Abstract
The shortest path problem (SPP) is an optimization problem of determining a path between specified source vertex s and destination vertex t in a fuzzy network. Fuzzy logic can handle the uncertainties, associated with the information of any real life problem, where conventional mathematical models may fail to reveal proper result. In classical SPP, real numbers are used to represent the arc length of the network. However, the uncertainties related with the linguistic description of arc length in SPP are not properly represented by real number. We need to address two main matters in SPP with fuzzy arc lengths. The first matter is how to calculate the path length using fuzzy addition operation and the second matter is how to compare the two different path lengths denoted by fuzzy parameter. We use the graded mean integration technique of triangular fuzzy numbers to solve this two problems. A common heuristic algorithm to solve the SPP is the genetic algorithm. In this manuscript, we have introduced an algorithmic method based on genetic algorithm for determining the shortest path between a source vertex s and destination vertex t in a fuzzy graph with fuzzy arc lengths in SPP. A new crossover and mutation is introduced to solve this SPP. We also describe the QoS routing problem in a wireless ad hoc network.
Highlights
The shortest path problem (SPP), which concentrates on obtaining a shortest path between a specified starting node and other destination nodes, is one of the most important and fundamental combinatorial network optimization problems [6,8] in graph theory that has been appeared in several real life applications as a sub problem. fields including routing, transportation, supply chain management, communications and wireless network
The motivation of this study is to present an extension of fuzzy SPP (FSPP) which will be efficient but simple enough to use in real life scenarios
We present the SPP from a specified source vertex to destination nodes on a fuzzy graph in which a trapezoidal fuzzy number is assigned to each edge as its edge weight
Summary
The shortest path problem (SPP), which concentrates on obtaining a shortest path between a specified starting node and other destination nodes, is one of the most important and fundamental combinatorial network optimization problems [6,8] in graph theory that has been appeared in several real life applications as a sub problem. fields including routing, transportation, supply chain management, communications and wireless network. They have presented a fuzzy membership function of the shortest path by using a linear programming method They have proposed an algorithmic technique for determining the single most significant edge in a fuzzy graph. Many scientists have worked a lots to present the exact algorithmic method, mathematical technique, heuristic technique and metaheuristic techniques for this SPP in fuzzy environment Both mathematical technique and exact algorithmic technique are used to get the exact shortest path from a source node to a destination node of a graph within a very short time. To the best of our information, most of the researches done on FSPP with several different types of fuzzy numbers as the edge weights and there is no research work on the SPP based on genetic algorithm with fuzzy numbers as arc lengths in the literature till now. Begin: A initial population of size N which consists of chromosomes is created
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