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GENETIC ALGORITHM TO SOLVE TRAVELING SALESMAN PROBLEM (TSP) Oloruntoyin Sefiu Taiwo, Olukehinde Olutosin Mayowa & Kolapo Bukola Rukayat Department of Computer Science & Engineering Faculty of Engineering & Technology Ladoke Akintola University of Technology, Ogbomoso, Nigeria.ABSTRACT: This research investigated the application of genetic algorithms in solving the traveling salesmanproblem (TSP). Genetic algorithms are able to generate successively shorter feasible tours by using informationaccumulated in the form of a pheromone trail deposited on the edges of the TSP graph. Computer simulationsdemonstrate that the genetic algorithm is capable of generating good solutions to both symmetric and asymmetricinstances of the TSP.The method is an example, like simulated annealing, neural networks, and evolutionarycomputation, of the successful use of a natural metaphor to design an optimization algorithm. A study of the geneticalgorithms explains its performance and shows that it may be seen as a parallel variation of tabu search, with animplicit memory. Genetic algorithm is the most efficient in computational time but least efficient in memoryconsumption. In this Paper we present a Genetic Algorithm for solving the Travelling Salesman problem (TSP).Genetic Algorithm which is a very good local search algorithm is employed to solve the TSP by generating a presetnumber of random tours and then improving the population until a stop condition is satisfied and the bestchromosome which is a tour is returned as the solution. Analysis of the algorithmic parameters (Population,Mutation Rate and Cut Length) was done so as to know how to tune the algorithm for various problem instances.Keywords:Travelling Salesman Problem (TSP), Genetic Algorithms, Simulated Annealing, Symmetric TSP,Asymmetric TSP.I. INTRODUCTION case a good but not necessarily optimal solution is The traveling salesman problem (TSP) is a sufficient (Johnson & McGeoch, 2002).well-known and important combinatorial In this Research Work, genetic algorithm isoptimization problem. The goal is to find the shortest used to solve Travelling Salesman Problem. Genetictour that visits each city in a given list exactly once algorithm is a technique used for estimating computerand then returns to the starting city. In contrast to its models based on methods adapted from the field ofsimple definition, solving the TSP is difficult since it genetics in biology (Michael & Kurt, 2011). To useis a Negative-Positive (NP) complete problem this technique, one encodes possible model behaviors(Michael & Kurt, 2011). into genes". After each generation, the current Apart from its theoretical approach, the TSP models are rated and allowed to mate and breedhas many applications. Some typical applications of based on their fitness. In the process of mating, theTSP include vehicle routing, computer wiring, genes are exchanged, crossovers and mutations cancutting wallpaper and job sequencing. The main occur. The current population is discarded and itsapplication in statistics is combinatorial data analysis, offspring forms the next generation (Holland 1975).e.g., reordering rows and columns of data matrices or Also, Genetic Algorithm describes a variety ofidentifying clusters. The NP-completeness of the TSP modeling and optimization techniques. Typically, thealready makes it more time efficient for small-to- object being modeled is represented in a fashion thatmedium size TSP instances to rely on heuristics in is easy to modify automatically. Then a large number of candidate models are generated and tested against 1
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the current data. Each model is scored and the "best" used as a benchmark for many optimization methods.models are retained for the next generation (Whitley Even though the problem is computationally difficult,et al. 1991). These models are then randomly a large number of heuristics and exact methods areperturbed (as in asexual reproduction) and the known, so that some instances with tens of thousandsprocess is repeated until it converges. If the model is of cities can be solved.constructed so that they have "genes," the winnerscan "mate" to produce the next generation. The TSP has several applications even in its purest formulation, such as planning, logistics, and II. RELATED WORK the manufacture of microchips. Slightly modified, it The traveling salesman problem (TSP) appears as a sub-problem in many areas, such as(Lawler, et al. 1985), (Gutin & Punnen, 2002) is a DNA sequencing. In these applications, the conceptwell-known and important combinatorial city represents, for example, customers, solderingoptimization problem. The goal is to find the shortest points, or DNA fragments, and the concept distancetour that visits each city in a given list exactly once represents travelling times or cost, or a similarityand then returns to the starting city. Formally, the measure between DNA fragments. In manyTSP can be stated as follows. The distances between applications, additional constraints such as limitedn cities are stored in a distance matrix D with resources or time windows make the problemelements dij where i; j = 1… n and the diagonal considerably harder.elements dii are zero. A tour can be represented by a In the theory of computational complexity,cyclic permutation of {1, 2…n} where the decision version of the TSP (where, given arepresents the city that follows city i on the tour. The length L, the task is to decide whether any tour istraveling salesman problem is then the optimization shorter than L) belongs to the class of NP-completeproblem to find a permutation that minimizesthe problems. Thus, it is likely that the worst-caserunninglength of the tour denoted by time for any algorithm for the TSP increases exponentially with the number of cities. [1] III. SYMMETRIC TSP In symmetric travelling salesman the For this minimization task, the tour length of distance from A to B is equal to the distance from B(n - 1)! Permutation vectors have to be compared. to A. Many TSPs are symmetric. Given a set of nThis results in a problem which is very hard to solve nodes and costs associated with each pair of nodes,and in fact known to be NP complete (Johnson and find a closed tour of minimal total costs that containsPapadimitriou 1985). However, solving TSPs is an each node exactly once, the cost associated with theimportant part of applications in many areas node pairs { i, j } and { j, i } being equal, that is , forincluding vehicle routing, computer wiring, machine any two cities A and B, the distance from A to B issequencing and scheduling. the same as that from B to A. In this case, we will get exactly the same tour length if we reverse the order in The travelling salesman problem (TSP) is an which they are visited- so there is no need toNP-hard problem in combinatorial optimization distinguish between a tour and its reverse, and we canstudied in operations research and theoretical leave off the arrows on the tour diagram. (Syswerda,computer science. Given a list of cities and their 1991).pairwise distances, the task is to find the shortestpossible route that visits each city exactly once andreturns to the origin city. It is a special case of the IV. ASYMMETRIC TSPtravelling purchaser problem. In most cases, the distance between two nodes in the TSP network is the same in both The problem was first formulated as a directions-the special case where the distance from Amathematical problem in 1930 and is one of the most to B is not equal to the distance from B to A is calledintensively studied problems in optimization. It is Asymmetric TSP. An example of a practical 2
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application of an asymmetric TSP is route for GAs. However, because of its outstandingoptimization using street-level routing (asymmetric performance in optimization, GAs have been wronglydue to one- way streets, slip-roads and motorways). regarded as a function optimizer. In fact, there are(Syswerda, 1991). many ways to view genetic algorithms. Perhaps most users come to GAs looking for a problem solver, butThe Algorithm Consists Of the Following this is a restrictive view (Jong, 1993).Fundamental Steps GA was introduced as a computationalInitialization: Chromosomes are randomly created. analogy of adaptive systems. They are modelledAt this point it is very important that the population is loosely on the principles of the evolution via naturaldiverse otherwise the algorithm may not produce selection, employing a population of individuals thatgood solutions. undergo selection in the presence of variation-Evaluation: Each chromosome is rated how well the inducing operators such as mutation andchromosome solves the problem at hand. A fitness recombination (crossover). A fitness function is usedvalue is assigned to each chromosome. to evaluate individuals, and reproductive success varies with fitness (Grefenstette, 1987).Selection: Fittest chromosomes are selected for Genetic algorithms are algorithms developedpropagation into the future generation based on how from the concept of genetics. In genetics, it isfit they are. understood that very living being‟s structure was developed from a „data bank‟ of information (itsRecombination: Individual chromosomes and pairs gene, or genetic code) that controls the being‟s formof chromosomes are recombined and modified and and type. For example, a human being‟s genethen put back in the population. contains his genetic information, which includes his skin colour, his eye hue, his teeth structure and evenMethods through which Travelling salesman problem his brain capacity. The gene is so minute, it takes acan be solved include simulated annealing, Genetic very powerful microscope to view it, and its „coding‟algorithm to mention but few. is to complex; it took quite a while- and a whole brood of scientists years to decode it (Holland 1975). V. GENETIC ALGORITHMS Genetic Algorithms (GAs) are adaptive VI. SIMULATED ANNEALINGheuristic search algorithm premised on theevolutionary ideas of natural selection and genetic Simulated annealing (SA) is a random-(Holland 1975). The basic concept of GAs is search technique which exploits an analogy betweendesigned to simulate processes in natural system the way in which a metal cools and freezes into anecessary for evolution, specifically those that follow minimum energy crystalline structure (the annealingthe principles first laid down by Charles Darwin of process) and the search for a minimum in a moresurvival of the fittest. As such they represent an general system; it forms the basis of an optimizationintelligent exploitation of a random search within a technique for combinatorial and other problems.defined search space to solve a problem. Simulated annealing was developed in 1983 First pioneered by John Holland in the 60s, to deal with highly nonlinear problems. SAGenetic Algorithms has been widely studied, approaches the global maximization problemexperimented and applied in many fields in similarly to using a bouncing ball that can bounceengineering worlds. Not only does GAs provide over mountains from valley to valley. It begins at aalternative methods to solving problem, it high "temperature" which enables the ball to makeconsistently outperforms other traditional methods in very high bounces, which enables it to bounce overmost of the problems link. Many of the real world any mountain to access any valley, given enoughproblems involved finding optimal parameters, which bounces. As the temperature declines the ball cannotmight prove difficult for traditional methods but ideal 3
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bounce so high and it can also settle to becometrapped in relatively small ranges of valleys.Agenerating distribution generates possible valleys orstates to be explored. An acceptance distribution isalso defined, which depends on the differencebetween the function value of the present generatedvalley to be explored and the last saved lowest valley.The acceptance distribution decides probabilisticallywhether to stay in a new lower valley or to bounceout of it. All the generating and acceptancedistributions depend on the temperature. It has beenproved that by carefully controlling the rate ofcooling of the temperature, SA can find the globaloptimum. In 1983, Kirkpatrick and co-workersproposed a method of using a Metropolis Monte Figrure 3.1 program interfaceCarlo simulation to find the lowest energy (moststable) orientation of a system. Their method is based Graph Name: This is name of each graphupon the procedure used to make the strongest represented. It contains nodes and edges.possible glass. This procedure heats the glass to ahigh temperature so that the glass is a liquid and the Graph Description: This shows how each nodeatoms can move relatively freely. The temperature of (each city) in the graph are connected to each otherthe glass is slowly lowered so that at each i.e. which loaction is linked with another as show intemperature the atoms can move enough to begin the screen shot of the interface. This also shows theadopting the most stable orientation. If the glass is way or manner in which the locations are visited.cooled slowly enough, the atoms are able to “relax‟into the most stable orientation. This slow cooling Node Visited: This show the node that is visited atprocess is known as annealing, and so their method least once.known as Simulated Annealing. IX. ENVIRONMENT VII. MODELLING THE GENETIC ALGORITHMSThis section explains the design of genetic algorithmin order to imitate the collective behaviour of theTSP. VIII. INTERFACE The interface is designed by using the C#environment. All the elements of the environmentwere modelled using C#. Since these elements aresimple, the implementation becomes simple. Manyproperties and methods of C# are specific for theinterface.The interfaces include: Figure 3.3 interface with generated cities/nodes 4
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EXPLANATIONCritically looking at the environment, someparameters used need to be well understood indetermining a successful and accurate tour.CITIES: The number of cities specified willdetermine the number of node to be generated on thegraph. The minimum and maximum number of citiesthat can be accomodated by this application are 5 and200 respectively.POPULATION SIZE: This specifies the maximumnumber of parent gene that can be combined in theprocess of mutation. The maximum number it canaccommodate is 100.GREEDY CROSS-OVER: This feature see to the Figure 3.4 interface for the plotted cities generatedmass selection of parent gene out the population size.It takes more parent to work upon when the feature is Considering the graph above and theactivated. foraging behaviour of genetic algorithm, it is easy to identify the similarity between these two problems.ITERATION: This is the number of time it uses to While genetic algorithm try to find the best andcompute the graph. shortest route between two places within an environment, a graph search algorithm tries to findPATH LENGTH: The is the total distance between the shortest path connecting two node within a graph.each node/cities. The main idea of the system proposed in this project is simply to put genetic algorithm on a graph to After the number of cities is known and observe the accuracy with time.written into the column provided, the “generate” keywill be press to generate the cities using nodes on the A graph‟s node is taken at different places wherescreen. Then the number of performance needed in genetic algorithm could stop during a travel and wethe computation of the tour i.e iteration is also written call them cities. The edges of the graph will representin the slot provided. The population size is also the route connecting cities. This virtual environmentincluded after which it should be decided whether to will be populated by individual genetic algorithm.mark of unmark the the greedy-crossover column. X. CONCLUSION The Genetic Algorithm is a very efficient and accurate optimization approach. Unfortunately it requires a large amount of memory and parallel system architecture for its complete implementation. The genetic algorithm was found to still be the best in solving combinatorial optimization problems (as it is general agreed by researcher around the world). We have successfully drawn out a model for the 5
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genetic algorithms, the design and the [13] Fogel, J., „Autonomous Automata‟. (1962). Ind. Res. 4: 14–19. implementation. It can be used successfully in large industries like Coca-Cola, Dangote Flour [14] Fogel, D., An Evolutionary Approach to the Traveling Mill to solve distribution problems, electronic Salesman Problem. Biological Cybernetics (1988).60: industries like Intel® for circuit board drilling, 139–144. order picking in a warehouse e.tc [15] Fogel, D., „A Parallel Processing Approach to a Multiple Traveling Salesman Problem Using Evolutionary Programming‟. Canter, L., (ed.) „Proceedings on the Fourth AnnualParallel Processing Symposium’, 318–326.Fullterton, (1990). CA. REFRENCES[1] Ackley, D. „A Connectionist Machine for Genetic Hill [16] Fogel, D., „Applying Evolutionary Programming to climbing‟ Kluwer Academic Publishers. (1987). Selected Traveling Salesman Problems. Cybernetics and Systems’ (1993).24: 27–36.[2] Ambati, B., Ambati, J., & Mokhtar, M., „Heuristic Combinatorial Optimization by Simulated Darwinian [17] Fox, M., & McMahon, M., „Genetic Operators for Evolution: A Polynomial Time Algorithm for the Sequencing Problems‟ In Rawlings, G. (ed.) Traveling Salesman Problem Biological Cybernetics‟ Foundations of Genetic Algorithms: First Workshop on (1991). 65: 31–35. the Foundations of Genetic Algorithms and Classifier Systems, (1987). 284–300. Los Altos, CA: Morgan[3] Asymmetric Travelling Salesman Problems‟, KaufmannPublishers. International Conference On Evolutionary Computation 616-621, (1996). [18] Freisieben, B., and Merz, P.: „New Genetic Local Search Operator for Travelling salesman Problem,[4] Banzhaf, W., The “Molecular” Traveling Salesman. Conference on Parallel Problem Solving From Biological Cybernetics(1990). 64: 7–14. nature,‟(1996). app. 890-899.[5] Beyer, H., Some Aspects of the „Evolution Strategy‟ for [19] Gunnels, J., Cull, P., & Holloway, J., Genetic Solving TSP Like Optimization Problems Appearing at Algorithms and Simulated Annealing for Gene the Design Studies of the 0.5 Linear Collider. (1992). Mapping. In Grefenstette, J. J. (ed.) Proceedings of the First IEEE Conferenceon Evolutionary Computation,[6] Brady, M., „Optimization Strategies Gleaned from 385–390. Florida: IEEE. (1994). Biological Evolution Nature’ (1985). 317: 804–806. [20] Goldberg, D., & Lingle, R., Alleles, „Loci and the[7] Braun H., “On Solving Travelling Salesman Problem TSP‟. In Grefenstette, J., (ed.) „Proceedings of the First by Genetic Algorithm”, in Parallel Problem- Solving International Conference on Genetic Algorithms and from Nature, Lecture Notes in Computer Science 496, Their Applications’, (1985).154–159. Hillsdale, New H.P. Schwefel and R. Manner Eds, Springer-Verlag, Jersey: Lawrence Erlbaum. app. 129-133. [21] Goldberg, D., „Genetic Algorithms in Search,[8] Bremermann, J., Rogson, M. & Salaff, S., „Search by Optimization and Machine Learning’. Reading, MA: Evolution‟. In Max field, CallahanA, M., & Fogel, L., Addison Wesley. (1989). (eds.) „Biophysics and Cybernetic Systems’, 157– 167.Washington: Spartan Books. (1965). [22] Goldberg, D., Computer-Aided Pipeline Operations Using Genetic Algorithms and Rule[9] Davis, L., „Applying Adaptive Algorithms to Epistatic Domains. Proceedings of the International Joint [23] Goldberg, D., „Genetic Algorithm in Search, Conference on Artificial Intelligence’, (1985).162–164. Optimization and Machine Learning. Addison-Wesley‟, 1989.[10] Davis, L., (ed.) „Genetic Algorithms and Simulated Annealing’, 42–60. Los Altos, CA: Morgan Kaufmann. [24] Gorges, M., „ASPARAGOS An Asynchronous Parallel Genetic Optimization Strategy‟ In Schaffer, J., (ed.)[11] Davis, L., (ed.) „Handbook of Genetic Algorithms‟ New „Proceedings on the Third International Conference on York: Van Nostrand Reinhold (1991). GeneticAlgorithm’s, 422–427. (1989). Los Altos, CA: Morgan Kaufmann Publishers.[12] Dorigo, M., and Gambradella, L., „Ant Colony System: A Cooperative Learning Approach to the Travelling [25] Greffenstette, J., „Genetic Algorithms Made Easy‟, Salesman Problem, JEEE Transaction on Evolutionary (1991). Computation‟, (1997).Vol. l, pp. 53-66,. [26] Grefenstette, J., Gopal, R., Rosmaita, B. & Van Gucht, D. „Genetic Algorithms for the TSP‟. In Grefenstette, 6
7.
J., (ed.) „Proceedings of the First International [39] Larra˜naga, P., Kuijpers, C., Poza, M., & Murga, R., Conference on Genetic Algorithms and Their „Decomposing Bayesian Networks: Triangulation of the Applications’, (1985).160–165. Hillsdale, New Jersey: Moral Graph with Genetic Algorithms‟. „Statistics LawrenceErlbaum. andcomputing’ (to be published). (1996a)[27] Grefenstette, J., (ed.) „Genetic Algorithms and Their [40] Larra˜naga, P., Kuijpers, C., Murga, R., & Yurramendi, Applications: Proceedings of the Second International Y., „Searching for the Best Ordering in the Structure Conference’. (1987a). Hillsdale, New Jersey: Lawrence Learning of Bayesian Networks‟. IEEE Transactions Erlbaum. onSystems, Man and Cybernetics(1996b). 26(4): 487– 493.[28] Grefenstette, J., „Incorporating Problem Specific Knowledge into Genetic Algorithms‟. (1987b). [41] Larra˜naga, P., Inza, I., Kuijpers, C., Gra˜na, M., & Lozano, J., „Algorithms Gen´eticos en el Problema[29] Holland, J., „Adaptation in Natural and Artificial delViajante de Comercio. Informatica Automatica’ Systems‟. Ann Arbor: University of Michigan Press. (submitted) (1996c). (1975). [42] Lauritzen, S., & Spiegel, D., „Local Computations with[30] Homaifar, A., & Guan, S., „A New Approach on the Probabilities on Graphic Structures and Their Traveling Salesman Problem by Genetic Algorithm’. Application on Expert Systems‟. „Journal of the Royal Technical Report, North Carolina A&T State Statistical Society’, Series B(1988). 50(2): 157–224. University. (1991). [43] Lawler, L., Lenstra, K., Rinnooy, A., & Shmoys, D.,[31] Homaifar, A., Guan, S., & Liepins, G., „A New (eds.) „The Travelling Salesman Problem: A Guided Approach on the Traveling Salesman Problem by Tour of Combinatorial Optimization’. Chichester: Genetic Algorithms‟. In Forrest, S. (ed.) „Proceedings Wiley (1985). of the Fifth InternationalConference on Genetic Algorithms’, (1993). 460–466. [44] Lawrence E. „Travelling Salesman TypeCombinatorial Problems and Their Relation to theLogistics of[32] Jog P., Suh, J., & Gucht, D., „The Effects of Population Regional Blood Banking’. PhD Thesis, Northwestern Size, Heuristic Crossover and Local Improvement on a University. (1976). Genetic Algorithm for the Traveling Salesman Problem‟. In Schaffer, J., (ed.) „Proceedings on the [45] Lidd, M., „The Travelling Salesman Problem Domain Third International Conference on Genetic Algorithms’ Application of a Fundamentally New Approach to 1989)., 110–115. Los Altos, CA: Morgan Kaufmann Utilizing Genetic Algorithms’. (1991). Technical Publishers. Report, MITRE Corporation.[33] Johnson D, McGeoch L (2002). Experimental Analysis [46] Liepins, E., Hilliard, R., Palmer, M., & Morrow, M., of Heuristics for the STSP."In Gutinand Punnen (2002), „Greedy Genetics‟. In Grefenstette, J., (ed.) „Genetic chapter 9, 369-444. Algorithms and Their Applications: Proceedings of the Second International Conference’, 90–99. Hillsdale,[34] Johnson, S., „Local Optimization and the Traveling New Jersey: Lawrence Erlbaum. (1987). Salesman Problem. Proc. 17th Colloq. Automata, Languages and Programming’ (1990). Springer Verlag. [47] Lin, S. „Computer Solutions on the Travelling Salesman Problem‟. Bell Systems Techn.(1965). J. 44:[35] Kirkpatrick, S., Gelatt, C., & Vecchi, M., „Optimization 2245–2269. by Simulated Annealing‟. Science (1983).220: 671– 680. [46] Lin, S., & Kernighan, B., „An Effective Heuristic Algorithm for the Traveling Salesman Problem‟.[36] Koza, J., „Genetic Programming: On the Programming Operations Research (1973).21: 498–516. of Computers by Means of Natural Selection’ (1992). MIT Press. [47] Lin, T., Kao, C., & Hsu, C., „Applying the Genetic Approach to Simulated Annealing in Solving NP Hard[37] Laarhoven, P., and Aarts, E.: „Simulated Annealing: Problems‟. IEEE Transactions on Systems, Man, and Theory and Applications, Kluwer Academic‟. (1987). Cybernetics (1993).23(6): 1752–1767.[38] LARRAN, P., KUIJPERS, C., MURGA, R., INZA I., [48] Lozano, A., Larra˜naga, P., & Gra˜na, M., „Partitional and DIZDAREVIC S., „Genetic Algorithms for the Cluster Analysis with Genetic Algorithms: Searching Travelling Salesman Problem‟: A Review of for the Number of Clusters‟. Fifth Conference of Representations and Operators. Artificial Intelligence InternationalFederation of Classification Societies, Review 13: 129–170, 1999 Kluwer Academic (1996).251–252. Kobe, Japan. Publishers(1999). [49] Learning. Part I; Genetic Algorithms in pipeline Optimization, Engineering with Computer 3, (1987). 7
8.
Problem. In Albrecht, R., Reeves, C., & Steele,[50] Matthews, R. A. J. „The Use of Genetic Algorithms in N.,(eds.) „Artificial Neural Nets and Genetic Cryptanalysis‟ Cryptologia (1993). XVII (2): 187–201. Algorithms’, (1993). 559–566. Wien: Springer Verlag.[51] Manner, R., & Manderick, B., (eds.) „Parallel Problem [64] Rechenberg, I., „Optimierung Technischer Systeme Solving from Nature 2, 361–370. Amsterdam: North Nach Prinzipien der Biologischen Informatio’n. Holland‟. Stuttgart: Frommann Verlag. (1973).[52] Michael H., & Kurt H., TSP – Infrastructure for [65] Reinelt, G., TSPLIB – A Traveling Salesman Library. Travelling Salesman Problem (2011). ORSA Journal on Computing (1991).3(4): 376–384.[53] Michalewicz, Z., Genetic Algorithms + Data Structures [66] Schaffer, J., (ed.) „Proceedings on the Third = Evolution Programs. Berlin Heidelberg: Springer International Conference on Genetic Algorithms’. Los Verlag. (1992). Altos, CA: Morgan Kaufmann Publishers. (1989).[54] Muhlenbein, H., Schleuter, M., & Kramer, O., „New [67] Schwefel, H., „Evolutions strategie und Numerische Solutions to the Mapping Problem of Parallel Systems: Optimierung’. Doctoral Thesis Diss. D 83, TU Berlin. The Evolution Approach. Parallel Computing’(1987). (1975). 4: 269–279. [68] Seniw, D. (1991). „A Genetic Algorithm for the[55] Muhlenbein, H., Gorges Schleuter, M. & Kr¨amer, O. Traveling Salesman Problem’.MSc Thesis, University Evolution Algorithms in Combinatorial of North Carolina at Charlotte. Optimization. Parallel Computing (1988).7: 65–85. [69] Seniw D., „A Genetic algorithm for the Travelling[56] Muhlenbein, H., „Parallel Genetic Algorithms, Salesman Problem‟, MSc Thesis, University of North Population Genetics and Combinatorial Optimization‟. Carolina, at Charlotte. In Schaffer, J. (ed.) „Proceedings on the Third http://www.heatonresearch.com/articales/65/page1.html International Conference onGenetic Algorithms’, . (1996). (1989).416–421. Los Altos, CA: Morgan Kaufmann Publishers. [70] Spillman, R., Janssen, M., Nelsonn B., & Kepner, M., (1993). „Use of a Genetic Algorithm in the[57] Muhlenbein, H., & Kindermann, J., „The Dynamics of Cryptanalysis Simple Substitution Ciphers.Cryptologia Evolution and Learning – Towards Genetic Neural XVII (1): 31–44. SPSSX, User‟s Guide (1988).3rd Networks‟. In Pfeiffer, J,. (ed.) „Connectionism in Edition. Perspectives. (1989). [71] Starkweather, T., McDaniel, S., Mathias, K., Whitley,[58] Muhlenbein, H. „Evolution in Time and Space – The C., & Whitley, D., „A Comparison of Genetic Parallel Genetic Algorithm. In Rawlins, G., (ed.) Sequencing Operators‟. In Belew, R., & Booker, L., Foundations of Genetic Algorithms’. Los Altos, CA: (eds.) „Proceedings on theFourth International Morgan Kaufmann. (1991). Conference on Genetic Algorithms’, 69–76. Los Altos, CA: Morgan Kaufmann Publishers. (1991).[59] Naef-Taher R., and Jalal, A., „Solving TSP problem using New Operator in Genetic Algorithms, American [72] Suh, J., & Gucht, D., „‟Incorporating Heuristic Journal of Applied Sciences‟ (2009) 6(8):1586-1590. Information into Genetic Search‟. In Grefenstette, J. J. (ed.) „Genetic Algorithms and Their Applications:[60] Nygard, K., and Yang, C., „Genetic Algorithm for the Proceedings of theSecond International Conference’, Travelling Salesman Problem With Time Window‟, in 100–107. Hillsdale, New Jersey: Lawrence Erlbaum. Computer Science and Operations Research: New (1987). Development in their Interface, O. Balci, R. Sharda and S. A. Zenios Eds, Pergamon Press, pp. 411-423. [73] Syswerda, G., „Schedule Optimization Using Genetic Algorithms‟. In Davis, L.(ed.) „Handbook of Genetic[61] Oliver, M., Smith, D., & Holland, C., „A Study of Algorithms’, (1991). 332–349. New York: Van Permutation Crossover Operators on the TSP‟. In Nostrand Reinhold. Grefenstette, J., (ed.) „Genetic Algorithms and Their Applications: Proceedings of the Second International [74] Ulder, J., Aarts, L., Bandelt, J., Laarhoven M., & Pesch, Conference’, (1987). 224–230. Hillsdale, New Jersey: E., „Genetic Local Search Algorithms for the Traveling Salesman Problem‟. In Parallel ProblemSolving from[62] Poli R. et. al. (Eds), „Evolutionary Image Analysis, Nature, 106–116. Berlin Heidelberg: Springer Verlag. Signal Processing and Telecommunications Springer‟, (1990). (1999). [75] Whitley, D., Starkweather, T., „Fuquay Scheduling[63] Prinetto, P., Rebaudengo, M., & Sonza, M., „Hybrid Problems and Travelling Salesman: The Genetic Edge Genetic Algorithms for the Traveling Salesman Recombination Operator. In Schaffer, J. (ed.) 8
9.
Proceedingson the Third International Conference on Genetic Algorithms, 133–140. Los Altos, CA: Morgan Kaufmann Publishers. (1989).[76] Whitley, D., Starkweather, T., & Shaner, D., „The Traveling Salesman and Sequence Scheduling: Quality Solutions Using Genetic Edge Recombination‟. In Davis, L. (ed.) „Handbook of Genetic Algorithms’, 350– 372. New York: Van Nostrand Reinhold. (1991).[77] Huai-Kuang Tsai. "Improving EAX with restricted 2- opt", Proceedings of the 2 9
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