A Fast TSP Solver Using GA on JAVA - Gcd.org

A<strong>Fast</strong> <strong>TSP</strong> <strong>Solver</strong> <strong>Using</strong> <strong>GA</strong> on JAVAHiroaki SENGOKUSystems Development Laboratory,Hitachi, Ltd.Kawasaki, 215 JAPANIkuo YOSHIHARAGraduate School of Engineering,Tohoku UniversitySendai, 980-77 JAPANAbstractAhybrid algorithm using <strong>GA</strong> and heuristics forrapid solution of Travelling Salesperson Problems(<strong>TSP</strong>) is presented. We developed a JAVA programbased on this algorithm. Visiting our web pages, everyonecan easily try our <strong>TSP</strong> solver. Since JAVAprograms are executable on many platforms, any onewho design a new <strong>TSP</strong> algorithm can compare his ownsolver and ours on the same machine. <strong>Using</strong> our programas the criterion, <strong>TSP</strong> researchers can evaluatetheir algorithms objectively.1 IntroductionThe Genetic Algorithm (<strong>GA</strong>)[1] is an optimizing algorithmthat is modeled after the evolution of organisms.Some of the <strong>GA</strong>'s merits are that it can be easilydeveloped because it does not require detailed knowledgeabout the problem, it can search globally, and itcan adapt to the changing conditions in the problem.Despite of these merits, <strong>GA</strong> is often slower thanconventional methods such as heuristic searches. Thisis because <strong>GA</strong> does not utilize explicitly the knowledgeof how to search for the solutions. Therefore, hybridmethods that combine <strong>GA</strong> with other techniques havebeen attempted.The <strong>TSP</strong> solver we presented[2] is one of the hybridmethods. It uses <strong>GA</strong> and the 2opt method (section2.1). Owing to the 2opt, it is much faster thanother <strong>TSP</strong> solvers based on <strong>GA</strong> alone.Generally, it is dicult to compare two approximatealgorithms objectively. Some algorithms have the advantageof high speed, but have the disadvantage of alow success rate in nding the optimum solution.The best way to compare two methods is by runningthem both on the same machine. But there are fewsuch programs in public domain[4], and we have notfound a program that can solve the problems listed in<strong>TSP</strong>LIB[5].Consequently, we wrote a JAVA 1 program based on1JAVA is a trademark of Sun Microsystems, Inc.our proposed method, and made it public on our pageon the World Wide Web (WWW). JAVA programs areexecutable on many platforms, so people who have developeda <strong>TSP</strong> solver can compare their programs andours by running both programs on the same machineunder equal conditions.We also developed a graphical user interface. Peoplecan freely situate towns by using a mouse, andsolve the problem they constructed. It's easy and fun.It's helpful for students studying the basics of computeralgorithms.In section 2, we explain our method. In section 3,we show the WWW page in which we present our <strong>TSP</strong>program.2 A <strong>TSP</strong> solver: 2opt<strong>GA</strong>We present a hybrid method[2] that uses <strong>GA</strong> andthe 2opt method. In our <strong>GA</strong>, the 2opt method providesmutation, while the crossover operator providesthe capability of jumping out from the local minima,where the solution often falls where only 2opt is used.The algorithm consists of the following steps.individuals ran-Initialization: Generation of Mdomly.Natural Selection: Eliminate p e % individuals. Thepopulation decreases by M 2 p e =100.Multiplication: Choose M 2p e =100 pairs of individualsrandomly and produce an ospring from eachpair of individuals. The population reverts to theinitial population M.Mutation by 2opt: Choose p i % of individuals randomlyand improve them by the 2opt method.The elite individual, or the individual that hasthe best tness value in the population, is alwayschosen. If the individual is already improved, donothing, because it cannot be further improvedby 2opt.We now describe the detail of each step.

2.1 Mutation by 2optThe 2opt method is one of the most well-known localsearch algorithms among <strong>TSP</strong> solving algorithms.It improves the tour edge by edge and reverses the orderof the subtour. For example, imagine a tour asshown in the upper part of Fig. 1. Remove the twoedges ab and cd, and reverse the order of the subtour(from b to c), and add the two edges ac and bd. Thisgives us a tour as shown in the lower part of Fig. 1.The lower tour is shorter than the upper one becauseab + cd > ac + bd.tour lengthcrossoverchildtourFigure 2: Pop-up from local minima.adcbg =(D; H; B; A; C; F; G; E) means that the salespersonvisits towns D, H, B, A, ..., E, successively, andreturns to town D.adFigure 1: The 2opt methodWe check every pair of edges, for example, ab andcd. Ifab + cd > ac + bd holds, we improve them in thesame way as shown in Fig. 1. Actually, ifbothac > aband bd > cd hold, then it is not necessary to check theedges. Therefore we can skip the pairs whose edgesare far away from each other.We repeat the procedures described above until nofurther improvement can be made.2.2 Crossover in MultiplicationWhen we apply the 2opt method to a solution, thesolution often falls into a local minimum. Then it cannotbe improved further by the 2opt method. Considertwo solutions that have fallen into dierent localminima. Potentially, each solution may have the bestsubtour for a dierent part of the tour. Then, we canmake a better solution if we combine those best subtoursappropriately. We cannot say, of course, whichof the subtours are good, but after many trials, we canexpect ospring solutions to be located in the valleyof the global minimum as shown in Fig. 2.We propose a new crossover operator that acquiresthe longest possible sequence of parents' subtours.We named it `Greedy Subtour Crossover (GSX)'. Weshowed[3] by experiments that using the GSX, the solutioncan pop up from local minima more eectivelythan by using simulated annealing (SA) methods.In the GSX, we use the path representation fora genetic coding. For example, the chromosomecbParentsPick up alphabets from the parents alternately.D H B A C F G EB C D G H F E AH B A C D GChildFigure 3: Greedy Subtour CrossoverAdd the rest of alphabetsin the random order.F EAlgorithm: Greedy Subtour CrossoverInputs: Chromosomes g a = (a 0 ;a 1 ;:::;a n01) andg b =(b 0 ;b 1 ;:::;b n01).Outputs: The ospring chromosome g.procedure crossover(g a ,g b ) ff a truef b truechoose town t randomlychoose x, where a x = tchoose y, where b y = tg tdo fx x 0 1 (mod n),y y +1 (mod n).if f a = true then fif a x 62 g then fg a x 1 g,g else ff a false.ggif f b = true then fif b y 62 g then fg g 1 b y ,g else f

gf b false.ggg while f a = true or f b = trueif jgj < jg a j then fadd the rest of townsto g in the random ordergreturn gNote that \1" in\g a x 1 g" is the concatenationoperator, and that sentence means to add a x beforethe chromosome g.An example is shown in Fig. 3. Suppose that chromosomesof parents are g a =(D;H; B; A; C; F; G; E)and g b =(B;C; D; G; H; F; E; A). First, choose onetown at random. In this example, town C is chosen.Then, x = 4 and y = 1 because a 4 = C and b 1 = Crespectively. Now the child g is (C).Next, pick up towns from the parents alternately.Begin with a 3 (town A) because x 4 0 1=3,andnext is b 2 (town D) because y 1 + 1 = 2. The childbecomes g =(A; C; D).In the same way, add a 2 (town B), b 3 (townG), a 1 (town H), and the child becomes g =(H; B; A; C; D; G). Now the next town is b 4 = H andtown H has already appeared in the child (rememberthe salesperson may not visit the same town twice), sowe can't add any more towns from parent g b .Therefore we add towns from parent g a . The nexttown is a 0 = D, but D is already used. Thus we can'tadd towns from parent g a , either.Then, we add the rest of the towns, i.e., E and F,to the child in the random order. Finally the child isg =(H; B; A;C; D; G; F; E).3 JAVA applet3.1 Applet on WWWWe developed a JAVA applet based on the proposedmethod, and put it on our WWW page:http://www.hitachi.co.jp/Div/sdl/e-naiyo/e-seika22/<strong>TSP</strong>.htmlIt is easy to use. First, using a mouse, put \towns"in the rectangular eld. Then, push the \start" button.The applet solves the problem you've just madeand displays the tour and its length (Fig. 4). Theparameters of the 2opt<strong>GA</strong> method, i.e., \Population"(M), \Selection" (p e ), \2opt" (p i ), are adjustable bylling out each eld on the applet.We announced the web page on the Net News inJapanese (fj.* newsgroups) only once in July 1996.Since then, the page has been accessed more than 200times a month.One person who sent us comments had mistakenlythought that <strong>GA</strong> was a tool only for experiments, andthat <strong>GA</strong> would be so slow that it could not be appliedto practical problems. But after accessing our webpages, he could understand why <strong>GA</strong> is usable.2.3 Natural SelectionEliminate R = M 2 p e =100 individuals. In the socalledsimple <strong>GA</strong>, the possibility of survival is proportionalto the tness value, but we pay more attentionto the diversity of the population. We eliminatesimilar individuals to maintain the diversity in orderto avoid the immature convergence that is one of thewell-known problems in <strong>GA</strong>.First, sort the individuals in tness-value order.Compare the tness value of adjoining individuals. Ifthe dierence is less than " (a small positive real number),eliminate preceding individual while the numberof eliminated individuals is less than R. Let r be thenumber of eliminated individuals.Next, if r

Netscape is a trademark of Netscape Communications Corporation.Figure 4: applet on WWWing new problems one after another and executing theapplet. Some of those problems are shown in Fig. 5.Some students added and removed a few towns intheir problems and watched the transformation of thesolution. Some tried to alter the parameters of the2opt<strong>GA</strong> and observe what would happen. Throughmany trials, they got the feeling of the diculty ofthe <strong>TSP</strong>, and gained a concrete understanding of theoptimization problems.A summary of comments from the students follows: Good Points{ It's easy and fun.{ I can see the progress of solving the problemsimultaneously.{ I had thought that web pages were only forreading, so I was surprised to see your pagefor solving the <strong>TSP</strong>.

Bad Points{ No grid. The coordinates of the mouse arenot displayed, so I cannot put towns accuratelyinto the places I want.{ I can't try pre-dened problems. I can't saveproblems. (Note: the current version of theapplet can read the <strong>TSP</strong> le that containsthe coordinates of towns.){ I want to see the convergence graph. I wishto check not only the elite but also the otherindividuals in the population. It would beconvenient if the CPU time consumed tosolve the problem was displayed.We also present the following sample problems inour WWW pages:and sample problems listed in Table 2 from the URL:http://www.hitachi.co.jp/Div/sdl/e-naiyo/e-seika22/Table 1: Class les to download.File nameCities.classCity.class<strong>GA</strong>.classGene.classSort.classSortable.class<strong>TSP</strong>.classTspRoute.classSize3k1k3k3k1k1k8k4k randomly located 100 towns double circle 192 towns (\C"-type) double circle 192 towns (\O"-type)Double circle problems were used as benchmarksby Yamamura et al.[6] The minimum solutions of theproblems are known and they are either \C"-type or\O"-type.Table 2: Sample problems.File name Problemgr96.data Africa-Subproblem of 666-city <strong>TSP</strong>gr202.data Europe-Subproblem of 666-city <strong>TSP</strong>test100.dat Randomly located 100 townsc192-4682.dat Double circle 192 towns (\C"-type)c192-4684.dat Double circle 192 towns (\O"-type)‘C’-type‘O’-typeinner / outer = 3/5 inner / outer = 4/5Figure 6: the minimum solutions of the double circleproblemsAs shown in Fig. 6, the type of solution that willyield the minimum, depends on the number of townsand the ratio of the radii of the inner and outer circles.3.2 Stand-alone ApplicationThe applet can also be used as a stand-alone application.First, download class les listed in Table 1\Africa-Subproblem of 666-city <strong>TSP</strong>" and \Europe-Subproblem of 666-city <strong>TSP</strong>" are contained in<strong>TSP</strong>LIB[5].\<strong>TSP</strong>.class" contains the main method, so executethat le as a JAVA application with arguments `lename'and `generation'. The rst argument, `lename',is the name of the <strong>TSP</strong> le that contains thelocation of towns of the <strong>TSP</strong> to be solved. The secondarguments, `generation', is the maximum generationto be evolved. Optional ags listed in Table 3 can beused.Each line in the <strong>TSP</strong> le represents the coordinatesof a town and consists of two elds separated by acomma. The rst eld means the X coordinate of thetown and the second eld means the Y coordinate.The cost between two towns is measured as the Euclideandistance between the coordinates of each town.If the <strong>TSP</strong> le contains the lineEDGE_WEIGHT_TYPE: GEOthe cost is dened by the geographical distance oftwo towns, i.e., the distance on the earth, instead of

Table 3: Optional ags of <strong>TSP</strong>.classOption Meaning Default-p M The population 100-e p e Eliminate p e % 30individuals-R p i Improve p i % individuals 20by 2opt-v Increment verbositylevelthe Euclidean distance. The X and Y coordinates describedabove mean the latitude and the longitude respectively.If the <strong>TSP</strong> le contains the lineMin:nwhere n is the positive real number, the programwill stop when the elite, whose tour length is n, isfound.On Windows 2 95 with the Java Development Kit(JDK)[7], for example, type the following commandin the \MS-DOS 2 Prompt" window.java <strong>TSP</strong> -v -p 200 gr96.data 300That command means to solve the <strong>TSP</strong> storedin the le `gr96.data' (Africa-Subproblem of 666-city<strong>TSP</strong> in <strong>TSP</strong>LIB[5]), where the population is 200, andthe maximum generation is 300.The output from the command follows (partiallyomitted):C:\>java <strong>TSP</strong> -v -p 200 gr96.data 300File: gr96.dataPopulation: 200Selection: 30 %2 opt: 20 %1:3[93 94 92 91 76 11111111111111111111 27 64 95]572842:253[94 93 95 64 65 1111111111111111 76 91 92]558403:119[37 34 33 32 10 1111111111111111 31 35 36]5548119:1251[49 51 52 54 50 111111111111 47 46 45]5533244:2646[93 94 92 91 76 111111111111 65 64 95]55322120:7132[41 40 39 38 78 1111111111 49 48 42]55291165:9780[49 45 46 47 53 1111111111 54 52 51]55259180:10692[52 51 49 45 46 11111111 48 50 54]55209The program outputs the elite, when a new elite is2MS-DOS and Windows are registered trademarks of MicrosoftCorporationfound, in the following format:g : i [ tour ] `where g, i, tour, and ` mean the generation, theID of the individual, the tour (a list of towns in thevisiting order) that corresponds to the individual, andthe length of the tour.The last line, for example, says that a new elite isfound at 180th generation, it is the 10692nd individual,and the length of the tour is 55209. According to<strong>TSP</strong>LIB, it is the minimum solution.4 ConclusionWe presented an algorithm for rapid solution of the<strong>TSP</strong> that combines the <strong>GA</strong> and the 2opt method. Wepublished a program based on the algorithm on ourweb pages. Because the program is written in JAVAlanguage and can be executed on many platforms, anyonecan verify the eciency of our algorithm. Anyonewho designed a new algorithm can compare his own<strong>TSP</strong> solver with ours by running both programs on thesame machine. Our <strong>TSP</strong> solver is useful as a criterionfor evaluating the performance of <strong>TSP</strong> solvers.References[1] Holland, J.H.: Adaptation in Natural and ArticialSystems, Univ. of Michigan Press (1975)[2] Sengoku, H., Yoshihara, I.: A<strong>Fast</strong> <strong>TSP</strong> Solutionusing Genetic Algorithm (Japanese), InformationProcessing Society of Japan 46th Nat'lConv. (1993)[3] Sengoku, H., Yoshihara, I.: An Evaluation of OptimizingCapability of Genetic Algorithm | <strong>GA</strong>vs SA | (Japanese), Information Processing Societyof Japan 47th Nat'l Conv. (1993)[4] Moscato, P.: <strong>TSP</strong>BIB, http://www.ing.unlp.edu.ar/cetad/mos/<strong>TSP</strong>BIB_home.html[5] Reinelt, G.: <strong>TSP</strong>LIB, ftp://softlib.rice.edu/pub/tsplib/tsplib.tar[6] Yamamura, M., Ono, T., Kobayashi, S.:Character-Preserving Genetic Algorithms forTraveling Salesman Problem (Japanese), Journalof Japanese Society for Articial IntelligenceVol.7, No.6 (1992)[7] Cornell, G., Horstmann, C.S.: core JAVA, Sun-Soft Press (1996)