Group decision-making analysis based on distance measures under rough environment

A rough set as a superset of a crisp set is a mathematical tool to cope with uncertainty using initial data without additional assumptions and pre-defined parameters. Using the technique of upper and lower approximations, rough models provide a complete description of the problem. This paper aims to introduce the notion of distance function, which is a metric, in rough graphs. We establish formulae of distance function, degree and radius of certain products of rough graphs in terms of initial given rough graphs. We discuss the concepts of weak isomorphism and isomorphism in rough graphs, and describe the properties of isomorphic rough graphs. We demonstrate the significance and importance of the distance function with an application to a decision making problem concerning organ trafficking networks. An application of rough matrices in steam valve systems is discussed, and we show how they can help to identify the potential failures and risk components during the power generation process. The results obtained under the rough model are compared with existing extensions of graphs and fuzzy based approaches.