Diversified binary relation-based fuzzy multigranulation rough set over two universes and application to multiple attribute group decision making

•An arbitrary fuzzy relation over diversified attribute set.•A diversified binary relation-based fuzzy multigranulation rough set over two universes.•A detailed investigates of the theoretical aspect for the established model.•A method of multiple attribute group decision making with different evaluation index set. This article considers a kind of multiple attribute group decision making problem which the decision-makers have different evaluation index set over the attribute set for all candidate alternatives. We firstly transform the multiple attribute group decision making into a fuzzy multigranulation decision making problem under the framework of different universes. We then construct a diversified binary fuzzy relation over the evaluation attribute index set associated to different decision-makers. Furthermore, using the idea of multigranulation methodology over two different universes, we define the lower and upper approximations of any fuzzy decision making object based on the diversified binary fuzzy relation. Subsequently, we establish the diversified binary fuzzy relation-based optimistic and pessimistic fuzzy multigranulation rough set over two universes. At the same time, Several interesting and important properties and conclusions are presented for the defined two types multigranulation fuzzy rough set over two universes in detail. Meanwhile, we propose an approach to multiple attribute group decision making with different evaluation index set by using the diversified binary fuzzy relation-based fuzzy multigranulation rough set over two universes. At last, a detailed comparison to the traditional studies is given and a numerical example is tested to illustrate the idea and the process of the decision making given in the new model and method.