Three-way decisions with decision-theoretic rough sets in multiset-valued information tables

The goal of three-way decisions is to divide a universe into three pair-wise disjoint regions and to act on some or all of these regions using appropriate strategies. The decision theoretic rough set model, a typical three-way decision model, trisects a universe using a pair of thresholds computed from loss functions. Previous studies of decision-theoretic rough set models do not consider loss functions based on multiset values and are unable to deal with multiset-valued information tables. In this paper, two generalized decision-theoretic rough set models involving multiset-valued data are proposed, namely a multiset-decision-theoretic rough set model and a multiset-fuzzy-decision-theoretic rough set model. Two methods are introduced that compute expected costs from loss functions. The first method is based on the mutiset addition and a new normal-multiset multiplication. The second method is based on a new concept known as the p-length of finite normal multisets. These two methods offer different ways of building multiset-decision-theoretic rough set models. By integrating the multiset-decision-theoretic rough set model with the fuzzy decision-theoretic rough set model, a multiset-fuzzy-decision-theoretic rough set model is created, a model which considers fuzzy relations in multiset-valued information tables. An example that recommends different home options is given to illustrate these models. (C) 2018 Elsevier Inc. All rights reserved.