Analysis of chaotic maps applied to self-organizing maps for the Traveling Salesman Problem

Chaotic maps are an alternative for calculating pseudorandom numbers which have created an increased interest among researchers dealing with stochastic search and optimization algorithms in the recent past. This interest is based on promising results with respect to both the quality of the results as well as the running time of the optimization algorithms compared to the usually used standard pseudorandom number generators. In this paper we investigate the influence of nine different chaotic maps on the quality of the results obtained by a self-organizing map (SOM) which has been used to solve the Traveling Salesman Problem (TSP). The investigation is based on various sizes of both the problem instances as well as the number of iterations where all nine chaotic maps are compared against the pseudorandom number generation. As a result it is proven that chaotic maps are significantly better in several cases. Finally, possible reasons for both the superiority and inferiority of chaotic maps compared to pseudorandom number generation are analyzed and discussed.

[1]  J. M. Burgers,et al.  Mathematical Examples Illustrating Relations Occurring in the Theory of Turbulent Fluid Motion , 1995 .

[2]  Takuji Nishimura,et al.  Mersenne twister: a 623-dimensionally equidistributed uniform pseudo-random number generator , 1998, TOMC.

[3]  John Fulcher,et al.  Computational Intelligence: An Introduction , 2008, Computational Intelligence: A Compendium.

[4]  Thomas Hanne,et al.  Optimization of the picking sequence of an automated storage and retrieval system (AS/RS) , 2014, 2014 IEEE Congress on Evolutionary Computation (CEC).

[5]  Thomas Hanne,et al.  A genetic algorithm with an embedded Ikeda map applied to an order picking problem in a multi-aisle warehouse , 2014, 2014 IEEE Symposium on Computational Intelligence in Production and Logistics Systems (CIPLS).

[6]  Xin-She Yang,et al.  Efficiency Analysis of Swarm Intelligence and Randomization Techniques , 2012, 1303.6342.

[7]  Jörg Kopecz,et al.  Neuronale Netze: Grundlagen und Anwendungen , 1995 .

[8]  J. Yorke,et al.  Chaos: An Introduction to Dynamical Systems , 1997 .

[9]  Xin-She Yang,et al.  Nature-Inspired Metaheuristic Algorithms , 2008 .

[10]  Zhang Tao,et al.  A new chaotic PSO with dynamic inertia weight for economic dispatch problem , 2009, 2009 International Conference on Sustainable Power Generation and Supply.

[11]  Robert C. Hilborn,et al.  Chaos And Nonlinear Dynamics: An Introduction for Scientists and Engineers , 1994 .

[12]  Luigi Fortuna,et al.  Chaotic sequences to improve the performance of evolutionary algorithms , 2003, IEEE Trans. Evol. Comput..

[13]  Takao Enkawa,et al.  A self‐organizing neural network approach for multiple traveling salesman and vehicle routing problems , 1999 .

[14]  Oliver Kramer.  Überblick Computational Intelligence , 2009 .

[15]  Pierre L'Ecuyer,et al.  Testing random number generators , 1992, WSC '92.

[16]  Charles Herring,et al.  Random number generators are chaotic , 1989, CACM.

[17]  Hans Hahn,et al.  Ergebnisse eines mathematischen Kolloquiums , 1932 .

[18]  I. Osman,et al.  A neural network algorithm for the traveling salesman problem with backhauls , 2003 .

[19]  Thomas Hanne,et al.  The way to an open-source software for automated optimization and learning — OpenOpal , 2010, IEEE Congress on Evolutionary Computation.

[20]  Elmetwally M. Elabbasy,et al.  Bifurcation Analysis, Chaos and Control in the Burgers Mapping , 2007 .

[21]  Ying Wang,et al.  Chaotic differential evolution methods for dynamic economic dispatch with valve-point effects , 2011, Eng. Appl. Artif. Intell..

[22]  Xiaohui Yuan,et al.  Hydrothermal scheduling using chaotic hybrid differential evolution , 2008 .

[23]  Roman Senkerik,et al.  Chaos driven evolutionary algorithms for the task of PID control , 2010, Comput. Math. Appl..

[24]  Teuvo Kohonen,et al.  Self-organized formation of topologically correct feature maps , 2004, Biological Cybernetics.

[25]  Danijel Koržinek,et al.  Kohonen Self-Organizing Map for the Traveling Salesperson Problem , 2007 .

[26]  M. Hénon,et al.  A two-dimensional mapping with a strange attractor , 1976 .

[27]  Günter Daniel Rey,et al.  Neuronale Netze - Eine Einführung in die Grundlagen, Anwendungen und Datenauswertung , 2008 .

[28]  Robert C. Hilborn,et al.  Chaos and Nonlinear Dynamics , 2000 .

[29]  Marco Budinich,et al.  A Self-Organising Neural Network for the Travelling Salesman Problem that is Competitive with Simulated Annealing , 1994 .