A Graph Theory Approach for Optimizing the Circulation Problem in Networks

Abstract

In this thesis we used an approach is suggested to define and apply graph theory which is different form of the classical one in the literature, this approach is used to illustrate and represent different circulation problems in network. The thesis will show that the new definition is more appropriate to put the network circulation problem in more rigors formal expression which will make a computer based technique to solve this problem more adequate. These days, graph theory is one of the most popular and fertile branches in mathematics and computer science. One important reason for this renewed interest in graph theory is its applicability to many of the complex and wide-ranging problems of modern society in such diverse fields as economics, facility location, management science, marketing, energy modeling, transmission of information, and transportation planning to name a few. Quite often such problems can be modeled as a graph or network. In this context graph theory is used first and foremost as a tool for formulating problems and defining structural inter relationships. Once a problem is formulated in graph-theoretical language, it becomes relatively easy to comprehend it in its generality. The next step will, of course, be to exploring avenues to seek a solution to the problem. The field of graph theory has two different braches: the algebraic aspects & the optimization aspects (the area of network optimization, which is greatly advanced by the advent of the computer). The thesis has been organized in six sections. The first is introduction, the second is the new definition of graph theory, the third is the directed graph, the fourth flow in network, the five circulation problems the sixth applications with reseal directions. Keywords (graph theory, network, flow, circulation, optimization, short path, assignment)