Direktorat Pengembangan Bakat Minat dan Kesejahteraan Mahasiswa - Materi Workshop Penulisan Jurnal Mahasiswa 2012 Ridwan paper.pdf
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 wellknown 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
All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means
without the written permission of TTP, www.ttp.net. (ID: 202.162.37.68-05/12/12,04:05:46)
Advanced Materials Research Vol. 576 719
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 multidepot multiple TSP [7]. In the fixed destination 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.
Continuing to the work on PTSP, Liu (2010) investigated the application of GA to solve PTSP
[9]. In that investigation, the GA is ran using 3 different initial solution which are generated
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.
Based on review on related works above, it can be concluded, from the characteristic of the
case, the proposed research is similar with research proposed by Chao et al. (2007). The
difference of proposed research with the previous research is, in the investigated mrTSP, each
cluster or sub tour has only one interconnection point and only one salesman will perform all of
sub tours. The salesman also need to go back to initial point of big tour through the
interconnection point of each
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:
D
=∑
n
ld + ( ) i
=
21
i
(1)
=∀= D
1
kk
K (2)
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.
Crossover. Crossover operator used in this research is based on two cut points crossover with
modification, which is based on heuristic local shortest path algorithm. The algorithm tries to
arrange locus in a chromosome so that 2 adjacent locus has shortest distance. Algorithm to run
the proposed crossover can be explained as follow:
Advanced Materials Research Vol. 576 721
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
Case No of Point
mGA GA Comp. Time
Distance
Comp. Time
Distance
1 (20, 25, 32, 15, 35) 7.23s 2235,53 7.03s 2237,47 2 (25, 32, 17, 38, 25, 15, 28) 10.21s 2731,12
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.
Acknowledgment This work has been supported by superior scholarship P2D from Bureau of
Planning and Cooperation of Foreign Affairs (BPKLN) under Ministry of National Education,
Republic of Indonesia.
722 Advances in Manufacturing and Materials Engineering
References [1] Cheng, C.H., Gupta, Y.P., Lee, W.H. & Wong K.F. A TSP-based heuristic for
forming machine groups and part families. International Journal of Production Research, 36
(1998) 1325-1337. [2] Phillips, J.M., Punnen, A.P. & Kabadi, S.N. A linear time algorithm for the
bottleneck travelling salesman problem on a Halin graph. Information Processing Letters, 67
(1998) 105- 110. [3] Ozgur, C.O. & Brown, J.R. A two-stage travelling salesman procedure for
the single machine
sequence-dependent scheduling problem. Omega, Int. J. Mgmt. Sci, 23 (1995) 205-219. [4]
Schneider, J.J & Kirkpatrick, S. Stochastic optimization. Springer-Verlag Berlin HeidelbergJerman, 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
engineering. System Engineering Procedia, 4 (2012) 226-235.
Advances in Manufacturing and Materials Engineering
10.4028/www.scientific.net/AMR.576
Solving Multiple Routes Travelling Salesman Problem Using Modified Genetic
Algorithm 10.4028/www.scientific.net/AMR.576.718
(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 wellknown 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
All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means
without the written permission of TTP, www.ttp.net. (ID: 202.162.37.68-05/12/12,04:05:46)
Advanced Materials Research Vol. 576 719
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 multidepot multiple TSP [7]. In the fixed destination 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.
Continuing to the work on PTSP, Liu (2010) investigated the application of GA to solve PTSP
[9]. In that investigation, the GA is ran using 3 different initial solution which are generated
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.
Based on review on related works above, it can be concluded, from the characteristic of the
case, the proposed research is similar with research proposed by Chao et al. (2007). The
difference of proposed research with the previous research is, in the investigated mrTSP, each
cluster or sub tour has only one interconnection point and only one salesman will perform all of
sub tours. The salesman also need to go back to initial point of big tour through the
interconnection point of each
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:
D
=∑
n
ld + ( ) i
=
21
i
(1)
=∀= D
1
kk
K (2)
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.
Crossover. Crossover operator used in this research is based on two cut points crossover with
modification, which is based on heuristic local shortest path algorithm. The algorithm tries to
arrange locus in a chromosome so that 2 adjacent locus has shortest distance. Algorithm to run
the proposed crossover can be explained as follow:
Advanced Materials Research Vol. 576 721
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
Case No of Point
mGA GA Comp. Time
Distance
Comp. Time
Distance
1 (20, 25, 32, 15, 35) 7.23s 2235,53 7.03s 2237,47 2 (25, 32, 17, 38, 25, 15, 28) 10.21s 2731,12
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.
Acknowledgment This work has been supported by superior scholarship P2D from Bureau of
Planning and Cooperation of Foreign Affairs (BPKLN) under Ministry of National Education,
Republic of Indonesia.
722 Advances in Manufacturing and Materials Engineering
References [1] Cheng, C.H., Gupta, Y.P., Lee, W.H. & Wong K.F. A TSP-based heuristic for
forming machine groups and part families. International Journal of Production Research, 36
(1998) 1325-1337. [2] Phillips, J.M., Punnen, A.P. & Kabadi, S.N. A linear time algorithm for the
bottleneck travelling salesman problem on a Halin graph. Information Processing Letters, 67
(1998) 105- 110. [3] Ozgur, C.O. & Brown, J.R. A two-stage travelling salesman procedure for
the single machine
sequence-dependent scheduling problem. Omega, Int. J. Mgmt. Sci, 23 (1995) 205-219. [4]
Schneider, J.J & Kirkpatrick, S. Stochastic optimization. Springer-Verlag Berlin HeidelbergJerman, 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
engineering. System Engineering Procedia, 4 (2012) 226-235.
Advances in Manufacturing and Materials Engineering
10.4028/www.scientific.net/AMR.576
Solving Multiple Routes Travelling Salesman Problem Using Modified Genetic
Algorithm 10.4028/www.scientific.net/AMR.576.718