An Improved Greedy Genetic Algorithm for Solving Travelling Salesman Problem

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1 2009 Fith International Conerence on Natural Computation An Improved Greedy Genetic Algorithm or Solving Travelling Salesman roblem Zhenchao Wang, Haibin Duan, and Xiangyin Zhang School o Automation Science and Electrical Engineering, Beihang University Beiing, , R China wzc88@163.com Abstract Genetic algorithm (GA) is too dependent on the initial population and a lac o local search ability. In this paper, an improved greedy genetic algorithm (IGAA) is proposed to overcome the above-mentioned limitations. This novel type o greedy genetic algorithm is based on the base point, which can generate good initial population, and combine with hybrid algorithms to get the optimal solution. The proposed algorithm is tested with the Traveling Salesman roblem (TS), and the experimental results demonstrate that the proposed algorithm is a easible and eective algorithm in solving complex optimization problems. Keywords-Genetic algorithm(ga); Greedy algorithm; Improved greedy genetic algorithm(igaa); Traveling Salesman roblem; Base point I. INTRODUCTION Genetic algorithm (GA) was irstly put orward by J. Holland in 1970s to study the sel adaptation behavior o natural system [1]. It is a random search algorithm and global optimization algorithm. It has strong ault tolerance and strong robustness. It is also easy to combine with other methods in optimization, which is suited to solve complex combinatorial optimization problem system. However, genetic algorithm is too dependent on the initial population, is easy to premature convergence, and has insuicient capacity o local optimization. Greedy algorithm, at the time o solving the problem, always maes the choice which at present appears to be the best. In other words, it is not optimal while the whole is taen into account, and what it has got is the best in only the local optimal solution. The Travelling Salesman roblem (TS) [2] is one o the most nown combinatorial optimization problems. These problems require very usually a colossal amount o computing resources to be solved eiciently because the solutions space that should be explored gains exponentially when the size o the problem increases[3]. In this paper, in order to overcome the disadvantages o genetic algorithm and greedy algorithm, an improved greedy genetic algorithm (IGGA) is proposed. Genetic algorithm is too dependent on the initial population. IGGA improved the greedy algorithm and made it search based on the base point. The experiments show that the algorithm reduced the number o cross-routes, and strengthened the capacity optimization. TS which is a amous classic N problem and a typical combinatorial optimization problem is oten used to veriy the validity o intelligent optimization algorithm. Through solving the TS problem, and compared with other algorithms, it demonstrates that IGGA is a easible and eective algorithm to solve complex combinatorial optimization problems. II. 2. MAIN ROCESS OF GA AND GREEDY ALGORITHM A. The main principle o GA[3] Selection. Selection is to evaluate each individual and eeps only the ittest ones among them. In addition to those ittest individuals, some less it ones could be selected according to a small probability. The others are removed rom the current population. Crossover. The crossover recombines two individuals to have new ones which might have a better perormance. Formula (1) shows the single-point crossover o the binary-coded genetic algorithm as ollows Δ Δ (1) Mutation. The mutation operator induces changes in a small number o chromosomes units. Its purpose is to maintain the population diversiied enough during the optimization process. Formula (2) shows the mutation o the binary-coded genetic algorithm as ollows B. The basic idea o greedy algorithm From the initial solution to a successive approximation solution o one problem, it achieves as quicly as possible to a better solution. When the algorithm cannot continue to move orward, the algorithm stops. The algorithm has the ollowing questions: (1) it cannot guarantee that the inal solution is the best; (2)it cannot be applied to see the maximum or minimum solution o the problems;(3) it can only see to the scope o the easible solution which satisy certain constraints. III. THE IMROVED GREEDY GENETIC ALGORITHM A. The basic idea o the proposed algorithm This paper presents an improved greedy genetic algorithm (IGGA), which maes greedy algorithm search or a one- (2) /09 $ IEEE DOI /ICNC

2 way based on the base point. In this way, it can improve the traditional greedy algorithm, strengthen its ability o global optimization while ensuring that their advantage in terms o speed, generate more good gene ragments at the same time, and speed up the generation o good population. Through the experiments, in solving the CHN31, the average route length o the initial population is between *10 ~ 5 *10,which comes rom the initial population that generated randomly,while it is between 2 *10 ~ 2.*10,which comes rom the initial population generated by the greedy algorithm based on the base point.however,the best solution is between 1.5*10 ~ 1.6*10.In addition, the genetic algorithm in the early evolution o population should be conducted to ensure that the search is overall in order to avoid premature convergence. At the evolution o the late, while approaching the optimal solution, the search should be ocusing on the local scope but not the global scope, as ar as possible, to improve the accuracy. Thereore, this paper quoted the seladaptive crossover and mutation strategies []. ( c _ max c _ min ) c _ max * iter... > avg c = it max c _ max... <= avg c c _ max c _ min m = m _ min m _ min ( ) m _ max m _ min + * iter... > it max... <= : The crossover probability : The maximal cross probability : The minimal cross probability m : The mutation probability m _ max : The maximal mutation probability m _ min : The minimal mutation probability Iter max : The maximal generation Iter: The current generation avg : The average itness o the species : The itness o one o the parent : The itness o the one or mutation In addition, in order to urther avoid the bad inluence o greedy algorithm in the process o the local optimization, this paper adds the 2-opt algorithm. avg avg (3) () Fig. 1 The 2-opt algorithm B. The greedy algorithm based on one-way and the base point The improved greedy algorithm starts rom the base point and then searches in accordance with the rules. The standard o selecting the base point: Choosing the largest city (or the smallest) among the abscissa o the cities. Search rules: Followed by the size o the order in accordance with the abscissa to search the nearest city and to put it into the tour while ensuring the new length o the tour is the shortest. Through this way, it can greatly reduce the number o cross-routes, which were ust because o the choice o the random initial points. And it can reduce the length o the tour, and reduce the computation greatly.it is the same as the ordinate. In this paper we choose the point, whose abscissa is the smallest, as the base point, and then ind two points which are nearer to the base point, and constitute a triangle with the three points. Then ollow the search rules, satisy the ollowing ormula, mae the d is the minimal [5]: d(, + 1) + d ( + ) d(, + 1) d = d(, + 1) + d ( + ) (5) J: The current city to insert K: The city in ront o J in the tour K-1: The city behind J in the tour d( + ): The distance between City J and City K For a more intuitive description o the advantages o the proposed algorithm, the ollowing igures show us the comparison. 375

3 Fig.2 The route with ive cities Fig.3 The route with eleven cities Fig.7 The route with eight cities Fig.8 The route with ten cities Fig. The route with twelve cities Fig.5 The route with ourteen cities Fig.2~Fig.5 show the evolution o the greedy algorithm based on one-way and the base point. Fig.6~Fig.9 show the evolution o the traditional greedy algorithm Fig.6 The route with three cities Fig. 9 The route with twenty-nine cities Through the comparison, greedy algorithm based on base point is very eective. In addition, greedy algorithm based on the base point, can generate better gene ragments during the search, which will contribute to accelerate the speed o search, and will save the search time. C. The process o our proposed hybrid improved algorithm or solving TS can be described as ollows. The process o the proposed IGGA or solving TS can be described as ollows: Step1. Initialization o parameters: set the current number o iteration iter=1; set the maximum number o iteration itmax=; set the number o population as Nn, and the number o selected population as Nq; set c _ max min =0.9, c _ max =0.8; m _ min =0.8, m _ =0. Step2. Generate initial population. Through the improved greedy algorithm based on the base point and one-way, it gives a solution, as well as the city route. Then use the 2- opt algorithm to optimize the route o cross route, ust because greedy algorithm is easy to all into the local 376

4 optimum.repeat Nn times. Then we get the population with Nn individuals. Step3. Choose the better population.in order to reduce the computation, choose the excellent population rom all. And then we get the new population with Nq individuals. Step. The introduction o new population.in order to avoid alling into the local optimum, we need introduce new population. The new population, which is with Nq individuals, are chosen rom randomly generated population with Nn individual. Up to now, we get the population we want, which is with Nq*2 individuals. Generate initial population Size: Nn Choose the better population Size: Nq Introduce new population Randomly generate another population. Size:Nn the better population to substitute or the original. Then select randomly ragment and reverse it. Step8. Enter the cycle. Repeat step3, until iter=itmax or other deined termination condition is satisied. Step9. Output the best tour and the shortest length. The above process can also be described with Fig 10. IV. EXERIMENTAL RESULTS A. Optimal route and the average evolution curve To veriy the superiority o this algorithmthis paper selected 31 provincial capital cities and municipalities to carry out tests to veriy IGGA. Experiments show that the optimal route length obtained by IGGA is The sub-optimal length: The optimal route: Get the population. Size: N q *2 Sel-adaptive crossover Choose the better population rom Nn individuals. Now size: Nq. Sel-adaptive mutation Accelerate the evolution with Hybrid algorithm x 10 Fig.11 The optimal route by IGGA The Length O Route Enter the cycle The average length by IGGA N iter<it max? 2 Y Get the best solution. Fig.10 Flow chart o the proposed IGGA Step5. Sel-adaptive crossover. Ater the crossover process inished, we choose the better ones rom parents and their ospring. Step6. Sel-adaptive mutation. Ater the mutation, we choose the better ones rom parents and their ospring. Step7. Accelerate the evolution with hybrid algorithms. Firstly, we exchange two cities every time, and then choose The shortest length by IGGA iter Fig.12 The evolutionary Curve o optimal route by IGGA B. The sub-optimal solution o length by IGGA The sub-optimal length: The sub-optimal route:

5 Figre.13 The sub-optimal route by IGGA 2.8 x 10 The Length O Route The average length by IGGA The optimal length by IGGA iter Fig.1 The evolutionary Curve o sub-optimal route by IGGA C. Experimental comparison with other algorithms in solving the CHN31 Table.1 The name and length o other algorithms The length o Name o the algorithm the optimal route Standard Genetic Algorithm[6] Geometric Algorithms[7] 1592 Genetic Algorithm with combinations o the ameliorated OX operator, greed recessive variation, 150 and combined variation[8] A illed unction method[9] 150 The geometric region-divided method[10] 150 Man-machine combination optimization method quality exhaustive method[11] In the Hopield neural networ methods, together with the constraint condition o the triangle about both sides with in place o one side[11] A Heuristic greedy method[12] 1509 Simulated annealing algorithm[6] It is obvious that IGGA can ind the best solution than the standard Genetic Algorithm and many other algorithms. IGGA has better convergence, and has a stronger ability o global optimization. V. CONCLUSIONS This paper proposed an improved greedy genetic algorithm or solving combinatorial optimization problem, and this algorithm can also solve other complex optimization problems eectively. ACKNOWLEDGMENT This wor was supported by Natural Science Foundation o China under grant # , Aeronautical Science Foundation o China under grant #2006ZC51039, Beiing NOVA rogram Foundation o China under grant #2007A0017 and Science Research Training rogram Foundation o Ministry o Education o China. The authors want to show thans to the anonymous reerees or their valuable comments and suggestions. REFERENCES [1] Holland J., Adaptation in Natural and Artiicial Systems, The University o Michigan ress,ann Arbor, MI,1975. [2] Lawler E.L., Lenstra J.K., Rinnooy Kan A.H.G., et al., The Traveling Salesman roblem, John Wiley & Sons, Chichester,1985. [3] Hichem Talbi, Amer Draa, Mohamed Batouche, A New Quantum- Inspired Genetic Algorithm or Solving the Travelling Salesman roblem. 200 IEEE International Conerence on Industrial Technology, 200, Vol.3, pp [] Shisanu T, rabhas C. arallel genetic algorithm with parameter adaptation, Inormation rocessing Letters, 2002, Vol.82, No.1, pp.7-5. [5] Jingun Zhang, Yuzhen Dong, Ruizhen Gao, Yanmin Shang. An Improved Genetic Algorithm Based on Fixed oint Theory or Function Optimization, 2009 International Conerence on Computer Engineering and Technology,2009,Vol.1, pp [6] Yiwen Zhong,Zhengyuan Ning,Yinghua Cai, et al. An Improved Discrete article Swarm Optimization Algorithm,Mini-Micro systems, 2006, Vol.27, No.10, pp [7] eide Zhou. A Geometrical Algorithm or Solving the Travelling Salesman roblem, Journal o Beiing Institute o Technology, 1995, Vol.15, No.1, pp [8] Chunsong Feng, Junyu Wang, Songsheng Zhou, et al. Ameliorative Genetic Algorithm o TS,Journal o Wuhan University o Technlogy,2006, Vol.28, No., pp ,130. [9] Wenxing Zhu, Qingxiang Fu. A Filled Function Method or the Traveling Salesman roblem, Chinese J.Computers, 2002, Vol.25, No.7, pp [10] Mutian Chen, Hexi Cai. The Geometric Region-Divided Method or Solving the China TS roblem, Computer Engineering&Science,1998, Vol.20, No.1, pp [11] Fan Jin, Junbo Fan, Yongdong Tan. Neural networs and Neural Computers, Southwest Jiaotong University ress, Chengdu,China, July,1991. [12] Lideng an, Xiaoeng Huang. A Heuristic Greedy Method or the Traveling Salesman roblem, Journal o Beiing University o Chemical Technology, 1998, Vol.25, No.2, pp