AN OPTIMAL SOLUTION TO MULTIPLE TRAVELLING SALESPERSON PROBLEM USING MODIFIED GENETIC ALGORITHM

AN OPTIMAL SOLUTION TO MULTIPLE TRAVELLING SALESPERSON PROBLEM USING MODIFIED GENETIC ALGORITHM Varunika Arya 1, Amit Goyal 2 and Vibhuti Jaiswal 3 1,2,3, Department of Electronics and Communication Engineering, Maharishi Markandeshwar University,Sadopur,Ambala,Haryana Abstract Travelling Salesman Problem (TSP) is an optimization problem. In its simplest form, a salesperson must make a circuit through a certain number of cities, visiting each only once, while minimizing the total distance travelled. The multiple Traveling Salesman Problem (mtsp) is the generalization of TSP and is a complex combinatorial optimization problem, where one or more salesmen can be used in the solution. The Constraint in the optimization task is that, each salesman returns to starting point at end of trip, travelling to a unique set of cities in between and except for the first, each city is visited by exactly one salesman. The Cost Function is to search for the shortest route i.e. the least distance needed for each salesman to travel from the start location to individual cities and back to the original starting place. mtsp is a complex, namely an NP-Hard problem and has a multiplicity of applications mostly in the areas of routing and scheduling. The amount of computation time to solve this problem grows exponentially as number of cities so, the heuristic optimization algorithms, like genetic algorithms (GAs) need to take into account. GA generates a population of solutions at each iteration & the best point in the population approaches an optimal solution. The aim of this paper is to review how genetic algorithm can be applied to solve these problems and propose an efficient solution to mtsp. Keywords: Multiple Travelling Salesperson Problem(MTSP), Optimization, Genetic Algorithm(GA), NP-Hard problems. 1. INTRODUCTION The multiple Traveling Salesman Problem (mtsp) is the generalization of TSP and is a complex combinatorial optimization problem, where one or more salesmen can be used in the solution. The Constraints in the optimization task are: 1. Each salesman returns to starting point at end of trip 2. Each salesman travels to a unique set of cities in between 3. Except for the first, each city is visited by exactly one salesman The Cost Function is to search for the shortest route i.e. the least distance needed for each salesman to travel from the start location to individual cities and back to the original starting place. Hence the total cost of visiting all nodes is minimized. In the MTSP, the n nodes must be partitioned into m tours, with each tour resulting in a TSP for one salesman. The MTSP is more difficult than the TSP because it requires assigning nodes to each salesman, as well as the optimal ordering of the nodes within each salesman s tour. The techniques used for solving the MTSP can be categorized into exact, heuristic and metaheuristic algorithms. Out of these, the exact approaches are used only for relatively small problems, but they guarantee optimality. These techniques apply algorithms that generate a lower and an upper bound on the true minimum value of the problem instance. Although the MTSP is conceptually simple, it is difficult to obtain an optimal solution. In other words, when the problem size is increased, the exact methods cannot solve it. So, heuristic or metaheuristic methods are necessary to be used for solving them in a reasonable amount of time particularly with large sizes. Some of the well-known heuristic algorithms are gravitational emulation search and local search. Metaheuristics methods tries to combine basic heuristic methods in higher level frameworks which aim at efficient and effective exploration of a search space. The term metaheuristic derives from the composition of two Greek words. Heuristic stems from the verb heuriskein which means to find, while the prefix meta means beyond in an upper level. Before this term was widely adopted, metaheuristics were often called modern. In general, it is vital to use metaheuristic algorithms to solve complex optimization problems when dealing with them. Since the metaheuristic approaches are very efficient for escaping from local optimum, they are one of the best group algorithms for solving combinatorial optimization problem. Some of the metaheuristic approaches which may be used are genetic algorithm(ga), memetic algorithm (MA), ant system (AS) and particle swarm optimization (PSO). Volume 3, Issue 1, January 2014 Page 425

In this paper, a modified GA is used for solving the MTSP. In the following sections, a mathematical model of MTSP is presented, along with the basic GA and explanation of the proposed algorithm. Analysis of the result obtained is also mentioned in the subsequent section. 2. MATHEMATICAL MODEL Consider an undirected connected graph M(A,B) with a vertex set A={0,1,2,.,p} and an edge set B={ (i, j) : i, j A, i j }. The following variables are used for representing the integer linear programming model for MTSP: p = number of nodes(cities) for each instance. q = number of salesmen used for each instance. C = cost matrix on graph M which is symmetric & true in triangle inequality. i.e. c ij = c ji and c ij + c ji c ik for each ( i,j,k = 1,2,.,p). So, the integer programming formulation for MTSP can be written as: min c ij *xij for i,j=1 to p (1) j=2,,p (2) j=1 (3) i=2,3,.,p (4) i=1 (5) 1 (ɸ S N = {2,,p}), S 2 (6) i S j N-S 1 (ɸ S N = {2,,p}), S 2 (7) i N-S j S xij {0,1} (8) The objective function (1) minimizes the total distance traveled in a tour. Constraint sets (2) and (3) ensure that the salesmen arrive once at each node and q times at the depot. Constraint sets (4) and (5) ensure that the salesmen leave each node once and the depot q times. Constraint sets (6) and (7) are to avoid the presence of sub-tours for each salesman. And the Constraint set (8) defines binary conditions on the variables. 3. METHODOLOGY 3.1 Genetic Algorithm Genetic algorithm (GA) as a computational intelligence method is a search technique used in computer science to find approximate solutions to combinatorial optimization problems. It includes the survival of the fittest idea algorithm. The idea is to first guess the solutions and then combining the fittest solution to create a new generation of solutions which should be better than the previous generation. The genetic algorithm uses three main types of rules at each step to create the next generation from the current population: Selection rules select the individuals, called parents that contribute to the population at the next generation. Crossover rules combine two parents to form children for the next generation. Mutation rules apply random changes to individual parents to form children. In detail, the genetic algorithm process consists of the following: 1. Encoding: A suitable encoding is found for the solution to our problem so that each possible solution has unique encoding and the encoding is some form of a string. Volume 3, Issue 1, January 2014 Page 426

2. Evaluation/Selection: The initial population is then selected, usually at random though alternative techniques using heuristics have also been proposed. The fitness of each individual in the population is then computed that is, how well the individual fits the problem and whether it is near the optimum compared to the other individuals in the population. 3. Crossover: The fitness is used to find the individual s probability of crossover. Crossover is where the two individuals are recombined to create new individuals which are copied into the new generation. 4. Mutation: Some individuals are chosen randomly to be mutated and then a mutation point is randomly chosen. The character in the corresponding position of the string is changed. 5. Decoding: A new generation has been formed and the process is repeated until some stopping criteria has been reached. At this point the individual who is closest to the optimum is decoded and the process is repeated until we get an individual with high fitness value. 3.2 Proposed Algorithm The proposed algorithm is a modified genetic algorithm in which only one kind of chromosome is commonly employed for solving the MTSP. This technique involves using a single chromosome of length p + q and is referred to as the one chromosome technique. In this technique, the nodes are represented by a permutation of the integers from 1 to p. This permutation is partitioned into q sub-tours by the insertion of q negative integers (from 1 to q) that represent the change from one salesman to the next. A modified crossover is proposed based on the Order crossover. In this crossover, a randomly chosen crossover point divides the parent strings in left and right substrings. The right substrings of the parents are selected. After selection of nodes the process is the same as the order crossover. The only difference is that instead of selecting random several positions in a parent tour, all the positions to the right of the randomly chosen crossover point are selected. Two mutations are used in the proposed algorithm. These operators select randomly two points in the string, and it replaces together or reverses the substring between these two cut points. This algorithm is coupled with a local search technique to obtain a better solution as compared to previous iterations. The search tries to improve route by replacing its two non-adjacent edges by two other edges. It should be noted that there are several routes for connecting nodes and producing the tour again, but a state that satisfies the problem s constraints is acceptable. So if, first, the above constraints are not violated and, second, the new tour produces a better value for the problem than the previous solution then only the unique tour will be accepted. The process is repeated until there is no possibility of reduction of route length. 4. COMPUTATIONAL RESULTS The proposed algorithm is coded in MATLAB. To reveal the performance of the algorithm it was tested with a big number of problems, only an illustrative result is presented here in the fig.1. As it was, tiny refinements in constraints are in progress. The example represents a whole process of a real problem s solution as an example same number of 5 salesperson with each salesperson were allowed to travel a minimum of 3 cities and fixed number of cities. The first step is the determination of the distance matrix. The input data is given by a map as it can see on Fig. 1 and a portion of the resulted distance table is shown on Table 1. It contains 50 locations (with the depot). The task is to determine the optimal routes for these locations. After distance table determination, the optimizer algorithm can be executed to determine the optimal routes using the proposed algorithm. Fig.1 Plot of city locations Volume 3, Issue 1, January 2014 Page 427

Fig.2 The result of the optimization in MATLAB The first run optimized the solution to a distance of 67.9625 units after 1227 iterations. Table 1 shows the optimal solution for the MTSP with 50 cities traversed by 3 salespersons. Results reveal the minimum distances travelled by the salesperson along with the number of iterations. Table 1: Performance of MTSP based on Iterations Total Iteration Distanc e 68.4053 1638 67.5306 1816 67.4341 1804 67.9625 1227 Fig.3 The Solution History. 5. Analysis This paper presents the technique to solve the multiple travelling salesperson problem using a modified genetic algorithm. As per the requirement & the application, the minimum number of cities travelled by each salesperson can be varied and so could be the number of cities & the number of salesperson. As the number of cities is increased, the computational complexity increases. So the proposed work will provide an effective solution to the problem in decreased time. The time taken by the algorithm for the above mentioned iterations was also analyzed. In every case the running time was nearly 30 seconds. The algorithm is sensitive for the number of iterations. The running time is proportional to iteration number. 6. Conclusion & Future Directions Volume 3, Issue 1, January 2014 Page 428

In this paper, a modified Genetic Algorithm was proposed which, along with the local search technique, improvised the solution at each iteration of the modified GA. For large-size problems, this proposed algorithm yields better results and the run time is also optimized. It is also assumed that the combination of this algorithm with other algorithms including prioritized GA, simulated annealing will yield better results. This proposed algorithm is also suggested for future research in applications like vehicle routing, computer wiring, drilling of printed circuit boards, overhauling gas turbine engines, X-Ray crystallography, order-picking problem in warehouses, mask plotting in PCB production, interview scheduling problem, mission planning problem, design of global navigation satellite system surveying networks, manufacture of microchips and DNA sequencing. REFERENCES [1] Sabah Sadiq, The travelling Salesperson Problem: Optimizing Delivery Routes using Genetic Algorithm, 161, 2012 [2] Mohammad Sedighpour, Majid Yousefikhoshbakht, Narges Mahmoodi Darani, An effective Genetic Algorithm for Solving the Multiple Travelling Salesman problem, Journal of Optimization in Industrial Engineering, 73-79, 2011 [3] R. D. Angel, W. L. Caudle, R. Noonan, A. Whinson, Computer assisted school bus scheduling. Management Science, 18, 279-288, 1972. [4] S. R. Balachandar, K. Kannan, Randomized gravitational emulation search algorithm for symmetric traveling salesman Problem, Applied Mathematics and Computation. 192(2), 413-421, 2007. [5] T. Bektas, The multiple traveling salesman problem: an overview of formulations and solution procedures. Omega. 34, 209 219, 2006. [6] L. Bianchi, J. Knowles, N. Bowler, Local search for the probabilistic traveling salesman problem: Correction to the 2- p-opt and 1-shift algorithms. European Journal of Operational Research. 162(1), 206-219, 2005. [7] B. Bontoux, C. Artigues, D. Feillet, A Memetic algorithm with a large neighborhood crossover operator for the generalized traveling salesman problem. Computers & Operations Research. 37(11), 1844-1852, 2010. [8] Cavill R, Smith S, Tyrrell A (2005) Multi-chromosomal genetic programming. In: Proceedings of the 2005 conference on Genetic and evolutionary computation, ACM New York, NY, USA, pp 1753 1759 [9] N. Christofides, S. Eilon, An algorithm for the vehicle dispatching problem. Operations Research Quarterly. 20, 309-318, 1969. [10] J. F. Cordeau, M. Dell Amico, M. Iori, Branch-and-cut forthe pickup and delivery traveling salesman problem with FIFO loading. Computers & Operations Research. 37(5), 970-980, 2010. [11] Bhide S, John N, Kabuka MR (1993) A boolean neural network approach for the traveling salesman problem. IEEE Transactions on Computers 42(10):1271 [12] S. Ghafurian, N. Javadian, An ant colony algorithm for solving fixed destination multi-depot multiple traveling salesmen problems. Applied Soft Computing. 11(1), 1256-1262, 2011. [13] F. Glover, Future paths for integer programming and links to artificial intelligence. Computers Operations Research. 13(5), 533.549, 1986. [14] P. Junjie, W. Dingwei, An ant colony optimization algorithm for multiple traveling salesman Problem, In ICICIC 06: Proceedings of the First International Conference on Innovative Computing. Information and Control, 210 213, 2006. [15] D. Karapetyan, G. Gutin, Lin-Kernighan heuristic adaptations for the generalized traveling salesman problem. European Journal of Operational Research. 208, 221 232, 2011. [16] D. Kaur, M. M. Murugappan, Performance enhancement in solving traveling salesman problem using hybrid genetic algorithm. Fuzzy Information Processing Society, NAFIPS. 1-6, 2008. [17] Ali AI, Kennington JL (1986) The asymmetric m-traveling salesmen problem: a duality based branch-and-bound algorithm. Discrete Applied Mathematics 13:259 276 [18] T. Tang, J. Liu, multiple traveling salesman problem model for hot rolling scheduling in Shanghai Baoshan Iron & Steel Complex. European Journal of Operational Research. 24, 267-282, 2000. [19] S.Yadlapalli, W. A. Alik, S. Darbha, M. Pachter, A Lagrangian-based algorithm for a Multiple Depot. Multiple Traveling Salesmen Problem, Nonlinear Analysis: Real World Applications. 10(4), 1990-1999, 2009. [20] W. Zhong, J. Zhang, W. Chen, A novel discrete particle swarm optimization to solve traveling salesman problem.proc. Evolutionary Computation. 3283 3287, 2007. Volume 3, Issue 1, January 2014 Page 429

AUTHORS Varunika Arya received the B.Tech degree in Electronics and Communication Engineering from Kurukshetra University in 2010. She is currently pursuing M.Tech in Electronics and Communication Engineering at Maharishi Markandeshwar University, Sadopur, Ambala, Haryana, under the guidance of Mr. Amit Goyal. Amit Goyal received the B.Tech and M.Tech degrees in Electronics and Communication Engineering from Punjab Technical University in 2002 and 2010, respectively. During 2002-2010, he worked as Asstt. Prof. in Electronics and Communication Engineering Department. His area of specialization is Neural Networks. His other areas of interest includes Wireless & Mobile Communication, Microprocessors, Computer Networking etc. He is currently working as Associate Professor in the Department of Electronics and Communication Engineering at Maharishi Markandeshwar University, Sadopur, Ambala, Haryana Vibhuti Jaiswal received the B.Tech degree in Electronics Engineering from Kurukshetra University and M.Tech degree in Electronics and Communication Engineering from Maharishi Markandeshwar University. She is working as Asstt. Prof. in the Department of Electronics and Communication Engineering at Maharishi Markandeshwar University, Sadopur, Ambala, Haryana. Her areas of interest includes Neuro-fuzzy System, Signals & Sytems, Network theory. Volume 3, Issue 1, January 2014 Page 430