Certain Concepts in Directed Rough Fuzzy Graphs and Application to Mergers of Companies

Abstract

A directed rough fuzzy graph (DRFG) is a unique and innovative hybrid model because it deals with more complex problems of uncertainty in the presence of incomplete data information or rough universe. A DRFG can be obtained from two given DRFGs by union, Cartesian product and composition. When we study operations for DRFGs with a large number of vertices, the degree of vertices in a DRFG presents a confusing picture. Therefore, a mechanism for determining the degree of vertices for DRFG operations is needed. The main objective of this study is to analyze and investigate the degree of vertices in DRFGs formed by certain operations, which will provide clear explanations of operations on DRFGs and their effects on vertex degrees with examples. In this paper, we find the degree of a vertex in DRFGs formed by these operations in terms of the degree of vertices in the given DRFGs in some special cases. We explain these operations with some examples. In addition, we provide an application to the corporate merger problem to test our approach and obtain an optimal result. We have developed two algorithms to elaborate the procedure for our application. Finally, we created a comparison table comparing our results for Algorithms 1 and 2 for the same enterprise merger network. KEYWORDSdirected rough fuzzy graph; indegree of vertex; outdegree of vertex; union; composition G raph theory is an extremely useful tool for solving combinatorial problems in various fields such as geometry, algebra, number theory, topology, operations research, optimization, computer science, engineering, and physical, biological, and social systems. The theory of graphs seems to be important in tackling real life problems. The study of graphs is a simple way to describe data containing the relationships between different elements. Point-to-point interconnection networks for parallel and distributed systems are usually modeled by directed graphs (or digraphs).A digraph is a graph whose edges have directions, and arrows on the edges are used to encode the directional information. In modern science and technology, there are complicated processes and phenomena for which complete information is not always available. For such situations, mathematical models are developed to deal with systems that contain elements of uncertainty. A large number of these models are based on an extension of classical set theory, namely fuzzy set (FS), rough sets. A mathematical framework for describing indeterminacy phenomena in real situations was proposed by Zadeh [1,2] . Fuzzy models are becoming increasingly useful because they aim to reduce the differences between the traditional