A two-universe model of three-way decision with ranking and reference tuple

Abstract The theory of three-way decision, introduced for the needs of explaining the three regions of rough sets, has developed into a more general theory of three regions in recent years. For different types of problems, we should have different types of intentions of trisecting, and only by considering the specific intentions of trisecting can we get the most accurate three regions. This is the starting point of this article. From a new perspective of trisecting, we propose two concepts on two universes, namely rankings of a set of attributes and reference tuples. These two concepts are combined together to express the original intention of trisecting in a new general meaning. At the same time, an evaluation of matching degree is proposed to formulate the trisecting. Based on the above two concepts and one evaluation method, we construct a two-universe model of three-way decision with concrete formulations, and show that the rough-set-based model proposed by Yan et al. is only equivalent to one of the eight cases of our model, with the eight cases corresponding to eight different types of intentions and hence to eight different types of problems. Therefore, the present paper extends classical rough-set-based models to a more general level on two universes. Two algorithms are provided to compute the three regions of our model, with the second one also computing the ordering of objects and hence the optimal ones.