Real Time Multigraphs for Communication Networks: An Intuitionistic Fuzzy Mathematical Model

Abstract

Many problems of computer science, communication network, transportation systems, can be modeled into multigraphs (or graphs) and then can be solved. Nowadays, the networks are expanding very fast in huge volumes in terms of their nodes and the connecting links. For a given alive network, in many situations, its complete topology may not be always available to the communication systems at a given point of time because of the reason that few or many of its links (edges/arcs) may be temporarily disable owing to damage or attack or blockage upon them and of course they are under repair at that point of time. Such cases are now so frequent that it calls for rigorous attention of the researchers, in particular to those who are concerned with Quality of Service (QoS) while in a network. Even in most of the cases the cost parameters corresponding to its links are not crisp numbers, rather intuitionistic fuzzy numbers (or fuzzy numbers). Thus at any real time instant, the complete multigraph is not available but a submutigraph of it is available to the system for executing its communication or transportation activities. Under such circumstances, none of the existing algorithms on Shortest Path Problems (SPP) can work. In this study the authors propose a mathematical model for such types of multigraphs to be called by ‘Real Time Multigraphs’ (RT-multigraphs) in which all real time information (being updated every q quantum of time) are incorporated so that the communication/transportation system can be made very efficiently with optimal results. It is a kind of intuitionistic fuzzy mathematical model being the most generalized form of the crisp multigraphs. As a special case, RT-multigraphs reduce to the case of ‘RT-graphs’. Finally an intuitionistic fuzzy method is developed to solve the shortest path problem in a RT-Multigraph. As a special case the problem reduces to fuzzy shortest path problem in a RT-Multigraph.