Traveling Salesman Problem

The idea behind TSP was conceived by Austrian mathematician Karl Menger in mid 1930s who invited the research community to consider a problem from the everyday life from a mathematical point of view. A traveling salesman has to visit exactly once each one of a list of m cities and then return to the home city. He knows the cost of traveling from any city i to any other city j. Thus, which is the tour of least possible cost the salesman can take? In this book the problem of finding algorithmic technique leading to good/optimal solutions for TSP (or for some other strictly related problems) is considered. TSP is a very attractive problem for the research community because it arises as a natural subproblem in many applications concerning the every day life. Indeed, each application, in which an optimal ordering of a number of items has to be chosen in a way that the total cost of a solution is determined by adding up the costs arising from two successively items, can be modelled as a TSP instance. Thus, studying TSP can never be considered as an abstract research with no real importance

An improved discrete bat algorithm for symmetric and asymmetric traveling salesman problems

Bat algorithm is a population metaheuristic proposed in 2010 which is based on the echolocation or bio-sonar characteristics of microbats. Since its first implementation, the bat algorithm has been used in a wide range of fields. In this paper, we present a discrete version of the bat algorithm to solve the well-known symmetric and asymmetric traveling salesman problems. In addition, we propose an improvement in the basic structure of the classic bat algorithm. To prove that our proposal is a promising approximation method, we have compared its performance in 37 instances with the results obtained by five different techniques: evolutionary simulated annealing, genetic algorithm, an island based distributed genetic algorithm, a discrete firefly algorithm and an imperialist competitive algorithm. In order to obtain fair and rigorous comparisons, we have conducted three different statistical tests along the paper: the Student's $t$-test, the Holm's test, and the Friedman test. We have also compared the convergence behaviour shown by our proposal with the ones shown by the evolutionary simulated annealing, and the discrete firefly algorithm. The experimentation carried out in this study has shown that the presented improved bat algorithm outperforms significantly all the other alternatives in most of the cases

An improved discrete bat algorithm for symmetric and asymmetric traveling salesman problems

Bat algorithm is a population metaheuristic proposed in 2010 which is based on the echolocation or bio-sonar characteristics of microbats. Since its first implementation, the bat algorithm has been used in a wide range of fields. In this paper, we present a discrete version of the bat algorithm to solve the well-known symmetric and asymmetric traveling salesman problems. In addition, we propose an improvement in the basic structure of the classic bat algorithm. To prove that our proposal is a promising approximation method, we have compared its performance in 37 instances with the results obtained by five different techniques: evolutionary simulated annealing, genetic algorithm, an island based distributed genetic algorithm, a discrete firefly algorithm and an imperialist competitive algorithm. In order to obtain fair and rigorous comparisons, we have conducted three different statistical tests along the paper: the Student's $t$-test, the Holm's test, and the Friedman test. We have also compared the convergence behaviour shown by our proposal with the ones shown by the evolutionary simulated annealing, and the discrete firefly algorithm. The experimentation carried out in this study has shown that the presented improved bat algorithm outperforms significantly all the other alternatives in most of the cases

Using 2-Opt based evolution strategy for travelling salesman problem

Harmony search algorithm that matches the (µ+ 1) evolution strategy, is a heuristic method simulated by the process of music improvisation. In this paper, a harmony search algorithm is directly used for the travelling salesman problem. Instead of conventional selection operators such as roulette wheel, the transformation of real number values of harmony search algorithm to order index of vertex representation and improvement of solutions are obtained by using the 2-Opt local search algorithm. Then, the obtained algorithm is tested on two different parameter groups of TSPLIB. The proposed method is compared with classical 2-Opt which randomly started at each step and best known solutions of test instances from TSPLIB. It is seen that the proposed algorithm offers valuable solutions

Traveling Salesman Problem

This book is a collection of current research in the application of evolutionary algorithms and other optimal algorithms to solving the TSP problem. It brings together researchers with applications in Artificial Immune Systems, Genetic Algorithms, Neural Networks and Differential Evolution Algorithm. Hybrid systems, like Fuzzy Maps, Chaotic Maps and Parallelized TSP are also presented. Most importantly, this book presents both theoretical as well as practical applications of TSP, which will be a vital tool for researchers and graduate entry students in the field of applied Mathematics, Computing Science and Engineering

Population-Based Optimization Algorithms for Solving the Travelling Salesman Problem

[Extract] Population based optimization algorithms are the techniques which are in the set of the nature based optimization algorithms. The creatures and natural systems which are working and developing in nature are one of the interesting and valuable sources of inspiration for designing and inventing new systems and algorithms in different fields of science and technology. Evolutionary Computation (Eiben& Smith, 2003), Neural Networks (Haykin, 99), Time Adaptive Self-Organizing Maps (Shah-Hosseini, 2006), Ant Systems (Dorigo & Stutzle, 2004), Particle Swarm Optimization (Eberhart & Kennedy, 1995), Simulated Annealing (Kirkpatrik, 1984), Bee Colony Optimization (Teodorovic et al., 2006) and DNA Computing (Adleman, 1994) are among the problem solving techniques inspired from observing nature. In this chapter population based optimization algorithms have been introduced. Some of these algorithms were mentioned above. Other algorithms are Intelligent Water Drops (IWD) algorithm (Shah-Hosseini, 2007), Artificial Immune Systems (AIS) (Dasgupta, 1999) and Electromagnetism-like Mechanisms (EM) (Birbil & Fang, 2003). In this chapter, every section briefly introduces one of these population based optimization algorithms and applies them for solving the TSP. Also, we try to note the important points of each algorithm and every point we contribute to these algorithms has been stated. Section nine shows experimental results based on the algorithms introduced in previous sections which are implemented to solve different problems of the TSP using well-known datasets

Finding the best tour for travelling salesman problem using artificial ecosystem optimization

This paper presents a new method based on the artificial ecosystem optimization (AEO) algorithm for finding the shortest tour of the travelling salesman problem (TSP). Wherein, AEO is a newly developed algorithm based on the idea of the energy flow of living organisms in the ecosystem consisting of production, consumption and decomposition mechanisms. In order to improve the efficiency of the AEO for the TSP problem, the 2-opt movement technique is equipped to enhance the quality of the solutions created by the AEO. The effectiveness of AEO for the TSP problem has been verified on four TSP instances consisting of the 14, 30, 48 and 52 cities. Based on the calculated results and the compared results with the previous methods, the proposed AEO method is one of the effective approaches for solving the TSP problem