at abstracting fuzzy sets into three pair-wise disjoint categories which facilitate semantic-oriented interpretations and reduce computing burden. Shadowed sets are a schema of three-way approximations of fuzzy sets which are formed based on a specific optimization mechanism. Among different principles guiding the construction of shadowed sets, the criterion of minimum distance offers a new insight within the framework of three-way decision theory. In this paper, the essential mathematical properties of the objective function used as a criterion to construct three-way approximations of fuzzy sets based on the principle of minimal distance, as well the characteristics of the optimal solutions, are analyzed. It is demonstrated that this optimization objective function is continuous but nonconvex with respect to the optimized variables. The nonconvex property makes the solution difficult and different approximate region partitions are obtainable even under the same optimization model. Therefore, further criteria are required to select final partition thresholds and make the construction process well-defined. To address this limitation, the notion of constrained three-way approximations of fuzzy sets is proposed from the perspective of minimal distance. Moreover, a constructive algorithm is provided to obtain the proposed constrained three-way approximations rather than using a direct enumeration method, and its performance is illustrated by considering some typical fuzzy sets along with some data from UCI repository.