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Traveling Salesman Problem TRAVELING SALESMAN PROBLEM, THEORY AND APPLICATIONS Edited by Donald Davendra Traveling Salesman Problem, Theory and Applications Edited by Donald Davendra Published by InTech Janeza Trdine 9, 51000 Rijeka, Croatia Copyright © 2010 InTech All chapters are Open Access articles distributed under the Creative Commons Non Commercial Share Alike Attribution 3.0 license, which permits to copy, distribute, transmit, and adapt the work in any medium, so long as the original work is properly cited. After this work has been published by InTech, authors have the right to republish it, in whole or part, in any publication of which they are the author, and to make other personal use of the work. Any republication, referencing or personal use of the work must explicitly identify the original source. Statements and opinions expressed in the chapters are these of the individual contributors and not necessarily those of the editors or publisher. No responsibility is accepted for the accuracy of information contained in the published articles. The publisher assumes no responsibility for any damage or injury to persons or property arising out of the use of any materials, instructions, methods or ideas contained in the book. Publishing Process Manager Ana Nikolic Technical Editor Teodora Smiljanic Cover Designer Martina Sirotic Image Copyright Alex Staroseltsev, 2010. Used under license from Shutterstock.com First published December, 2010 Printed in India A free online edition of this book is available at www.intechopen.com Additional hard copies can be obtained from [email protected] Traveling Salesman Problem, Theory and Applications, Edited by Donald Davendra p. cm. ISBN 978-953-307-426-9 free online editions of InTech Books and Journals can be found at www.intechopen.com Contents Preface IX Chapter 1 Traveling Salesman Problem: An Overview of Applications, Formulations, and Solution Approaches 1 Rajesh Matai, Surya Prakash Singh and Murari Lal Mittal Chapter 2 The Advantage of Intelligent Algorithms for TSP 25 Yuan-Bin MO Chapter 3 Privacy-Preserving Local Search for the Traveling Salesman Problem 41 Jun Sakuma and Shigenobu Kobayashi Chapter 4 Chaos Driven Evolutionary Algorithm for the Traveling Salesman Problem 55 Donald Davendra , Ivan Zelinka, Roman Senkerik and Magdalena Bialic-Davendra Chapter 5 A Fast Evolutionary Algorithm for Traveling Salesman Problem 71 Xuesong Yan, Qinghua Wu and Hui Li Chapter 6 Immune-Genetic Algorithm for Traveling Salesman Problem 81 Jingui Lu and Min Xie Chapter 7 The Method of Solving for Travelling Salesman Problem Using Genetic Algorithm with Immune Adjustment Mechanism 97 Hirotaka Itoh Chapter 8 A High Performance Immune Clonal Algorithm for Solving Large Scale TSP 113 Fang Liu, Yutao Qi, Jingjing Ma, Maoguo Gong, Ronghua Shang, Yangyang Li and Licheng Jiao VI Contents Chapter 9 A Multi-World Intelligent Genetic Algorithm to Optimize Delivery Problem with Interactive-Time 137 Yoshitaka Sakurai and Setsuo Tsuruta Chapter 10 An Effi cient Solving the Travelling Salesman Problem: Global Optimization of Neural Networks by Using Hybrid Method 155 Yong-Hyun Cho Chapter 11 Recurrent Neural Networks with the Soft ‘Winner Takes All’ Principle Applied to the Traveling Salesman Problem 177 Paulo Henrique Siqueira, Maria Teresinha Arns Steiner and Sérgio Scheer Chapter 12 A Study of Traveling Salesman Problem Using Fuzzy Self Organizing Map 197 Arindam Chaudhuri and Kajal De Chapter 13 Hybrid Metaheuristics Using Reinforcement Learning Applied to Salesman Traveling Problem 213 Francisco C. de Lima Junior, Adrião D. Doria Neto and Jorge Dantas de Melo Chapter 14 Predicting Parallel TSP Performance: A Computational Approach 237 Paula Fritzsche, Dolores Rexachs and Emilio Luque Chapter 15 Linear Programming Formulation of the Multi-Depot Multiple Traveling Salesman Problem with Differentiated Travel Costs 257 Moustapha Diaby Chapter 16 A Sociophysical Application of TSP: The Corporate Vote 283 Hugo Hernandez-Salda ´ na˜ Chapter 17 Some Special Traveling Salesman Problems with Applications in Health Economics 299 Liana Lups¸ a, Ioana Chiorean, Radu Lups¸ a and Luciana Neamt¸ iu Preface Computational complexity theory is a core branch of study in theoretical computing science and mathematics, which is generally concerned with classifying computational problems with their inherent diffi culties. One of the core open problems is the resolu- tion of P and NP problems. These are problems which are very important, however, for which no effi cient algorithm is known. The Traveling Salesman Problem (TSP) is one of these problems, which is generally regarded as the most intensively studied problem in computational mathematics. Assuming a traveling salesman has to visit a number of given cities, starting and end- ing at the same city. This tour, which represents the length of the travelled path, is the TSP formulation. As the number of cities increases, the determination of the optimal tour (in this case a Hamiltonian tour), becomes inexorably complex. A TSP decision problem is generally classifi ed as NP-Complete problem. One of the current and best-known approaches to solving TSP problems is with the application of Evolutionary algorithms. These algorithms are generally based on natu- rally occurring phenomena in nature, which are used to model computer algorithms. A number of such algorithms exists; namely, Artifi cial Immune System, Genetic Algo- rithm, Ant Colony Optimization, Particle Swarm Optimization and Self Organising Migrating Algorithm. Algorithms based on mathematical formulations such as Dif- ferential Evolution, Tabu Search and Scatt er Search have also been proven to be very robust. Evolutionary Algorithms generally work on a pool of solutions, where the underlying paradigm att empts to obtain the optimal solution. These problems are hence classifi ed as optimization problems. TSP, when resolved as an optimization problem, is classifi ed as a NP-Hard problem. This book is a collection of current research in the application of evolutionary algo- rithms and other optimal algorithms to solving the TSP problem. It brings together researchers with applications in Artifi cial Immune Systems, Genetic Algorithms, Neu- ral Networks and Diff erential 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 X Preface tool for researchers and graduate entry students in the fi eld of applied Mathematics, Computing Science and Engineering. Donald Davendra Faculty of Electrical Engineering and Computing Science Technical University of Ostrava Tr. 17. Listopadu 15, Ostrava Czech Republic [email protected] 1 Traveling Salesman Problem: An Overview of Applications, Formulations, and Solution Approaches Rajesh Matai1, Surya Prakash Singh2 and Murari Lal Mittal3 1Management Group, BITS-Pilani 2Department of Management Studies, Indian Institute of Technology Delhi, New Delhi 3Department of Mechanical Engineering, Malviya National Institute of Technology Jaipur, India 1. Introduction 1.1 Origin The traveling salesman problem (TSP) were studied in the 18th century by a mathematician from Ireland named Sir William Rowam Hamilton and by the British mathematician named Thomas Penyngton Kirkman. Detailed discussion about the work of Hamilton & Kirkman can be seen from the book titled Graph Theory (Biggs et al. 1976). It is believed that the general form of the TSP have been first studied by Kalr Menger in Vienna and Harvard. The problem was later promoted by Hassler, Whitney & Merrill at Princeton. A detailed dscription about the connection between Menger & Whitney, and the development of the TSP can be found in (Schrijver, 1960). 1.2 Definition Given a set of cities and the cost of travel (or distance) between each possible pairs, the TSP, is to find the best possible way of visiting all the cities and returning to the starting point that minimize the travel cost (or travel distance). 1.3 Complexity Given n is the number of cities to be visited, the total number of possible routes covering all cities can be given as a set of feasible solutions of the TSP and is given as (n-1)!/2. 1.4 Classification Broadly, the TSP is classified as symmetric travelling salesman problem (sTSP), asymmetric travelling salesman problem (aTSP), and multi travelling salesman problem (mTSP). This section presents description about these three widely studied TSP. sTSP: Let Vv= { 1 ,......, vn} be a set of cities, ArsrsV=∈{( ,:,) } be the edge set, and ddrs= sr be a cost measure associated with edge (rs, )∈ A. The sTSP is the problem of finding a minimal length closed tour that visits each city once. In this case cities vVi ∈ are given by their coordinates (xii,y ) and drs is the Euclidean distance between r and s then we have an Euclidean TSP. 2 Traveling Salesman Problem, Theory and Applications aTSP: If ddrs≠ sr for at least one (rs, ) then the TSP becomes an aTSP. mTSP: The mTSP is defined as: In a given set of nodes, let there are m salesmen located at a single depot node. The remaining nodes (cities) that are to be visited are intermediate nodes. Then, the mTSP consists of finding tours for all m salesmen, who all start and end at the depot, such that each intermediate node is visited exactly once and the total cost of visiting all nodes is minimized. The cost metric can be defined in terms of distance, time, etc. Possible variations of the problem are as follows: Single vs. multiple depots: In the single depot, all salesmen finish their tours at a single point while in multiple depots the salesmen can either return to their initial depot or can return to any depot keeping the initial number of salesmen at each depot remains the same after the travel. Number of salesmen: The number of salesman in the problem can be fixed or a bounded variable. Cost: When the number of salesmen is not fixed, then each salesman usually has an associated fixed cost incurring whenever this salesman is used. In this case, the minimizing the requirements of salesman also becomes an objective.
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