Fuzzy rough set over multi-universes and its application in decision making

This paper presents a general framework for the study of rough approximation of a fuzzy (crisp) concept in multi-universes environment, which is an extension of the original Pawlak rough set theory. The Pawlak rough set is mainly concerned with the approximation of objects confined on the same universe by an arbitrary binary relation. However, the objects (concept) may be related with three or more different universes in reality of the decision-making. This paper presents the rough set model over multi-universes, where the concept approximations are defined by using a multiple relation on the multi-universes. We firstly present the rough approximation of a fuzzy concept over multi-universes by using the multiple fuzzy relation defined on the multi-universes, i. e., the fuzzy rough set over multi-universes. Then, a number of important properties of fuzzy rough set over multi-universes are obtained. It is shown that some of the properties of the fuzzy rough set on the same universe are special instances of those of fuzzy rough set over multi-universes. Furthermore, several special rough set models over multi-universes are derived from the fuzzy rough set over multi-universes. Subsequently, we give a new approach to multiple attribute decision making (MADM) with the characteristic of uncertainty, incomplete and inaccurate available information based on fuzzy rough set over multi-universes by combing the idea of uncertainty risk decision making. The decision steps and the algorithm of the decision method are also given. The proposed approach can obtain an objectively decision result with the data information owned by the decision problem only. Finally, the validity of the decision methods is tested by a numerical example with the background of clinical medical diagnosis decision making. The main contribution of this paper is twofold. One is to present a new perspective of the universe-oriented extension for classical rough set and then establish a new generalized rough set model under the framework of multiple different universes, i. e., fuzzy rough set model over multi-universes. Another is to present an approach to multiple attribute decision making problems based on fuzzy rough set over multi-universes.