A New Search Algorithm for Solving Symmetric Traveling Salesman Problem Based on Gravity

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1 Worl Applie Sciences Journal 16 (10): , 2012 ISSN IDOSI Publications, 2012 A New Search Algorithm for Solving Symmetric Traveling Salesman Problem Base on Gravity Aliasghar Rahmani Hosseinabai, mohamma yazanpanah an Ali Shokouhi Rostami Member of young research club Islamic Aza University, Behshahr branch, Behshahr, Iran Abstract: Traveling Salesman Problem (TSP) is a famous an classic operation for combination of optimization problems which is very use. Many complex issues can be moele as traveling salesman problems. Since TSP is a NP-complete problem, certain algorithms cannot be use for solving it. Hence heuristic methos are common to resolve these issues. This paper presents a new algorithm calle TSP-GSA for solving the traveling salesman problem by means of Gravitational Search algorithm or GSA. This algorithm has use 2 parameters out of 4 main parameters of velocity an gravitational force in physics base on ranom search concepts. The propose algorithm has been compare with the genetic algorithm [1] an experimental results showe that not only propose algorithm has better performance but also it takes less time to be solve. Key wors: Traveling Salesman Gravitational force Genetic algorithm Velocity Newton law INTRODUCTION In recent years, many algorithms have been propose to solve this problem. Some of these algorithms are as Traveling Salesman Problem consists of n cities an follows: between every two city there can exist a way (path). Each Fozia hanif khan, Nasiruin khan an sye of these paths has specifie istance or cost. Traveling inayatullah [2] have use a new metho to show salesman wants to start from one of these cities an then chromosome by using a binary matrix to solve either travel to all cities such that he passes through each city symmetrical or asymmetrical TSP algorithm. only once an finally, return to the origin. The aim (target) Nagham Azmi et al. [3] have suggeste a simple of the traveling salesman problem is fining the orer of stanar genetic which is a new approach of genetic the cities such that, in general, the total istance of the algorithm an is calle a social genetic algorithm for travel, or the cost of the trip, be reuce by the seller TSP. This algorithm shows that social genetic algorithm uring the trip [2-4]. works better than stanar genetic algorithm. TSP can be symmetrical or asymmetrical. In the Seye Reza Hejazi an Reza Soltani [4], use Ant symmetrical one, the istance between the two cities oes Colony an genetic algorithm an coul get better tour not epene on the istance of travel. For example, if we result for TSP. show the istance between two noes calle i an j by ij Alessio Plebe an Angelo Marcello Anile [5] have an if we have ji= ij TSP is symmetric otherwise it is presente a metho base on the neural network to solve asymmetrical. Although it is easy to unerstan traveling the Extra Double TSP such that in this metho, the final salesman problem, but if its size gets large, solving the result for the problem is one of the changes of basic TSP, problem will be ifficult an even in very large sizes, it is where these aims can be ene to two parallel operational almost impossible to solve it [4]. salesmen. Genetic algorithm is one of the most functional TSP is a complicate or a NP-compete problem that methos for solving TSP. It has ha some efects cannot be solve using traitional algorithms an usually inclue the high operational costs an the slow innovative methos are use to solve these kins of movement towars the optimal nationwie. In this article, problems. Some of the methos which are utilize to solve a new algorithm which is calle TSP-GSA has been such problems are genetic algorithm, simulate presente by using gravitational force to solve the annealing algorithm, hill climbing algorithm an etc. traveling salesman problem. Corresponing Author: Aliasghar Rahmani Hosseinabai, Member of young research club, Islamic Aza University, Behshahr branch, Behshahr, Iran. 1387

2 Worl Appl. Sci. J., 16 (10): , 2012 Laws of Motion: The real velocity of every particle is equal to the sum of the coefficient of the previous velocity of the particle an the change of its velocity. Change in velocity an acceleration, is equal to the impose force on the particle ivie to its gravitation mass. Fig. 1: An example of traveling salesman problem [1] The structure of article has been organize as follows: In the secon part, we offer the traveling salesman problem. In the thir part, we escribe GSA algorithm an in the fourth part, we explain GSA algorithm for solving traveling salesman problem in etails. In the fifth part, simulation results have been shown with ifferent efficiency parameters an then conclusion has been mentione. Problem Description: In TSP, there are several cities or several thousan cities in an area that the salesman must pass all these cities an then return to the origin city as he shouln t pass any of them twice an also the istance has been taken by the salesman shoul be minimize [1,6]. Figure1 shows an example of solving TSP. The more the number of cities is, the more complex the problem is an the more time is neee to solve it. GSA Algorithm Main Iea: The search space of a problem is assume as a multiimensional system with ifferent solutions to the problem. Each point in space is a unique solution to solve the problem an each solution possesses a "mass" through which the objective function will be compute. Any solution which is better, more objective function value is generate an thereby its mass will be more. Besies, search agents are collection of particles. After constituting search space, its rules will be recognize to govern it, with the assumption that only gravitation an motion laws govern it [7-10]. Gravitation Law: Each particle in search space will attract other particles towar itself. The amount of this force is proportional to the gravitational property of the particle an inversely to the istance between the two particles. Gravitational Search Algorithm: The search space is consiere as a set of m particles. Position of each particle in search space is a point in space an is taken in account as a solution to the problem. This position is obtaine from equation (1) in which position of particle i in imension is shown by. i x I D i i i i x = ( x,..., x,... x ) In this system in time t, a force is impose from particle ito each particle j in the irection of imension as much as () t. The amount of this force is compute Fij from equation (2) in which G(t) is gravitation constant in time t, R ij(t) is istance between two particles i an j in that time an is a small number. To obtain istance between the particles Eucliean istance has been use (Equation 3). Gt () Mi() t Mj() t Fij () t = ( xj () t xi ()) t R () t + ij R () t = X (), t X () t ij i j The impose force on the particle i in irection of imension in time t, () t, is equal to the total amount Fi of the forces of other particles of the system impose to it (Equation 4). m Fi () t = rj fij () t j= 1, j i Accoring to secon Newton s law, each particle gains acceleration in irection of imension an this acceleration is proportional to the impose force on the particle in the same imension ivie by the particle gravitational mass. The particle acceleration in irection of imension in time t is shown by obtaine from Equation 5. 2 ai (1) (2) (3) (4) () t an it will be 1388

3 Worl Appl. Sci. J., 16 (10): , 2012 As mentione above, in TSP problem, there are several cities or several thousan cities in an area that the salesman must pass in a way that he oesn t pass each city twice. He also has to take the way with the shortest istance between the cities an come back to the start point again. We have use GSA algorithm because our cities must be within a large array as non-repetitive an then the amount of istance cost between the cities as current response shoul be compute. To solve the problem of TSP, three matrices of istance an initial velocity matrices are prouce ranomly. In the velocity matrix, an initial velocity is given to each city which is consiere as a mass. The velocity will change in later missions. The time matrix can be attaine with the relation (9). Fig. 2: The gravitational force algorithm [10] Fi () t ai () t = M () t Velocity of each particle is equal to the sum of a coefficient of the present velocity of the particle an acceleration of the particle that is obtaine from Equation 6. The new position of the particle i in the imension is equal to the sum of its present position an its velocity that computes by Equation 7. i i i i V ( t+ 1) = r V ( t) + a () t i j i i i X ( t + 1) = r X ( t) + V ( t + 1) rianrjare the ranom numbers by uniform istribution in the interval of (0,1) which have been use to keep the ranom property of search. Equation 8 is use in orer to ajust the gravitational constant. Gravitation constant tens to ecline exponentially. a 1 Gt () = T Propose Algorithm: In this paper, propose metho was taken from GSA Algorithm as a strategy for solving TSP problem. The goal of this algorithm is, to fin out minimum paths between cities for traveling salesman an make it possible to be use for solving the problem in huge sizes for the least possible time. (5) (6) (7) (8) T = ( Y Y ) + ( X X ) 2 2 B A B A V ina, B Cities will be put ranomly in an array after proucing above matrices. Now, a solution is mae for the problem an its fitness will be calculate an efine as its mass. The best solution must have the greatest mass ue to gravity law. The next solution will be prouce ue to the current one on the basis of problem s restrictions an its fitness will be calculate an known as its mass. This solution may replace current if it is optimal. If it is not optimal, another solution will be mae ue to the current solution an it will continue on until the algorithm yiels the most optimal solution. The algorithm ens at two following states: (9) Each element in the initial velocity vector becomes zero Repetition times achieve the maximum. The explaine propose algorithm is shown as pseuo coe in Figure 3. Simulate Results: In orer to implement the algorithm, Matlab software has been use. The programs were performe in a computer with a processor of 2.4 GHz Pentium-IV an RAM 1 GB. Propose algorithm (TSP-GSA) has been compare with GA algorithm. For comparing those algorithms five ata tests have been esigne to cover small, meium an big systems. Those ata tests are entitle as Test_sh_c that sh is the number of the problem an c is the number of 1389

4 Worl Appl. Sci. J., 16 (10): , 2012 Table 1: The results of the simulation of the two algorithms of GA an GSA TSP-GSA GA Problem Time (t) Best Fitness Worst Fitness Time (t) Best Fitness Worst Fitness Test_01_20 Minimal Test_02_50 Negligible Test_03_ Test_04_ Test_05_ Best Fitness: The best value for the algorithm that returns the number of cities an the istance taken between the cities Worst Fitness: The worst value for the algorithm that returns the number of cities an the istance taken between the cities: Time: The require amount of time for implementing the algorithm Matrix_Distance = Create_Matrix_Distance Matrix_Spee = Create_Matrix_Spee Matrix_Time = Create_Matrix_Time (Matrix_Distance, Matrix_Spee) CU = Create_Parente CA = Create_ChilCU.Fit = Fittnes (CU ) CA.Fit = fittnes (CA) While ( Number Of City's Region) If AC.Fit<CU.Fit Then R = Calculate( CA, Matrix_Distance) F= 6.62 * ( CU.Fit - CA.Fit ) / ( R ^ 2 ) V = Matrix_Spee( CA.City_Current, CA.City_Ol ) V = V + F Fig. 3: The propose algorithm pseuo coe cities. For example, test_02_50 shows the secon ata test has 50 cities that the salesman must pass the istance between those 50 cities so that he mustn t pass a city twice an must fin the shortest istance between the cities. Table 1 compares the results between the two algorithms TSP-GSA an GA. This table epicts that the spent time is lower in the propose algorithm in comparison with the GA algorithm. The value of t shows the require time for implementing the algorithm. The results show that the propose algorithm nees much less time to be solve than in the GA algorithm. Matrix_Spee (CA.City_Current, CA.City_Ol)=V matrix_time (CA.City_Current, CA.City_Ol)= (Matrix_Distance( CA.City_Current, CA.City_Ol ) * 60) / V Swap( CU, CA) CA = Create_Chil CA.Fit = fittnes (CA) ELSE Empty (CA) CA = Create_Chil CA.Fit = fittnes (CA) END IfEND WhileDesign Graph Fig. 4: The propose algorithm Figure 5 shows the results of the comparison between the two algorithms by using Test_01_100. As it is shown in the figure, the propose algorithm nees much less time to be performe an also achieves better results. Figure 6 shows the results of the comparison between the two algorithms by using Test_02_1000 Figure 7 shows the results of the comparison between the two algorithms for the best values by using Test_01_

5 Worl Appl. Sci. J., 16 (10): , 2012 Fig. 5: Comparison of the two algorithms of GA an GSA by means of Test_01_100 Fig. 8: Comparison of the two algorithms of GA an GSA for the best values by means of Test_02_1000 Fig. 6: Comparison of the two algorithms of GA an GSA by means of Test_02_1000 Fig. 9: Comparison of the two algorithms of GA an GSA consiering the average time neee to be run Figure 8 shows the results of the comparison between the two algorithms for the best values by using Test_02_1000. Figure 9 shows the time require to implement the two algorithms on ifferent ata tests. CONCLUSIONS Fig. 7: Comparison of the two algorithms of GA an GSA for the best values by means of Test_01_100 In this article, a gravitational force algorithm calle TSP-GSA has been offere to solve traveling salesman problem. The avantages of this algorithm are velocity, implementation time an very little fitness amount. The goal of this algorithm is to reuce the implementation time an fin the shortest route between the cities by the salesman. 1391

6 Worl Appl. Sci. J., 16 (10): , 2012 Efficiency of this algorithm is compare with the GA 4. Hejazi, S.R. an R. Soltani, The implementation algorithm. of combine ant colony an genetic algorithm for The achieve results showe that the TSP_GSA solving traveling salesman problem. In the algorithm has improve for 15 percent in worst state. The proceeing of 4th International Inustrial amount of this improvement is more significant in larger Engineering Conference. systems. 5. Plebe, A. an A.M. Anile, A Neural Network As shown in the table of simulation, some of state Base Approach to the Double Traveling Salesman improvements of the TSP_GSA algorithm was about 50 Problem, Preprint of the extene article on Neural percent. The neee time for propose algorithm was Computation. much lower than genetic algorithm. It also requires much 6. Wang, Z., H. Duan an X. Zhang, An Improve less time which was about less than 1% of the genetic Greey Genetic Algorithm for Solving Travelling algorithm. For the reason why, the require amount of Salesman Problem. In the proceeing of 5th time for achieving the final response an fining the International Conference on Natural Computation. shortest route for solving TSP is of utmost importance. 7. Rashei, E., H. Nezamabai-pour an S. Saryaz, Spening a very short time is also one of the main GSA: A Gravitational Search Algorithm, avantages of the propose algorithm. Information Sci., 179(13): REFERENCES 8. Rashei, E., H. Nezamabai-pour an S. Saryazi, BGSA: Binary gravitational search algorithm, Natural Computing, 9(3): Moan, H. an Lifang Xu, Biogeography 9. Yin, M., Y. Hu, F. Yang, X. Li an W. Gu, A Migration Algorithm for Traveling Salesman Problem, novel hybri K-harmonic means an gravitational ICSI, pp: search algorithm approach for clustering, Expert 2. Foziahanif khan, Nasiruin khan an Systems with Applications, 38: Syeinayatullah, Solving TSP problem BY 10. Rostami, A.S., H.M. Bernety an A.R. Hosseinabai, using genetic algorithm. International J. Basic an A Novel an Optimize Algorithm to Select Applie Sciences IJBAS. 9: 10. Monitoring Sensors by GSA, ICCIA. 3. Azmi, N., AL-Mai an A. Tajuin Khaer, The Traveling Salesman Problem as a Benchmark Test for a Social-Base Genetic Algorithm, J. Computer Sci., 4(10):