Algorithms for some Fuzzy Network Problems using Ranking Function

Abstract

This thesis is devoted to different type of fuzzy networks which occur in real life problems. The analysis of these networks has been done from different angles. The main topics are Fuzzy minimal spanning tree problem, Fuzzy shortest path problem and Fuzzy maximal flow problem. The chapter-wise summary of the thesis is as follows: Chapter 1 is introductory in nature. This chapter includes basic definitions, operations and concepts used throughout the work. Chapter 2 presents a brief review of the work done in the area of fuzzy minimal spanning tree problem, fuzzy shortest path problem and fuzzy maximal flow problem. In Chapter 3 the fuzzy minimal spanning tree problem in a given connected graph is considered. It is assumed that the edge costs are not precisely known and they are specified as triangular fuzzy numbers. A new algorithm has been proposed to characterize the optimality of edges of the graph and to choose a spanning tree under fuzzy costs and results are discussed based on the present study, conclusions are drawn. In Chapter 4 shortest path problem in a given connected graph is considered. Shortest path problem where the approximate costs are known is one of the most studied problems in fuzzy sets and systems area. In this chapter, we have introduced an algorithm that assumes a ranking function for comparing the fuzzy numbers. In chapter 5 the problem of finding the maximum flow between a source and a destination node in a network with uncertainties in its capacities is considered and an algorithm based on the classical algorithm is proposed. This thesis is devoted to different type of fuzzy networks which occur in real life problems. The analysis of these networks has been done from different angles. The main topics are Fuzzy minimal spanning tree problem, Fuzzy shortest path problem and Fuzzy maximal flow problem. The chapter-wise summary of the thesis is as follows: Chapter 1 is introductory in nature. This chapter includes basic definitions, operations and concepts used throughout the work. Chapter 2 presents a brief review of the work done in the area of fuzzy minimal spanning tree problem, fuzzy shortest path problem and fuzzy maximal flow problem. In Chapter 3 the fuzzy minimal spanning tree problem in a given connected graph is considered. It is assumed that the edge costs are not precisely known and they are specified as triangular fuzzy numbers. A new algorithm has been proposed to characterize the optimality of edges of the graph and to choose a spanning tree under fuzzy costs and results are discussed based on the present study, conclusions are drawn. In Chapter 4 shortest path problem in a given connected graph is considered. Shortest path problem where the approximate costs are known is one of the most studied problems in fuzzy sets and systems area. In this chapter, we have introduced an algorithm that assumes a ranking function for comparing the fuzzy numbers. In chapter 5 the problem of finding the maximum flow between a source and a destination node in a network with uncertainties in its capacities is considered and an algorithm based on the classical algorithm is proposed. This thesis is devoted to different type of fuzzy networks which occur in real life problems. The analysis of these networks has been done from different angles. The main topics are Fuzzy minimal spanning tree problem, Fuzzy shortest path problem and Fuzzy maximal flow problem. The chapter-wise summary of the thesis is as follows: Chapter 1 is introductory in nature. This chapter includes basic definitions, operations and concepts used throughout the work. Chapter 2 presents a brief review of the work done in the area of fuzzy minimal spanning tree problem, fuzzy shortest path problem and fuzzy maximal flow problem. In Chapter 3 the fuzzy minimal spanning tree problem in a given connected graph is considered. It is assumed that the edge costs are not precisely known and they are specified as triangular fuzzy numbers. A new algorithm has been proposed to characterize the optimality of edges of the graph and to choose a spanning tree under fuzzy costs and results are discussed based on the present study, conclusions are drawn. In Chapter 4 shortest path problem in a given connected graph is considered. Shortest path problem where the approximate costs are known is one of the most studied problems in fuzzy sets and systems area. In this chapter, we have introduced an algorithm that assumes a ranking function for comparing the fuzzy numbers. In chapter 5 the problem of finding the maximum flow between a source and a destination node in a network with uncertainties in its capacities is considered and an algorithm based on the classical algorithm is proposed. This thesis is devoted to different type of fuzzy networks which occur in real life problems. The analysis of these networks has been done from different angles. The main topics are Fuzzy minimal spanning tree problem, Fuzzy shortest path problem and Fuzzy maximal flow problem. The chapter-wise summary of the thesis is as follows: Chapter 1 is introductory in nature. This chapter includes basic definitions, operations and concepts used throughout the work. Chapter 2 presents a brief review of the work done in the area of fuzzy minimal spanning tree problem, fuzzy shortest path problem and fuzzy maximal flow problem. In Chapter 3 the fuzzy minimal spanning tree problem in a given connected graph is considered. It is assumed that the edge costs are not precisely known and they are specified as triangular fuzzy numbers. A new algorithm has been proposed to characterize the optimality of edges of the graph and to choose a spanning tree under fuzzy costs and results are discussed based on the present study, conclusions are drawn. In Chapter 4 shortest path problem in a given connected graph is considered. Shortest path problem where the approximate costs are known is one of the most studied problems in fuzzy sets and systems area. In this chapter, we have introduced an algorithm that assumes a ranking function for comparing the fuzzy numbers. In chapter 5 the problem of finding the maximum flow between a source and a destination node in a network with uncertainties in its capacities is considered and an algorithm based on the classical algorithm is proposed.