Advanced Online available Materials since Research 2012/Oct/08 Vol. 576 at www.scientific.net (2012) pp 718-722
© (2012) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMR.576.718
Solving Multiple Routes Travelling Salesman Problem using Modified Genetic Algorithm Muhammad Ridwan Andi Purnomo1,a, Mohammad Iqbal2,b, Mila Faila Sufa3,c 1Department of Industrial Engineering, Faculty of Industrial Technology, Universitas Islam Indonesia, Indonesia 2Department of
Manufacturing and Material Engineering, Faculty of Engineering International Islamic University of Malaysia, Malaysia 3Department of Industrial Engineering, Faculty of Engineering Universitas Muhammadiyah Surakarta, Indonesia aridwan_ie@uii.ac.id, bmohammad_iqbal@iium.edu.my,
cmfsisonline@gmail.com
Keywords: multiple routes, TSP, modified GA, heuristic crossover, heuristic mutation.
Abstract. The multiple routes travelling salesman problem (mrTSP) is an extension of the well-known travelling salesman problem (TSP), where there are several points clusters to be visited by salesman. The problem to be solved is how to define the best route in every cluster and initial position of each routes as interconnection points for the salesman. In this paper, modified genetic algorithm (mGA) is proposed in order to solve the mrTSP problem. In the proposed mGA, new heuristic algorithm for crossover and mutation operator based on local shortest path algorithm is proposed in order to assist the mGA to improve 'best solution so far'. Numerical examples are also given to test the performance of proposed mGA when solving mrTSP. The result of the study shows that the mGA is superior compared to conventional GA.
Introduction Travelling salesman problem (TSP) based research has received a great deal of attention from researchers. There are several real world problem that can be modelled using the TSP concept, such as manufacturing cell formation [1], optimisation of Halin graph [2], machine scheduling problem [3] and supply chain distribution network design [4]. In conventional TSP, the decision variable is order of each point to be visited by the salesman with minimum total distance. The salesman needs to visit all of the points and go back to initial point.
One of the extensions of TSP problem is multiple routes TSP (mrTSP). In the mrTSP, there are several clusters of points to be visited by the salesman. Hence, the decision variables are the sequence of each point in a cluster and the initial point of each cluster as the terminal point for the salesman. The decision variables determination is subject to minimum total distance. Figure 1 shows the example of mrTSP solution.
Fig. 1: Example of mrTSP solution In conventional TSP, the salesman needs to go back to initial position, hence, the last visited point will be connected directly to initial position. In the mrTSP, the salesman will go back to initial position through the terminal point of each points clusters.
Therefore, mrTSP problem will become
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more complicated since the determination of terminal points must consider not only the distance to go to another cluster but also the distance to go back through the same way. In this research, a modified genetic algorithm (mGA) is proposed to overcome the problem complexity of mrTSP.
Related works. Until now, there is no formal algorithm to solve TSP and also mrTSP. Hence, several heuristic algorithms have been investigated by previous researchers. Besides, the use of artificial intelligent techniques have been widely investigated to solve TSP and mrTSP. The development of TSP research is not only from the tools for problem-solving but also from the TSP case itself and leads to more complicated to be solved.
Paletta (2002) proposed simple heuristic algorithm to solve period TSP [5]. The concept of proposed algorithm is generating random tour and embedding a procedure to improve the generated tour. The embedded improvement procedure is executed based on remove-insertion technique by considering improvement value. If improvement value is increased then the new solution based on the embedded improvement procedure will be accepted as new solution, and vice versa. Based on the investigation, it shows that the new algorithm finds a larger number of best solutions than other extant algorithms. Chao et al. (2007) investigated clustered TSP and solve it using two level GA (TLGA) [6]. In the proposed TLGA, the objective of the lower level is to produce clusters of Hamiltonian cycle like in standard GA when used to solve TSP. The higher level GA has objective to form an entire tour that is as short as possible based on the tours generated in the lower level. Based on performance analysis that has been conducted, the performance of proposed TLGA is superior compared to conventional GA and comparable to other algorithm when used to solve large scale multi routes TSP.
Ghafurian & Javadian (2007) applied ant colony algorithm to solve fixed destination multi-depot multiple TSP [7]. In the fixed destination multi multi-depot multiple TSP, there are several sub tours to be visited by several salesman. In each sub tour, the salesman will start from a point and will going back to the initial point after performing the tour. In that investigation, the performance of proposed ant colony algorithm is compared with an existing commercial software, which is Lingo 8.0, and it shows that the performance of proposed ant colony algorithm is superior. Liu (2007) investigated about probabilistic TSP (PTSP), which each node has probability to being visited in a range of 0 to 1 [8]. PTSP is not like TSP, which each node absolutely certain to be visited. In that investigation, a hybrid scatter search (HSS) algorithm, which incorporating the nearest neighbour rule, threshold accepting and edge recombination, is proposed to solve the PTSP. Based on the investigation, it shows that the proposed HSS can effectively solve the PTSP and the incorporating threshold accepting into the scatter search framework can increase the computation efficiency while maintaining solution quality. These findings show the potential of the proposed HSS in solving the large-scale PTSP.
based on nearest neighbourhood (NN) type 1, type 2 and random solution. The difference between NN type 1 and type 2 is at the insertion technique. Based on that investigation, it shows that in GA to solve PTSP, different initial solution resulting different solution. In the investigated case, NN type 1 is superior compared to others while RAN is the worst one. Comparison study on the performance of several intelligent algorithm to solve TSP has been conducted by Hui (2012) [10]. In that study, GA, Hopfield neural network and ant colony algorithm have been compared in the perspectives of time complexity, space complexity, the advantages and disadvantages of the calculation results, and difficulty level of realisation. A comprehensive evaluation from engineering angle for each intelligent algorithm has also been conducted based on paired comparison matrix and it shows that ant colony algorithm is superior followed by GA and Hopfiled neural network.
sub tour. The used tool for problem solving is also different. In proposed research, modification of GA is conducted at crossover and mutation operation. Heuristic algorithm based on local shortest path is applied at crossover and mutation operation in order to "guide" the GA to improve best solution so far.
The mGA. Basically, the process of mGA is similar with the standard GA. Following subsections explain the process of mGA to be used to solve mrTSP.
Solution encoding. In GA, solution of a problem needs to be encoded in the form of chromosome. A chromosome could consist of one or more than one genes depend on the type of solution to be searched. Since each points cluster can be identified clearly, then solution for each points cluster will be encoded in a gene separately. In the gene, there will be locus with the number of locus is same with the number of points in the cluster. The gene will be followed by another gene that consists of only one locus, to be used to encode interconnection point in a points cluster. It can be concluded, if there are n points cluster, then there will be n x 2 genes in a chromosome. Fig. 2 shows the chromosome and the encoded solution.
720 Advances in Manufacturing and Materials Engineering
Fig. 2: Chromosome and the encoded solution
Fitness function. Fitness function used in this research is the function of total distance of all points cluster added with double distance between each interconnection point in each points cluster. The concept of GA is keeping strong chromosome, improve its performance until last generation. Hence, it can be viewed as maximisation case. Contrary to the mrTSP as minimisation case, then fitness function of each chromosome is the inverse value of the total distance. It can be formulated mathematically as follow:
where: D = total distance
d = distance of points cluster i l = total distance of interconnection points Fitness
k
Fitness
k
...3,2,1,,
= fitness value of chromosome k i = index of points cluster n = number of points cluster k = index of chromosome K = number of chromosome in a population
Reproduction and selection. In this research, reproduction and selection mechanism uses well known technique which is roulette wheel selection. This technique enables proportional selection in a chromosome population based on its fitness value.
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Step 1 : Define first and second cut points randomly Step 2 : Set sub chromosome for the first child chromosome from the first parent chromosome
with refer to the both cut points. Step 3 : For each locus in second parent chromosome which is not around in the first child chromosome, will be placed at the first child chromosome based on the shortest distance from the most right locus or the most left locus (if any) Step 4 : Going back to Step 2 to be applied for second parent chromosome and second child
chromosome The proposed crossover algorithm will produce two child chromosomes with local shortest distance with still considering randomness factor in GA, which is one of the advantages of GA. The crossover operator is applied only for gene which represents tour sequence of a points cluster.
Mutation. Mutation operator used in this research is insertion mutation based on local shortest path as well. Algorithm of the proposed mutation operator is as follow:
Step 1 : Define position of locus to be mutated randomly Step 2 : Insert selected locus to a position which resulting shortest distance between the locus and the locus to the left and the locus to the right.
The algorithm above is applied only for gene which represents tour sequence of a points cluster. Mutation operator for gene represents interconnection point of each points cluster is simple flip mutation, that will flip value of selected gene to another value.
Sampling space. To keep all of the information produced by new child chromosomes, then enlarged sampling space is used. All of produced child chromosomes are gathered with parent chromosomes and will be selected based on the performance. If the main sampling space is already full, then the rest of the chromosomes will be removed and considered as not-survived chromosomes.
Performance evaluation. Five cases of mrTSP have been used to evaluate the performance of proposed mGA. The evaluation included comparison with conventional GA. Both mGA and GA are coded using Microsoft Visual C# and has been ran in a personal computer with I7 2.4GHZ processor speed and 8G of RAM. Table 1 shows the evaluation result.
Table 1: Performance comparison of mGA and GA for solving mrTSP
9.97s 2817,87 3 (15, 27, 35, 27, 29, 32, 37, 32) 13.85s 2987,08 13.57s 2990,21 4 (23, 18, 35, 30, 17, 28, 29, 31, 19) 17.02s 3212,12 16.89s 3315,47 5 (30, 15, 35, 17, 30, 23, 19, 23, 28, 18) 22.16s 3845,64 21.82s 4215,98
Conclusion The local shortest path algorithm used in the crossover and mutation is able to assist mGA to improve the chromosomes through local paths distance minimisation. Further, the chromosomes are improved continuously through evolution process in the mGA. Based on the conducted experiments, it can be concluded that the performance of proposed mGA is superior compared to conventional GA. The proposed mGA also have potentiality to be used to solve large scale mrTSP problem if it has been analysed from the computational time and also quality of the solution.
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sequence-dependent scheduling problem. Omega, Int. J. Mgmt. Sci, 23 (1995) 205-219. [4] Schneider, J.J & Kirkpatrick, S. Stochastic optimization. Springer-Verlag Berlin
Heidelberg-Jerman, 2006. [5] Paletta, G. The period traveling salesman problem: a new heuristic algorithm. Computer &
Operation Research, 29 (2002) 1343-1352. [6] Chao, D., Ye, C. & Miao, H. Two-level genetic algorithm for clustered traveling salesman problem with application in large-scale TSPs. Tsinghua Science and Technology, 12 (2007) 459-465. [7] Ghafurian, S. & Jafarian, N. An ant colony algorithm for solving fixed destination multi-depot
multi traveling salesman problem. Applied Soft Computing, 11 (2007) 1256-1262. [8] Liu, Y.H. A hybrid scatter search for the probabilistic traveling salesman problem. Computer & Operation Research, 34 (2007) 2949-2963. [9] Liu, Y.H. Different initial solution generators in genetic algorithms for solving the probabilistic traveling salesman problem. Applied Mathematics and Computation, 216 (2010) 125-137. [10] Hui, W. Comparison of several intelligent algorithms for solving TSP problem in industrial
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