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  1. 1. 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
  2. 2. 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
  3. 3. 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
  4. 4. 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
  5. 5. 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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