Research Article The Ordered Clustered Travelling Salesman Problem: A Hybrid Genetic...

14
Research Article The Ordered Clustered Travelling Salesman Problem: A Hybrid Genetic Algorithm Zakir Hussain Ahmed Department of Computer Science, Al Imam Mohammad Ibn Saud Islamic University (IMSIU), P.O. Box 5701, Riyadh 11432, Saudi Arabia Correspondence should be addressed to Zakir Hussain Ahmed; [email protected] Received 28 August 2013; Accepted 30 December 2013; Published 19 February 2014 Academic Editors: M. Rˇ adulescu, L. Scrimali, and W. Szeto Copyright © 2014 Zakir Hussain Ahmed. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. e ordered clustered travelling salesman problem is a variation of the usual travelling salesman problem in which a set of vertices (except the starting vertex) of the network is divided into some prespecified clusters. e objective is to find the least cost Hamiltonian tour in which vertices of any cluster are visited contiguously and the clusters are visited in the prespecified order. e problem is NP-hard, and it arises in practical transportation and sequencing problems. is paper develops a hybrid genetic algorithm using sequential constructive crossover, 2-opt search, and a local search for obtaining heuristic solution to the problem. e efficiency of the algorithm has been examined against two existing algorithms for some asymmetric and symmetric TSPLIB instances of various sizes. e computational results show that the proposed algorithm is very effective in terms of solution quality and computational time. Finally, we present solution to some more symmetric TSPLIB instances. 1. Introduction e clustered travelling salesman problem (CTSP), intro- duced by Chisman [1], is a variation of the usual travelling salesman problem (TSP). It can be defined as follows: let = (,) be a complete undirected graph with vertex set and edge set . e vertex set ={V 1 , V 2 ,..., V }, except the starting vertex (depot) V 1 , is partitioned into prespecified clusters 1 , 2 ,..., . e number of vertices in the clusters (i.e., size of the clusters) is 1 , 2 ,..., , respectively. A cost matrix = [ ] representing travel costs, distances, or travel times is defined on the edge set = {(V , V ): V , V ∈, ̸ = }. Starting from the depot V 1 , the objective of the CTSP is to determine the least cost Hamiltonian tour on in which the vertices of any cluster are visited contiguously, and the clusters are visited in the order 1 , 2 ,..., . ere are several variants of the problem depending on whether the start and end vertices of a cluster as well as the number and order of clusters have been specified. If the number of clusters is either one or each cluster has only one vertex, then the problem becomes the usual TSP. If the number of clusters is two then the problem is called TSP with backhauls (TSPB) [2]. In the free CTSP, the cluster order is not prespecified and the problem is to simultaneously determine the optimal cluster order as well as the routing within and between clusters. is paper focuses on the variant with specified order of clusters and unspecified end vertices of the clusters, which is called ordered CTSP (OCTSP). For simplicity, we label the vertices as natural numbers from 1 to and, thus, assume that the label of the vertices of any cluster is less than the label of the vertices of the following clusters. Since all the variations are generalization of the usual TSP, they all are NP-hard [3]. e CTSP has many appli- cations in real life, for example, in automated warehouse routing [1, 4], in production planning [4], and in vehicle routing, manufacturing, computer operations, examination timetabling, cytological testing, and integrated circuit testing [5, 6]. Chisman [1] showed that the CTSP can be transformed into a TSP by adding or subtracting an arbitrarily large constant to or from the cost of each intercluster edge. erefore, aſter the transformation, any exact algorithm for the TSP can be applied to solve the problem exactly. However, Hindawi Publishing Corporation e Scientific World Journal Volume 2014, Article ID 258207, 13 pages http://dx.doi.org/10.1155/2014/258207

Transcript of Research Article The Ordered Clustered Travelling Salesman Problem: A Hybrid Genetic...

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Research ArticleThe Ordered Clustered Travelling Salesman ProblemA Hybrid Genetic Algorithm

Zakir Hussain Ahmed

Department of Computer Science Al ImamMohammad Ibn Saud Islamic University (IMSIU) PO Box 5701Riyadh 11432 Saudi Arabia

Correspondence should be addressed to Zakir Hussain Ahmed zhahmedgmailcom

Received 28 August 2013 Accepted 30 December 2013 Published 19 February 2014

Academic Editors M Radulescu L Scrimali and W Szeto

Copyright copy 2014 Zakir Hussain Ahmed This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The ordered clustered travelling salesman problem is a variation of the usual travelling salesman problem in which a set ofvertices (except the starting vertex) of the network is divided into some prespecified clusters The objective is to find the leastcost Hamiltonian tour in which vertices of any cluster are visited contiguously and the clusters are visited in the prespecified orderThe problem is NP-hard and it arises in practical transportation and sequencing problems This paper develops a hybrid geneticalgorithm using sequential constructive crossover 2-opt search and a local search for obtaining heuristic solution to the problemThe efficiency of the algorithm has been examined against two existing algorithms for some asymmetric and symmetric TSPLIBinstances of various sizes The computational results show that the proposed algorithm is very effective in terms of solution qualityand computational time Finally we present solution to some more symmetric TSPLIB instances

1 Introduction

The clustered travelling salesman problem (CTSP) intro-duced by Chisman [1] is a variation of the usual travellingsalesman problem (TSP) It can be defined as follows let119866 = (119881 119864) be a complete undirected graph with vertex set 119881

and edge set 119864 The vertex set 119881 = V1 V2 V

119899 except the

starting vertex (depot) V1 is partitioned into 119898 prespecified

clusters 1198811 1198812 119881

119898 The number of vertices in the clusters

(ie size of the clusters) is 1198991 1198992 119899

119898 respectively A cost

matrix 119862 = [119888119894119895] representing travel costs distances or travel

times is defined on the edge set119864 = (V119894 V119895) V119894 V119895

isin 119881 119894 = 119895Starting from the depot V

1 the objective of the CTSP is to

determine the least cost Hamiltonian tour on 119866 in which thevertices of any cluster 119881

119896are visited contiguously and the

clusters are visited in the order 1198811 1198812 119881

119898

There are several variants of the problem depending onwhether the start and end vertices of a cluster as well asthe number and order of clusters have been specified If thenumber of clusters is either one or each cluster has onlyone vertex then the problem becomes the usual TSP If the

number of clusters is two then the problem is called TSPwith backhauls (TSPB) [2] In the free CTSP the clusterorder is not prespecified and the problem is to simultaneouslydetermine the optimal cluster order as well as the routingwithin and between clustersThis paper focuses on the variantwith specified order of clusters and unspecified end verticesof the clusters which is called ordered CTSP (OCTSP) Forsimplicity we label the vertices as natural numbers from 1 to119899 and thus assume that the label of the vertices of any clusteris less than the label of the vertices of the following clusters

Since all the variations are generalization of the usualTSP they all are NP-hard [3] The CTSP has many appli-cations in real life for example in automated warehouserouting [1 4] in production planning [4] and in vehiclerouting manufacturing computer operations examinationtimetabling cytological testing and integrated circuit testing[5 6] Chisman [1] showed that the CTSP can be transformedinto a TSP by adding or subtracting an arbitrarily largeconstant 119872 to or from the cost of each intercluster edgeTherefore after the transformation any exact algorithm forthe TSP can be applied to solve the problem exactly However

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 258207 13 pageshttpdxdoiorg1011552014258207

2 The Scientific World Journal

as the size increases finding exact optimal solution to theCTSP instances becomes impractical and hence heuristicmust be used

We seek approximate solution using heuristic algorithmfor the OCTSP For the TSP and related problems well-known heuristic algorithms are genetic algorithms (GAs)tabu search (TS) artificial neural network (ANN) simu-lated annealing (SA) approximate algorithms and so forthAmongst the heuristics GAs are found to be the bestalgorithms for the TSP and its variations Since OCTSP is avariation of the usual TSP therefore we develop a hybrid GA(HGA) using sequential constructive crossover [7] and 2-optsearch and a local search [8] to obtain heuristically optimalsolution to the problem The efficiency of our algorithmhas been examined against partitioning algorithm [9] forsome medium sized asymmetric TSPLIB [10] instances andlexisearch algorithm [11] for some small sized symmetricTSPLIB [10] instances The computational experiments showthe effectiveness of our proposed HGA Finally we presentsolution to some medium sized symmetric TSPLIB [10]instances However to the best of our knowledge no litera-ture presents solution to these symmetric instances Hencewe could not provide any comparative study of these results

This paper is organized as follows Section 2 presents adetailed literature review to the problem A hybrid geneticalgorithm is developed and reported in Section 3 Com-putational experiment for the algorithm is presented inSection 4 Finally Section 5 presents comments and conclud-ing remarks

2 Literature Review

Chisman [1] transformed the CTSP to the usual TSP and thenapplied branch and bound approach [12] to solve the problemexactly but did not obtain very good resultsThereafter Lokin[4] and Jongens and Volgenant [13] applied exact algorithmsto find exact optimal solution to the problem Aramgiatisiris[9] developed an exact partitioning algorithm (LBDCOMPtherein) by transforming the problem to the TSPB thensolving independently linehaul and backhaul subproblemsand finally reformulating as a direct shortest path on thebipartite graph problems However the algorithm does notreally obtain exact solutions for many instances [11]

An approximation algorithm with good empirical per-formance [14] was developed to solve the problem with aprespecified order on the clusters Also three more heuristicswere proposed and compared among them As reportedthe best results were obtained by the heuristic that firsttransforms the problem into a TSP and then applies theGENIUS heuristic

Laporte et al [5] proposed a TS heuristic that combinedwith a phase of diversification using a GA to solve theproblem with a prespecified order of visiting the clusters Asreported the TS outperforms the GA [15] that exploits order-based crossover operators and local search heuristics How-ever when comparing TS with a postoptimization procedure[14] TS obtained better quality of solutions but requiredmorecomputational time

Another GA was developed for the problem [16] that firstfinds intercluster paths and then intracluster paths Finallya comparative study of the GA was presented against aGENIUS heuristic [14] and lower bounds [13] As reportedthe GA could solve instances up to 500 vertices with 4 and10 clusters and obtained solutions within 55 of the lowerbound

An approximation algorithm of 53 performance ratiohas been developed for the OCTSP with a prespecifiedvisiting sequence for the clusters [17] Another approximationalgorithm has been proposed which guarantees boundedperformance for some variants of the CTSP [3] For theproblemwith unspecified end vertices its algorithm first usesamodified Christofidesrsquo algorithm [18] to get the shortest freeendsHamiltonian paths in each cluster After the first step thetwo end vertices for each cluster and the intracluster paths arespecifiedThen a rural postman problem algorithm is used toconnect all the intracluster paths to form a whole tour Thisalgorithm favours the intracluster Hamiltonian paths whichimplies that the inter-cluster pathsmay be sacrificedwhen theend vertices in each cluster are already determined

A two-level-TSP hierarchical algorithm that favoursintercluster paths has been proposed for the CTSP [19] Firstthe shortest intercluster paths connecting every cluster havebeen specified then the start and end vertices are specified foreach cluster Next a modified Christofidesrsquo algorithm [18] isused to get the shortest Hamiltonian paths with two specifiedend vertices in each cluster At the end a whole tour is formedby combining the paths generated in both levels They alsoshowed that the penalties caused by favoring the intraclusterHamiltonian paths and the intercluster paths are comparable

A two-level GA (TLGA) has been developed for solvingCTSP with unspecified end vertices [20] The algorithm firstfinds the shortest Hamiltonian cycle for each cluster andthen connects all the intracluster paths in a certain sequenceto form a whole tour In the lower level a GA is used tofind the shortest Hamiltonian cycle rather than the shortestHamiltonian path for each cluster In the higher level amodified GA is designed to determine an edge that will bedeleted from the shortest Hamiltonian cycle for each clusterand the visiting sequence of all the clusters with the objectiveof shortest travelling tour for the whole problem The higherlevel algorithm has the freedom to delete any edge of theclusters while searching for the shortest complete tour Testresults demonstrate that the TLGA for large TSPs is moreeffective and efficient than the classical GA

3 A Hybrid Genetic Algorithm for the OCTSP

31 A Brief Overview of GAs GAs are structured yet ran-domized search methods based on mimicking the survivalof the fittest among the species generated by random changesin the gene structure of the chromosomes in the evolutionarybiology [21] They start with a population of chromosomes(solutions) that evolve from one generation to the next Eachgeneration consists of the following three operations

(a) Selection This procedure is a stochastic process thatmimics the ldquosurvival-of-fittestrdquo theory However here

The Scientific World Journal 3

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3 5

4

2

76

Figure 1 An example of result of the OCTSP using GA

no new chromosome is created Some of the chromo-somes are copied (even more than once) to the nextgeneration probabilistically based on their objectivefunction value whereas some other chromosomes arediscarded

(b) Crossover It is a binary operator that applies to twoparent chromosomes with a large probability whichcreates new offspring chromosome(s) It is a veryimportant operator in GAs Also crossover operatortogether with selection operator is found to be themost powerful process in the GA search

(c) Mutation It is a unary operator that applies to eachof the chromosomes with a small probability It is theoccasional random change of some selected gene(s) ofa chromosome to diversify the GA search space

Starting from a randomly generated or heuristicallygenerated initial population the GAs search repeated theabove three operators until the stopping criterion is satisfiedCrossover operator is a unique feature of GAs that wishesto combine good quality parent chromosomes to createone or more new offspring chromosome(s) However it isseen that the crossover alone cannot generate high qualitychromosomes for the combinatorial optimization problemslike the TSP and its variations Thus powerful local searchmethods are incorporated to improve the quality of offspringchromosomes [5] In hybrid GAs crossover operator gener-ates new starting solutions for the local search methods

GAs are found to be successful heuristic algorithms forsolving the usual TSP and its variations However GAs donot guarantee the optimality of the solution but they can findvery good near optimal solution in very short time We areapplying crossover mutation and local search methods foreach cluster in the prespecified order for the OCTSP Resultof the GA for a 7-vertex problem instance is a complete touras shown in Figure 1

32 Bias Removal Bias removal step is adopted in lexisearchalgorithm [11] and found effective for the CTSP We alsoconsider the bias removal step in ourGAThemain advantageof the bias removal is that a large amount of the solutionvalue is kept fixed and for the remaining small value wehave to search The process for bias removal of the costmatrix is as follows subtract each row-minima from its

Table 1 The cost matrix with row-minima and column-minima

Vertex 1 2 3 4 5 6 7 Row-minima1 999 75 99 9 35 63 8 82 51 999 86 46 88 29 20 203 100 5 999 16 28 35 28 54 20 45 11 999 59 53 49 115 86 63 33 65 999 76 72 336 36 53 89 31 21 999 52 217 58 31 43 67 52 60 999 32Column-minima 9 0 0 1 0 9 0

Table 2 The reduced cost matrix

Vertex 1 2 3 4 5 6 71 999lowast 67 91 0 27 46 02 22 999lowast 66 25 68 0 03 86 0 999lowast 10 23 21 234 0 34 0 999lowast 48 33 385 44 30 0 31 999lowast 34 396 6 32 68 9 0 999lowast 317 18 0 12 35 21 29 999lowast

(lowastElements are left as 999)

corresponding row elements repeat the same column-wiseon the resultant matrix The total of the row-minima and thesubsequent column-minima is called the ldquobiasrdquo of the matrixHowever we have not incorporated clusters precedencerelations in our cost matrix This does not affect the value ofa chromosome since while generating a chromosome theclusters precedence relations are taken care of

The bias of the cost matrix given in Table 1 is (row-minima + column-minima = 129 + 19 =) 148 The reducedcost matrix (ie after removing bias of the matrix) is shownin Table 2 We shall solve the problem with respect to thereduced cost matrix After we find solution value with respectto the reduced matrix we shall add the bias to the value forfinding the solution value with respect to the original costmatrix

33 Alphabet Table Alphabet matrix 119860 = [119886(119894 119895)] is asquare matrix of order 119899 formed by positions of elementsof the reduced cost matrix of order 119899 119862

1015840= [1198881015840

119894119895] when

they are arranged in the nondecreasing order of their costsAlphabet table ldquo[119886(119894 119895) minus 119888

1015840

119894119886(119894119895)]rdquo is the combination of

elements (vertices) of matrix A and their costs in the reducedmatrix [11] The alphabet table for the reduced cost matrixin Table 2 is shown in Table 3 This alphabet table also is nottaking care of the clusters precedence relations

34 Improved Initial Population The path representation fora chromosome is used in our GA In this representation anyvertex is assigned to a unique natural number from 1 to 119899that is genes are natural numbers The path of a salesmanis represented by a chromosome which is a permutation ofnumber genes A gene segment is defined as a permutation

4 The Scientific World Journal

Table 3 The alphabet table

Vertex 119881mdashCost 119881mdashCost 119881mdashCost 119881mdashCost 119881mdashCost 119881mdashCost 119881mdashCost1 4mdash0 7mdash0 5mdash27 6mdash46 2mdash67 3mdash91 1mdash9992 6mdash0 7mdash0 1mdash22 4mdash25 3mdash66 5mdash68 2mdash9993 2mdash0 4mdash10 6mdash21 5mdash23 7mdash23 1mdash86 3mdash9994 1mdash0 3mdash0 6mdash33 2mdash34 7mdash38 5mdash48 4mdash9995 3mdash0 2mdash30 4mdash31 6mdash34 7mdash39 1mdash44 5mdash9996 5mdash0 1mdash6 4mdash9 7mdash31 2mdash32 3mdash68 6mdash9997 2mdash0 3mdash12 1mdash18 5mdash21 6mdash29 4mdash35 7mdash999

of the vertices in a cluster A chromosome is a permutationof all the gene segments with one gene segment per clusterFor example let 1 2 3 4 5 6 7 be the vertices with 119881

1=

2 3 4 1198812

= 5 6 7 and 1198811is followed by 119881

2 in a 7-

vertex instance then starting from vertex 1 a complete tour1rarr 3rarr 4rarr 2rarr 6rarr 7rarr 5rarr 1 may be represented as thechromosome (1 3 4 2 6 7 5) where (3 4 2) and (6 7 5) arethe gene segments for cluster 1 and cluster 2 respectively

It is to be noted that starting from a good initialpopulation can deliver better quality of solutions quicklyand that is why many literatures report generating initialpopulation using heuristics Hence we are going to usesequential sampling algorithm for heuristically generatinginitial population that has been applied successfully to thebottleneck TSP [22]This algorithm is a simple version of thesequential constructive sampling algorithm [8] It is basicallya probabilistic method to generate a tour of the salesmanTheprobability of visiting each unvisited vertex of a cluster in arow of the alphabet table is assigned in such a way that thefirst unvisited vertex gets more probability than the secondone and so on Thereafter cumulative probability of eachunvisited vertex of a cluster is calculated Next a randomnumber 119903 isin [0 1] is generated and the vertex that representsthe chosen random number in the cumulative probabilityrange is accepted The probability of visiting each unvisitedvertex of a cluster is assigned as follows Suppose the numberof unvisited vertices of a cluster in a row of the alphabet tableis 119896 The probability of visiting the 119894th unvisited vertex is

119901119894=

2 (119896 minus 119894 + 1)

119896 (119896 + 1) (1)

The algorithm may be summarized as follows

Step 0 Construct the ldquoalphabet tablerdquo based on the reducedcost matrix Repeat the following steps for the fixed popula-tion size (119875

119904)

Step 1 Since ldquovertex 1rdquo is the starting vertex so initialize 119901 =

1 and go to Step 2

Step 2 Go to the 119901th row of the ldquoalphabet tablerdquo and visitprobabilistically by using (1) any unvisited vertex of the row(say vertex 119902) in the present cluster and go to Step 3

Step 3 Rename the ldquovertex 119902rdquo as ldquovertex 119901rdquo and go to Step 4

Step 4 If all vertices of the present cluster are visited then goto the next cluster in the order (if any) and go to Step 5 elsego to Step 2

Step 5 If all vertices of the network are visited then go toStep 1 for generating another chromosome in the populationelse go to Step 2

Let us illustrate the algorithm through the example givenin Table 1 with 119881

1= 2 3 4 119881

2= 5 6 7 and 119881

1is followed

by 1198812 We start from 1st row of the ldquoalphabet tablerdquo The

number of unvisited vertices of the 1st cluster in the row is3 which are 4 2 and 3 with cumulative probabilities 05000833 and 1000 respectively Suppose the vertex 4 is selectedrandomly then the partial chromosome will be (1 4) Nextwe go to 4th row of the ldquoalphabet tablerdquo and probabilisticallyselect another node and so on Proceeding in this way it maylead to a complete chromosome (1 4 2 3 6 5 7)

A preliminary study shows the effectiveness of the sam-pling algorithm for initial population However instead ofconsidering all unvisited vertices if we consider at most firstten vertices in a cluster then the algorithm generates verygood population A similar observation has been reportedfor the bottleneck TSP also [22] Hence we consider thisrestricted domain of unvisited vertices of a cluster for ourstudy Further to start with better population we apply 2-optsearch to each chromosome The 2-opt search removes twoedges and then replaces them by a different set of edges insuch a way so as to maintain the feasibility of the tour Let 120572

119894

120572119894+1

120572119895 and 120572

119895+1be four vertices in a cluster then if the edges

(120572119894 120572119894+1

) and (120572119895 120572119895+1

) are removed the only way to form anew valid tour is to connect 120572

119894to 120572119895and 120572

119894+1to 120572119895+1

35 Fitness Function and Selection Method The objectivefunction of each chromosome in the population is the cost ofthe tour represented by the chromosomeThe fitness functionof a chromosome is defined as multiplicative inverse of theobjective functionThere are various selectionmethods in theliteratureThe selection operation considered for our study isthe stochastic remainder selection method [23]

36 The Sequential Constructive Crossover Operation Sincecrossover operation plays main role in GAs hence severalcrossover operators have been proposed for the usual TSPwhich are then used for the variant TSPs also Out of themthe sequential constructive crossover (SCX) [7] is found to

The Scientific World Journal 5

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1

3

5

4

2

7

6

1

3

5

4

2

7

6

(a) P1 (1 2 4 3 6 7 5) (b) P2 (1 2 4 3 6 7 5) (c) Offspring (1 2 4 3 6 5 7)

Figure 2 Example of sequential constructive crossover operation

be one of the best crossover operators for the usual TSP Amultiparent extension of the SCX has been applied to theusual TSP and found good results [24]The SCXhas also beensuccessfully applied to the TSP with some other variations[22 25] In general it produces an offspring using betteredges of the parents However it does not depend only onthe parentsrsquo structure it sometimes introduces new but goodedges to the offspring which are not even available in thepresent population We modify the SCX operator for theOCTSP as follows

Step 1 Start from ldquovertex 1rdquo (ie current vertex 119901 = 1)

Step 2 Sequentially search both of the parent chromosomesand consider the first unvisited vertex of the present clusterappearing after ldquovertex 119901rdquo in each parent If no unvisitedvertex after ldquovertex 119901rdquo is available in any (or both) of theparents search sequentially from the starting of that parentand consider the first unvisited vertex of the cluster and goto Step 3

Step 3 Suppose the ldquovertex 120572rdquo and the ldquovertex 120573rdquo are foundin the 1st and 2nd parents respectively then for selecting thenext vertex in the offspring chromosome go to Step 4

Step 4 If 119888119901120572

lt 119888119901120573 then select ldquovertex 120572rdquo otherwise select

ldquovertex120573rdquo as the next vertex and concatenate it to the partiallyconstructed offspring chromosome and go to Step 5

Step 5 If there is not any vertex left in that cluster thengo to the next cluster if any If the offspring is a completechromosome then stop otherwise rename the present vertexas ldquovertex 119901rdquo and go to Step 2

Let a pair of parent chromosomes be 1198751 (1 2 4 3 6

7 5) and 1198752 (1 3 2 4 6 5 7) with costs 357 and 354

respectively with respect to the original cost matrix givenin Table 1 By applying above SCX we obtain the offspringchromosome (1 2 4 3 6 5 7) with cost 318 which is lessthan both parentsTheparent and the offspring chromosomesare shown in Figure 2 In general crossover operator inheritsparentsrsquo characteristics and the operator that preserves goodcharacteristics of parents in the offspring is said to be goodoperatorThe SCX is found to be excellent in this regard Boldedges in Figure 2(c) are the edges which are available either in

the first parent or in second parent For this given example alledges are selected from either of the parents

For the crossover operation a pair of parent chromo-somes is selected sequentially from the mating pool It isreported that the SCX gets stuck in local minimums quicklyfor the TSP [7] which is very often due to the identicalpopulation So to overcome this situation the selectedparents are checked for duplication If the selected parentsare found to be identical then the second parent is modifiedtemporarily by swapping some randomly chosen pair of genesin the chromosome and then the crossover operation isperformed To improve quality of the solution as well as havea mixture of parents and offspring in a population the firstparent is replaced by the offspring only if the offspring valueis better than the average value of the present populationIn this way the mixed population retains diversity alsoTo further improve the quality of the solution obtained bycrossover many researchers applied 2-opt search operator Toimprove solution quality we are going to use a local searchmethod that combines three mutation operators that will bediscussed in Section 38 However we are not applying thislocal search method to all of the offspring rather it is appliedonly to the offspring if its value is better than the averagepopulation valueNow since our crossover operator producesonly one offspring to keep population size fixed throughoutthe generations while pairing with the next chromosome inorder the present second original parent chromosomewill beconsidered as the first parent and so on

37 Mutation Operation The mutation operator randomlyselects a position in the chromosome and changes the cor-responding gene thereby modifying information The needfor mutation comes from the fact that as the less fit chromo-somes of successive generations are discarded some aspectsof genetic material could be lost forever By performingoccasional random changes in the chromosomes GAs ensurethat newparts of the search space are reachedwhich selectionand crossover could not fully guarantee In doing somutationensures that no important features are prematurely lost thusmaintaining the mating pool diversity For this investigationwe have considered reciprocal exchange mutation operatorthat selects two genes randomly of a chromosome in everycluster and swaps them The probability of mutation isusually chosen to be considerably less than the probability

6 The Scientific World Journal

1

3

5

4

2

7

6

1

3

5

4

2

7

6

(a) (1 2 4 3 6 7 5) (b) (1 4 2 3 6 5 7)

Figure 3 Example of reciprocal exchange mutation operation

of crossover So mutation plays a secondary role in the GAsearch For example let the chromosome (1 2 4 3 6 7 5)be selected for mutation and vertices 2 and 4 are swapped incluster 1 and vertices 7 and 5 are swapped in cluster 2 thenthe mutated chromosome becomes (1 4 2 3 6 5 7) whichis shown in Figure 3 Bold edges in Figure 3(b) are the newedges in the mutated chromosome

38 A Local Search Method We have considered the com-bined mutation operation as a local search method whichhas been successfully applied to the bottleneck TSP [822] and maximum TSP [25] It combines three mutationoperators insertion inversion and reciprocal exchange withcent percentage of probabilities Insertion operator selects avertex (gene) in a chromosome and inserts it in a randomplace and inversion operator selects two points along thelength of a chromosome and reverses the subchromosomesbetween these points This local search a modification ofthe hybrid mutation operator [26] is applied to a chro-mosome Recall that sizes of the clusters 119881

1 1198812 119881

119898are

1198991 1198992 119899

119898 respectively Suppose (1 = 120572

1 1205722 1205723 120572

119899) is

a chromosome then the local search for the OCTSP can bedeveloped as follows

Step 0 Set 119909 = 2 and 119910 = 1

Step 1 For 119894 = 1 to 119898 perform Step 2

Step 2 Set 119910 = 119899119894+ 119910 and go to Step 3

Step 3 For 119895 = 119909 to (119910 minus 1) perform Step 4

Step 4 For 119896 = (119895 + 1) to 119910 perform Step 5

Step 5 If inserting vertex 120572119895after vertex 120572

119896reduces the

present tour cost then insert the vertex 120572119895after vertex 120572

119896 In

either case go to Step 6

Step 6 If inverting subchromosome between the vertices120572119895and 120572

119896reduces the present tour cost then invert the

subchromosome In either case go to Step 7

Step 7 If swapping the vertices 120572119895and 120572

119896reduces the present

tour cost then swap them In either case go to Step 8

Step 8 Set 119909 = 119910 + 1 and go to Step 1

39 Immigration It is seen that sometimes GAs get stuckin local minimums for the combinatorial optimization prob-lems which is very often due to the identical populationSo to improve capability of GAs the population should bediversified To diversify the population immigrationmethodis also adopted where some randomly selected chromosomesare replaced by new chromosomes after some generations[22] We are also considering an immigration method Forour investigation 20 of the population is replaced ran-domly using sequential sampling algorithm as discussed inSection 34 if no improvement is found within the last 20generations Once the immigration is applied we wait for thenext 20 generations for any improvement Our hybrid GA(HGA) for the OCTSP may be summarized as in Figure 4[22]

4 Results and Discussions

We encoded our HGA in Visual C++ executed on a PC with340GHz Intel(R) Core (TM) i7-3770CPU and 800GBRAMunderMSWindows 7 operating system and testedwith someTSPLIB [10] instances

41 Parameter Setting GAs are well suited for the com-binatorial optimization problems They find near optimalsolution in reasonable time However they are guided bysuitable choice of parameters namely crossover probability(119875119888) mutation probability (119875

119898) population size (119875

119904) and

termination condition Successful working of GAs dependson a proper selection of these parameters [23] But there isnot any intelligent rule to set these parameters In generalvarious sets of the parameters are tested and then the bestone is selected We are also following a similar methodSo we set the parameters as follows a maximum of 20000generations as termination condition 20 as population size100 (100) as crossover probability and 20 independentruns for each setting However we are not reporting ourexperiments except for the mutation probability

To set mutation probability six mutation probabilities000 001 002 003 004 and 005 are considered andtested on five asymmetric TSPLIB instances with four clusters(1198991 1198992 1198993 and 119899

4) for each of the instances ftv110 ftv120

ftv130 ftv140 and ftv150 For example the 7-vertex instancewith two clusters (3 3)means119881

1= 2 3 4119881

2= 5 6 7 and

1198811is followed by 119881

2

Table 4 reports the mean and standard deviation (inparenthesis) of the best solution values over 20 trials on fiveinstances ftv110ndashftv150 for different mutation probabilitiesThe boldface denotes the best average solution value It is seenthat there is significant improvement of the solutions usingnonzero mutation probabilities over using zero mutationprobability It shows that mutation operation also plays animportant role in GAs Mutation probabilities 003 and 004

The Scientific World Journal 7

Table 4 Mean and standard deviation of best solution values on five asymmetric TSPLIB instances

Instance Clusters 119875119898

= 000 119875119898

= 001 119875119898

= 002 119875119898

= 003 119875119898

= 004 119875119898

= 005

ftv110 (29 27 27 27) 270978 (6750) 252634 (3184) 250800 (2521) 249080 (2351) 252835 (3456) 252867 (2479)ftv120 (30 30 30 30) 283967 (8294) 263634 (4194) 262878 (4089) 259623 (3020) 259568 (3536) 259579 (2722)ftv130 (34 32 32 32) 304411 (9377) 284489 (5532) 286056 (4003) 281756 (2442) 283578 (3423) 284166 (1967)ftv140 (35 35 35 35) 320445 (15065) 306831 (5807) 308629 (5450) 307126 (4318) 307002 (4943) 307741 (5558)ftv150 (39 37 37 37) 357625 (15455) 327566 (5772) 327682 (2714) 326526 (3560) 326555 (2964) 326939 (5968)

Start

Improved initial population

Evaluate the population and assign best chromosome cost as the best solution value

Is termination condition satisfied

Print the best solution value and

the best tour

Yes Selection operation

No

Sequential constructive crossover operation

StopMutation operation with

mutation probability

Evaluate the population

Is best population value better than

best solution value

No

Update best solution value

Yes

Local search to the best chromosome

Immigration

Yes

No

Is number ofgeneration gt20

till last update

Figure 4 Flowchart of our hybrid genetic algorithm

are competing Using 119875119898

= 003 the algorithm obtainsthe best average solution for the instances ftv110 ftv120 andftv150 For the remaining two instances the algorithmobtainsthe best average solution at 119875

119898= 004 However if we look

at the standard deviation solutions are relatively stable at119875119898

= 003Figure 5 plots the average best solution values for the five

instances obtained by the HGA using mutation probabilitiesfrom 000 to 005 The figure shows clearly the effectivenessof mutation operator It is seen that as mutation probabilityincreases solution quality also increases However after119875

119898=

004 solution quality is not found to be good From the tableand the figure we can conclude that 119875

119898= 003 is suitable for

our algorithm Hence we are going to use 119875119898

= 003 for ourfurther study

42 Comparative Study on Asymmetric Instances We presenta comparative study between HGA and LBDCOMP [9] forsome asymmetric TSPLIB instances of sizes from 34 to 171

with various clusters and different cluster sizes It is to bementioned that LBDCOMP [9] is claimed to find exactoptimal solution of the OCTSP instances which has beendisproved by showing results of some small sized instances[11] Anyway since no other literature reports the exactsolution for large size instances we are going to comparewith the LBDCOMP algorithm to see solution quality by ourHGA Table 5 shows this comparative study between HGAand LBDCOMPThe table reports results by LBDCOMP andbest solution value (BestSol) average solution value (AvgSol)in 20 runs average complete computational time (CTime)average computational time when final best solution is seenfor the first time (FTime) in twenty runs and percentage oferror (Error()) of the best solution obtained by our HGAThe percentage of error is calculated by the formula

Error () =119861119890119904119905119878119900119897 minus 119874119901119905119878119900119897

119874119901119905119878119900119897times 100 (2)

8 The Scientific World Journal

Table 5 A comparative study between LBDCOMP and HGA for asymmetric TSPLIB instances

Instance Clusters LBDCOMP HGASolution Time BestSol Error () AvgSol FTime CTime

ftv33(16 17) 1584 511 1501 minus524 150216 012 101(9 24) 1509 587 1501 minus053 150324 013 125(3 30) 1356 500 1356 000 135915 018 163

ftv35(17 18) 1747 1132 1731 minus092 173929 041 113(10 25) 1660 346 1660 000 166334 032 126(3 32) 1527 1389 1527 000 153313 042 141

ftv38(19 19) 1681 413 1681 000 168606 096 135(11 27) 1689 765 1689 000 169225 022 153(3 35) 1573 1766 1573 000 158600 016 211

ftv44(22 22) 1935 2492 1935 000 194023 017 175(13 31) 1830 790 1830 000 185212 047 201(4 40) 1670 4871 1670 000 168900 040 272

ftv47(23 24) 2470 1006 2470 000 252624 078 193(13 34) 2349 1049 2257 minus392 228127 115 242(4 43) 1957 593 1957 000 200631 152 317

ftv55(27 28) 2299 600 2219 minus348 224824 071 260(16 39) 1937 815 1937 000 198212 207 306(5 50) 1763 3206 1763 000 178824 116 448

ftv64(32 32) 2658 2854 2658 000 268615 215 368(19 45) 2383 6527 2383 000 249411 298 437(6 58) 2006 11093 2006 000 204821 224 638

ftv70(35 35) 2308 13586 2308 000 234130 130 411(21 49) 2244 10269 2244 000 226724 272 479(7 63) 2134 32331 2134 000 216332 460 737

ftv90(45 45) 1756 1771 1756 000 183378 528 693(27 63) 1710 5671 1710 000 178480 555 846(9 81) 1579 6824 1579 000 165322 579 1443

ftv100(50 50) 2008 2413 2008 000 208417 489 890(30 70) 1903 14351 1903 000 196966 759 1188(10 90) 1788 18749 1788 000 190415 1172 1927

ftv110 (29 27 27 27) 2410 28919 2411 004 249080 608 748ftv120 (30 30 30 30) 2571 8396 2571 000 259623 566 904ftv130 (34 32 32 32) 2747 33167 2751 015 281756 595 1120ftv140 (35 35 35 35) 2941 57144 2947 020 307126 1002 1323ftv150 (39 37 37 37) 3119 8142 3120 003 326526 794 1579ftv160 (40 40 40 40) 3561 75454 3580 053 369618 1251 1834ftv170 (44 42 42 42) 3927 29771 3891 minus092 399201 1536 2239Average 10548 minus038 356 635

whereBestSol denotes the best solution obtained byHGA andOptSol denotes the solution obtained by LBDCOMP

It is seen from Table 5 that our HGA finds bestoptimalsolution of thirty-two instances at least once in twenty runswhereas LBDCOMP could not find optimal solution for atleast six instancesmdashftv33 with clusters (16 17) and (9 24)ftv35 with clusters (17 18) ftv47 with clusters (13 34) ftv55with clusters (27 28) and ftv170 with clusters (44 42 42 42)That is for these six instances solution quality by HGA is

found to better On the other hand for five instances namelyftv110 ftv130 ftv140 ftv150 and ftv160 with four clusterseach solution quality by LBDCOMP is better than by ourHGA For these five instances percentage of error by HGA isat most 053 However on average solution quality byHGAis 038 better than that of by LBDCOMP

In terms of computational time we cannot directlycompare the algorithms because they are executed in differentmachines and it was not possible to access the original code

The Scientific World Journal 9

Table 6 A comparative study between LSA and HGA for symmetric TSPLIB instances

Instance Clusters LSA HGASolution Time BestSol Error () AvgSol FTime CTime

burma14 (6 7) 3621 000 3621 000 362100 000 025ulysses16 (7 8) 7303 000 7303 000 730300 000 030gr17 (8 8) 2517 000 2517 000 251700 000 032gr21 (10 10) 3465 000 3465 000 346500 000 048ulysses22 (10 11) 8190 017 8190 000 819000 000 054gr24 (11 12) 1558 014 1558 000 155800 032 063fri26 (12 13) 957 005 957 000 95700 000 062

bayg29 (14 14) 2144 2103 2144 000 214400 007 093(9 9 10) 2408 3522 2408 000 240800 000 065

bays29 (14 14) 2702 2733 2702 000 270200 000 094(9 9 10) 2991 2489 2991 000 299100 000 066

dantzig42(20 21) 699 44656 699 000 69900 000 151

(13 14 14) 699 102 699 000 69900 002 126(10 10 10 11) 699 517 699 000 69900 000 112

swiss42(20 21) 1605 1440000 1605 000 161233 077 275

(13 14 14) 1919 1440000 1919 000 192300 053 159(10 10 10 11) 1944 1440000 1944 000 194542 015 115

gr48(23 24) 6656 1440000 6433 minus335 643300 007 201

(15 16 16) 7466 1440000 7466 000 750472 004 158(11 12 12 12) 8554 1440000 8554 000 855400 038 143

eil51(25 25) 570 1440000 564 minus105 56400 073 227(16 17 17) 689 1440000 681 minus116 68100 015 173

(12 12 13 13) 714 1440000 714 000 71400 012 162Average 565920 minus024 015 115

2400

2600

2800

3000

3200

3400

3600

Aver

age b

est s

olut

ion

valu

e

ftv110ftv120ftv130

ftv140ftv150

Pm=000

Pm=001

Pm=002

Pm=003

Pm=004

Pm=005

Figure 5 Average best solution values on five asymmetric TSPLIBinstances using six mutation probabilities

of LBDCOMP However a large gap between computationaltime by LBDCOMP and HGA is seen in the table and HGAtakes much less time Further if FTime is considered forHGA then definitely it is found to be much better thanLBDCOMP It is interesting to see that for any of these

instances with the same number of clusters but differentcluster sizesHGA takes different computational times and asthe size of clusters becomesmore unbalanced computationaltime increases In an unbalanced clustered instance size ofthe clusters is not equal It is also seen that on averageHGA hits final best solution for the first time within 56of complete computational time This shows that HGA findsbest solution on average in the middle of the generations forthese asymmetric TSPLIB instances

43 Comparative Study on Symmetric Instances Now we aregoing to compare our HGA with lexisearch algorithm (LSA)[11] on some small sized symmetric TSPLIB [10] instanceswith various clusters and different cluster sizes It is to benoted that our HGA does not require any modification forsolving different types and cases of the instances Table 6shows comparative study between LSA and HGA The solu-tion quality by HGA is found to be insensitive to the numberof runs for most of the instances HGA finds bestoptimalsolution of twenty-three instances at least once in twentyruns whereas LSA could not find optimal solution for atleast three instances within four hours of computational timefor example the instances gr48 with clusters (23 24) andeil51 with clusters (25 25) and (16 17 17) Overall for thesesymmetric instances solution quality by HGA is found to

10 The Scientific World Journal

Table 7 Results on some symmetric TSPLIB instances using HGA

Instance Clusters BestSol AvgSol FTime CTime

berlin52 (51) 7542 (000) 754200 070 363(25 26) 10422 1042200 034 250

brazil58 (57) 25395 (000) 2539500 109 450(28 29) 34110 3411000 114 305

st70 (69) 675 (000) 67715 149 615(34 35) 916 91600 205 418

eil76 (75) 538 (000) 53926 242 733(37 38) 721 72312 199 511

pr76 (75) 108159 (000) 10825455 209 746(37 38) 120436 12058313 274 510

gr96 (95) 55209 (000) 5567285 589 1207(47 48) 56757 5676722 344 840

rat99 (98) 1211 (000) 121840 116 1291(49 49) 1346 134825 553 910

kroA100 (99) 21282 (000) 2132180 367 1302(24 25 25 25) 45733 4614795 340 677

kroB100 (99) 22141 (000) 2219315 490 1348(24 25 25 25) 45709 4581385 252 724

kroC100 (99) 20749 (000) 2078945 304 1262(24 25 25 25) 46388 4647535 410 690

kroD100 (99) 21294 (000) 2138911 464 1226(24 25 25 25) 45681 4595220 343 607

kroE100 (99) 22068 (000) 2211639 518 1452(24 25 25 25) 45431 4555925 324 727

rd100 (99) 7910 (000) 793270 475 1346(24 25 25 25) 15501 1552405 384 634

eil101 (100) 629 (000) 63275 579 1607(25 25 25 25) 1080 108000 380 892

lin105 (104) 14379 (000) 1441665 560 1439(26 26 26 26) 17584 1761820 213 825

pr107 (106) 44303 (000) 4440567 177 1456(26 26 27 27) 51487 5153880 214 779

gr120 (119) 6942 (000) 698695 616 2073(29 30 30 30) 13109 1312915 523 1048

pr124 (123) 59030 (000) 5918175 351 2077(30 31 31 31) 71295 7129500 112 1238

bier127 (126) 118282 (000) 11841960 933 2827(30 32 32 32) 174112 17425070 683 1954

ch130 (129) 6110 (000) 615050 1311 3012(32 32 32 33) 12000 1202205 566 2005

pr136 (135) 96772 (000) 9724080 1480 2836(33 34 34 34) 106605 10671840 814 2004

gr137 (136) 69853 (000) 7042950 1200 2816(34 34 34 34) 81628 8171501 422 1499

pr144 (143) 58537 (000) 5867119 618 3083(35 36 36 36) 69093 6912834 258 2022

kroA150 (149) 26524 (000) 2662965 1233 3585(37 37 37 38) 52824 5298840 1274 1880

kroB150 (149) 26130 (000) 2626423 1721 3807(37 37 37 38) 54008 5423775 1315 1916

The Scientific World Journal 11

Table 7 Continued

Instance Clusters BestSol AvgSol FTime CTime

ch150 (149) 6528 (000) 655631 2092 3859(37 37 37 38) 13042 1308525 1020 1904

pr152 (151) 73682 (000) 7401745 1192 3425(37 38 38 38) 79941 7994100 180 2428

u159 (158) 42080 (000) 4233610 1285 3887(39 39 40 40) 42287 4230245 329 2094

si175 (174) 21407 (000) 2141210 5119 9568(43 43 44 44) 22893 2291065 761 3472

brg180 (179) 1950 (000) 201035 1429 4458(44 45 45 45) 19430 2106020 690 1529

rat195 (194) 2323 (000) 236220 2713 7136(48 48 49 49) 2544 255172 2086 3524

d198 (197) 15800 (013) 1585875 2460 7779(49 49 49 50) 17320 1733950 2976 4392

kroA200 (199) 29420 (018) 2961880 2108 7041(49 50 50 50) 62514 6294175 2345 3981

kroB200 (199) 29463 (009) 2980700 2387 7501(49 50 50 50) 62842 6325311 1828 4140

gr202 (201) 40160 (000) 4041385 3696 9725(50 50 50 51) 44176 4424820 2486 4043

ts225 (224) 126643 (000) 12700683 2851 9385(56 56 56 56) 171269 17154330 2573 4877

tsp225 (224) 3923 (018) 396735 3829 10530(56 56 56 56) 5133 517115 2011 5491

pr226 (225) 80467 (012) 8095360 1628 8769(56 56 56 57) 96508 9651010 2838 6322

gr229 (228) 134957 (026) 13618435 3733 11979(57 57 57 57) 143028 14363245 3204 5731

gil262 (261) 2391 (055) 240315 5308 15126(65 65 65 66) 4874 490645 6433 9372

pr264 (263) 49219 (017) 4981445 4875 14999(65 66 66 66) 60161 6029415 2278 9720

a280 (279) 2585 (023) 261405 10525 18703(69 70 70 70) 2740 274375 3275 9428

pr299 (298) 48375 (038) 4885706 3811 20284(74 74 75 75) 55253 5595165 4536 12361

lin318 (317) 42301 (065) 4267965 8568 25392(79 79 79 80) 52578 5281135 6229 13335

rd400 (399) 15370 (058) 1545220 20327 54674(99 100 100 100) 30821 3100660 15140 25978

fl417 (416) 11930 (058) 1200463 22738 54417(104 104 104 104) 20457 2057624 20162 34601

gr431 (430) 173270 (108) 17604720 30035 71664(107 107 108 108) 185162 18666156 29181 40647

Average 2986 7016

be better and on average solution quality by HGA is 024better than that by LSA

In terms of computational time it can be easily concludedthat HGA ismuch better than LSA though LSAwas executedon slower machine (Pentium IV PC with speed 3GHz and

448 MB RAM) Of course the nature of LSA and HGA isnot the same LSA gives exact optimal solution whereas HGAgives heuristic solution It is also seen from the table thaton average HGA hits final best solution for the first timewithin 13 of complete computational time This shows that

12 The Scientific World Journal

HGA finds best solution on average in the beginning of thegenerations for these instances

44 Proposed Solution for Some More Symmetric InstancesTable 7 presents results for some more symmetric TSPLIBinstances of sizes from 52 to 431 with various clusters andcluster sizes Since to the best of our knowledge no literaturepresents solution for these instances hence we could notprovide any comparative study on these instances Howeverwe present the results for future study of the OCTSP on theseinstances For our self-comparison we provide solution valueand percentage of error (in parentheses) by our HGA forthe instances with one cluster which are of course usualTSP instances Out of forty-seven usual TSP instances HGAfinds exact optimal solution to thirty-three instances For theremaining instances maximum percentage of error is 108That means our algorithm can provide near exact solutionif not exact Treating this study as a base for effectiveness ofthe algorithm we can conclude that the reported solutionsare near exact solution if not exact It is also seen fromthe table that for the same instances as the number ofclusters increases solution value also increases On the otherhand as the number of clusters increases computational timedecreases In general computational time for solving a singleclustered instance (ie usual TSP instance) is more thanits corresponding multiclustered instances It seems that thestructures of these multiclustered instances are less complexand hence easier than their corresponding single clusteredinstances For these symmetric instances on average HGAhits final best solution for the first time within 43 ofcomplete computational time This shows that HGA findsbest solution for these instances on average in the middleof the generations

5 Conclusions

We presented a hybrid genetic algorithm using sequentialconstructive crossover 2-opt search a local search and animmigration method to obtain heuristic solution to theOCTSPWe have used a sequential samplingmethod for gen-erating initial population The efficiency of the hybrid GA tothe problemhas been examined against the exact partitioningalgorithm (LBDCOMP) [9] for some asymmetric TSPLIBinstances and the lexisearch algorithm (LSA) [11] for somesmall sized symmetric TSPLIB instances The computationalexperiments show that ourHGA is efficient in producing highquality of solution for the benchmark instances On the basisof solution quality our HGA is found to be better than theLBDCOMP and LSA In terms of computational time alsoour algorithm is found to be the best one Finally we presentsolution to the problem for some more symmetric TSPLIBinstances Since to the best of our knowledge no literaturepresents solution for these instances we could not confirmthe quality of our solutions for the instances However forthe symmetric instances of size up to 51 we found that ourHGA obtains exact optimal solution to the instances It isto be noted that HGA does not require any modification forsolving different types of TSPLIB instances

For any instance as the number of clusters increasesthe solution value also increases Computational time forsolving a single clustered instance (ie usual TSP instance) ismore than that for solving its corresponding multiclusteredinstances For any multiclustered instance as the clustersbecome more unbalanced computational time increases

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The author is very much thankful to the honourable review-ers for their constructive comments and suggestions Thisresearch was supported by the NSTIP Strategic Technologiesprogramno 10 in the Kingdomof Saudi Arabia via Award no11-INF1788-08 The author is also very much thankful to theNSTIP for its financial and technical supports

References

[1] J A Chisman ldquoThe clustered traveling salesman problemrdquoComputers and Operations Research vol 2 no 2 pp 115ndash1191975

[2] MGendreau AHertz andG Laporte ldquoThe traveling salesmanproblem with backhaulsrdquo Computers and Operations Researchvol 23 no 5 pp 501ndash508 1996

[3] N Guttmann-Beck R Hassin S Khuller and B RaghavacharildquoApproximation algorithms with bounded performance guar-antees for the clustered traveling salesman problemrdquo Algorith-mica vol 28 no 4 pp 422ndash437 2000

[4] F C J Lokin ldquoProcedures for travelling salesman problemswith additional constraintsrdquo European Journal of OperationalResearch vol 3 no 2 pp 135ndash141 1979

[5] G Laporte J-Y Potvin and F Quilleret ldquoTabu search heuristicusing genetic diversification for the clustered traveling salesmanproblemrdquo Journal of Heuristics vol 2 no 3 pp 187ndash200 1997

[6] G Laporte and U Palekar ldquoSome applications of the clus-tered travelling salesman problemrdquo Journal of the OperationalResearch Society vol 53 no 9 pp 972ndash976 2002

[7] Z H Ahmed ldquoGenetic algorithm for the traveling salesmanproblem using sequential constructive crossover operatorrdquoInternational Journal of Biometrics amp Bioinformatics vol 3 no6 pp 96ndash105 2010

[8] Z H Ahmed ldquoA hybrid sequential constructive samplingalgorithm for the bottleneck traveling salesman problemrdquoInternational Journal of Computational Intelligence Researchvol 6 no 3 pp 475ndash484 2010

[9] T Aramgiatisiris ldquoAn exact decomposition algorithm for thetraveling salesman problemwith backhaulsrdquo Journal of Researchin Engineering and Technology vol 1 pp 151ndash164 2004

[10] TSPLIB 1995 httpcomoptifiuni-heidelbergdesoftwareTSPLIB95

[11] Z H Ahmed ldquoAn exact algorithm for the clustered travelingsalesman problemrdquo Opsearch vol 50 no 2 pp 215ndash228 2013

[12] J D E Little K G Murthy D W Sweeny and C KarelldquoAn algorithm for the travelling salesman problemrdquo OperationsResearch vol 11 pp 972ndash989 1963

The Scientific World Journal 13

[13] K Jongens and T Volgenant ldquoThe symmetric clustered trav-eling salesman problemrdquo European Journal of OperationalResearch vol 19 no 1 pp 68ndash75 1985

[14] M Gendreau G Laporte and J Y Potvin ldquoHeuristics for theclustered traveling salesman problemrdquo Tech Rep CRT-94-54Centre de Recherche sur les Transports Universite deMontrealMontreal Canada 1994

[15] J-Y Potvin and F Guertin ldquoA genetic algorithm for theclustered traveling salesman problem with an a priori order onthe clustersrdquo Tech Rep CRT-95-06 Centre de recherchesur lestransports Universite de Montreal Montreal Canada 1995

[16] J-Y Potvin and F Guertin ldquoThe clustered traveling salesmanproblem a genetic approachrdquo in Meta-Heuristics Theory ampApplications I H Osman and J Kelly Eds pp 619ndash631 KluwerAcademic Norwell Mass USA 1996

[17] S Anily J Bramel andAHertz ldquo53-Approximation algorithmfor the clustered traveling salesman tour and path problemsrdquoOperations Research Letters vol 24 no 1 pp 29ndash35 1999

[18] N Christofides ldquoWorst-case analysis of a new heuristic for thetraveling salesmanproblemrdquoTech Rep 388Graduate School ofIndustrial Administration Carnegie-Mellon University Pitts-burgh Pa USA 1976

[19] W Sheng N Xi M Song and Y Chen ldquoRobot path planningfor dimensional measurement in automotive manufacturingrdquoJournal of Manufacturing Science and Engineering Transactionsof the ASME vol 127 no 2 pp 420ndash428 2005

[20] C Ding Y Cheng andM He ldquoTwo-level genetic algorithm forclustered traveling salesman problem with application in large-scale TSPsrdquo Tsinghua Science and Technology vol 12 no 4 pp459ndash465 2007

[21] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley New York NY USA 1989

[22] Z H Ahmed ldquoA hybrid genetic algorithm for the bottlenecktraveling salesman problemrdquo ACM Transactions on EmbeddedComputing Systems vol 12 no 1 article 9 2013

[23] K Deb Optimization for Engineering Design Algorithms andExamples Prentice Hall India New Delhi India 1995

[24] Z H Ahmed ldquoMulti-parent extension of sequential construc-tive crossover for the travelling salesman problemrdquo Interna-tional Journal of Operational Research vol 11 no 3 pp 331ndash3422011

[25] Z H Ahmed ldquoAn experimental study of a hybrid geneticalgorithm for the maximum travelling salesman problemrdquoMathematical Sciences vol 7 no 1 pp 1ndash7 2013

[26] C-X Wang D-W Cui Z-R Wang and D Chen ldquoA novel antcolony system based on minimum 1-tree and hybrid mutationfor TSPrdquo in Proceedings of the 1st International Conferenceon Natural Computation (ICNC rsquo05) LNCS pp 1269ndash1278Springer Changsha China August 2005

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Stochastic AnalysisInternational Journal of

Page 2: Research Article The Ordered Clustered Travelling Salesman Problem: A Hybrid Genetic ...downloads.hindawi.com/journals/tswj/2014/258207.pdf · 2019-07-31 · Research Article The

2 The Scientific World Journal

as the size increases finding exact optimal solution to theCTSP instances becomes impractical and hence heuristicmust be used

We seek approximate solution using heuristic algorithmfor the OCTSP For the TSP and related problems well-known heuristic algorithms are genetic algorithms (GAs)tabu search (TS) artificial neural network (ANN) simu-lated annealing (SA) approximate algorithms and so forthAmongst the heuristics GAs are found to be the bestalgorithms for the TSP and its variations Since OCTSP is avariation of the usual TSP therefore we develop a hybrid GA(HGA) using sequential constructive crossover [7] and 2-optsearch and a local search [8] to obtain heuristically optimalsolution to the problem The efficiency of our algorithmhas been examined against partitioning algorithm [9] forsome medium sized asymmetric TSPLIB [10] instances andlexisearch algorithm [11] for some small sized symmetricTSPLIB [10] instances The computational experiments showthe effectiveness of our proposed HGA Finally we presentsolution to some medium sized symmetric TSPLIB [10]instances However to the best of our knowledge no litera-ture presents solution to these symmetric instances Hencewe could not provide any comparative study of these results

This paper is organized as follows Section 2 presents adetailed literature review to the problem A hybrid geneticalgorithm is developed and reported in Section 3 Com-putational experiment for the algorithm is presented inSection 4 Finally Section 5 presents comments and conclud-ing remarks

2 Literature Review

Chisman [1] transformed the CTSP to the usual TSP and thenapplied branch and bound approach [12] to solve the problemexactly but did not obtain very good resultsThereafter Lokin[4] and Jongens and Volgenant [13] applied exact algorithmsto find exact optimal solution to the problem Aramgiatisiris[9] developed an exact partitioning algorithm (LBDCOMPtherein) by transforming the problem to the TSPB thensolving independently linehaul and backhaul subproblemsand finally reformulating as a direct shortest path on thebipartite graph problems However the algorithm does notreally obtain exact solutions for many instances [11]

An approximation algorithm with good empirical per-formance [14] was developed to solve the problem with aprespecified order on the clusters Also three more heuristicswere proposed and compared among them As reportedthe best results were obtained by the heuristic that firsttransforms the problem into a TSP and then applies theGENIUS heuristic

Laporte et al [5] proposed a TS heuristic that combinedwith a phase of diversification using a GA to solve theproblem with a prespecified order of visiting the clusters Asreported the TS outperforms the GA [15] that exploits order-based crossover operators and local search heuristics How-ever when comparing TS with a postoptimization procedure[14] TS obtained better quality of solutions but requiredmorecomputational time

Another GA was developed for the problem [16] that firstfinds intercluster paths and then intracluster paths Finallya comparative study of the GA was presented against aGENIUS heuristic [14] and lower bounds [13] As reportedthe GA could solve instances up to 500 vertices with 4 and10 clusters and obtained solutions within 55 of the lowerbound

An approximation algorithm of 53 performance ratiohas been developed for the OCTSP with a prespecifiedvisiting sequence for the clusters [17] Another approximationalgorithm has been proposed which guarantees boundedperformance for some variants of the CTSP [3] For theproblemwith unspecified end vertices its algorithm first usesamodified Christofidesrsquo algorithm [18] to get the shortest freeendsHamiltonian paths in each cluster After the first step thetwo end vertices for each cluster and the intracluster paths arespecifiedThen a rural postman problem algorithm is used toconnect all the intracluster paths to form a whole tour Thisalgorithm favours the intracluster Hamiltonian paths whichimplies that the inter-cluster pathsmay be sacrificedwhen theend vertices in each cluster are already determined

A two-level-TSP hierarchical algorithm that favoursintercluster paths has been proposed for the CTSP [19] Firstthe shortest intercluster paths connecting every cluster havebeen specified then the start and end vertices are specified foreach cluster Next a modified Christofidesrsquo algorithm [18] isused to get the shortest Hamiltonian paths with two specifiedend vertices in each cluster At the end a whole tour is formedby combining the paths generated in both levels They alsoshowed that the penalties caused by favoring the intraclusterHamiltonian paths and the intercluster paths are comparable

A two-level GA (TLGA) has been developed for solvingCTSP with unspecified end vertices [20] The algorithm firstfinds the shortest Hamiltonian cycle for each cluster andthen connects all the intracluster paths in a certain sequenceto form a whole tour In the lower level a GA is used tofind the shortest Hamiltonian cycle rather than the shortestHamiltonian path for each cluster In the higher level amodified GA is designed to determine an edge that will bedeleted from the shortest Hamiltonian cycle for each clusterand the visiting sequence of all the clusters with the objectiveof shortest travelling tour for the whole problem The higherlevel algorithm has the freedom to delete any edge of theclusters while searching for the shortest complete tour Testresults demonstrate that the TLGA for large TSPs is moreeffective and efficient than the classical GA

3 A Hybrid Genetic Algorithm for the OCTSP

31 A Brief Overview of GAs GAs are structured yet ran-domized search methods based on mimicking the survivalof the fittest among the species generated by random changesin the gene structure of the chromosomes in the evolutionarybiology [21] They start with a population of chromosomes(solutions) that evolve from one generation to the next Eachgeneration consists of the following three operations

(a) Selection This procedure is a stochastic process thatmimics the ldquosurvival-of-fittestrdquo theory However here

The Scientific World Journal 3

1

3 5

4

2

76

Figure 1 An example of result of the OCTSP using GA

no new chromosome is created Some of the chromo-somes are copied (even more than once) to the nextgeneration probabilistically based on their objectivefunction value whereas some other chromosomes arediscarded

(b) Crossover It is a binary operator that applies to twoparent chromosomes with a large probability whichcreates new offspring chromosome(s) It is a veryimportant operator in GAs Also crossover operatortogether with selection operator is found to be themost powerful process in the GA search

(c) Mutation It is a unary operator that applies to eachof the chromosomes with a small probability It is theoccasional random change of some selected gene(s) ofa chromosome to diversify the GA search space

Starting from a randomly generated or heuristicallygenerated initial population the GAs search repeated theabove three operators until the stopping criterion is satisfiedCrossover operator is a unique feature of GAs that wishesto combine good quality parent chromosomes to createone or more new offspring chromosome(s) However it isseen that the crossover alone cannot generate high qualitychromosomes for the combinatorial optimization problemslike the TSP and its variations Thus powerful local searchmethods are incorporated to improve the quality of offspringchromosomes [5] In hybrid GAs crossover operator gener-ates new starting solutions for the local search methods

GAs are found to be successful heuristic algorithms forsolving the usual TSP and its variations However GAs donot guarantee the optimality of the solution but they can findvery good near optimal solution in very short time We areapplying crossover mutation and local search methods foreach cluster in the prespecified order for the OCTSP Resultof the GA for a 7-vertex problem instance is a complete touras shown in Figure 1

32 Bias Removal Bias removal step is adopted in lexisearchalgorithm [11] and found effective for the CTSP We alsoconsider the bias removal step in ourGAThemain advantageof the bias removal is that a large amount of the solutionvalue is kept fixed and for the remaining small value wehave to search The process for bias removal of the costmatrix is as follows subtract each row-minima from its

Table 1 The cost matrix with row-minima and column-minima

Vertex 1 2 3 4 5 6 7 Row-minima1 999 75 99 9 35 63 8 82 51 999 86 46 88 29 20 203 100 5 999 16 28 35 28 54 20 45 11 999 59 53 49 115 86 63 33 65 999 76 72 336 36 53 89 31 21 999 52 217 58 31 43 67 52 60 999 32Column-minima 9 0 0 1 0 9 0

Table 2 The reduced cost matrix

Vertex 1 2 3 4 5 6 71 999lowast 67 91 0 27 46 02 22 999lowast 66 25 68 0 03 86 0 999lowast 10 23 21 234 0 34 0 999lowast 48 33 385 44 30 0 31 999lowast 34 396 6 32 68 9 0 999lowast 317 18 0 12 35 21 29 999lowast

(lowastElements are left as 999)

corresponding row elements repeat the same column-wiseon the resultant matrix The total of the row-minima and thesubsequent column-minima is called the ldquobiasrdquo of the matrixHowever we have not incorporated clusters precedencerelations in our cost matrix This does not affect the value ofa chromosome since while generating a chromosome theclusters precedence relations are taken care of

The bias of the cost matrix given in Table 1 is (row-minima + column-minima = 129 + 19 =) 148 The reducedcost matrix (ie after removing bias of the matrix) is shownin Table 2 We shall solve the problem with respect to thereduced cost matrix After we find solution value with respectto the reduced matrix we shall add the bias to the value forfinding the solution value with respect to the original costmatrix

33 Alphabet Table Alphabet matrix 119860 = [119886(119894 119895)] is asquare matrix of order 119899 formed by positions of elementsof the reduced cost matrix of order 119899 119862

1015840= [1198881015840

119894119895] when

they are arranged in the nondecreasing order of their costsAlphabet table ldquo[119886(119894 119895) minus 119888

1015840

119894119886(119894119895)]rdquo is the combination of

elements (vertices) of matrix A and their costs in the reducedmatrix [11] The alphabet table for the reduced cost matrixin Table 2 is shown in Table 3 This alphabet table also is nottaking care of the clusters precedence relations

34 Improved Initial Population The path representation fora chromosome is used in our GA In this representation anyvertex is assigned to a unique natural number from 1 to 119899that is genes are natural numbers The path of a salesmanis represented by a chromosome which is a permutation ofnumber genes A gene segment is defined as a permutation

4 The Scientific World Journal

Table 3 The alphabet table

Vertex 119881mdashCost 119881mdashCost 119881mdashCost 119881mdashCost 119881mdashCost 119881mdashCost 119881mdashCost1 4mdash0 7mdash0 5mdash27 6mdash46 2mdash67 3mdash91 1mdash9992 6mdash0 7mdash0 1mdash22 4mdash25 3mdash66 5mdash68 2mdash9993 2mdash0 4mdash10 6mdash21 5mdash23 7mdash23 1mdash86 3mdash9994 1mdash0 3mdash0 6mdash33 2mdash34 7mdash38 5mdash48 4mdash9995 3mdash0 2mdash30 4mdash31 6mdash34 7mdash39 1mdash44 5mdash9996 5mdash0 1mdash6 4mdash9 7mdash31 2mdash32 3mdash68 6mdash9997 2mdash0 3mdash12 1mdash18 5mdash21 6mdash29 4mdash35 7mdash999

of the vertices in a cluster A chromosome is a permutationof all the gene segments with one gene segment per clusterFor example let 1 2 3 4 5 6 7 be the vertices with 119881

1=

2 3 4 1198812

= 5 6 7 and 1198811is followed by 119881

2 in a 7-

vertex instance then starting from vertex 1 a complete tour1rarr 3rarr 4rarr 2rarr 6rarr 7rarr 5rarr 1 may be represented as thechromosome (1 3 4 2 6 7 5) where (3 4 2) and (6 7 5) arethe gene segments for cluster 1 and cluster 2 respectively

It is to be noted that starting from a good initialpopulation can deliver better quality of solutions quicklyand that is why many literatures report generating initialpopulation using heuristics Hence we are going to usesequential sampling algorithm for heuristically generatinginitial population that has been applied successfully to thebottleneck TSP [22]This algorithm is a simple version of thesequential constructive sampling algorithm [8] It is basicallya probabilistic method to generate a tour of the salesmanTheprobability of visiting each unvisited vertex of a cluster in arow of the alphabet table is assigned in such a way that thefirst unvisited vertex gets more probability than the secondone and so on Thereafter cumulative probability of eachunvisited vertex of a cluster is calculated Next a randomnumber 119903 isin [0 1] is generated and the vertex that representsthe chosen random number in the cumulative probabilityrange is accepted The probability of visiting each unvisitedvertex of a cluster is assigned as follows Suppose the numberof unvisited vertices of a cluster in a row of the alphabet tableis 119896 The probability of visiting the 119894th unvisited vertex is

119901119894=

2 (119896 minus 119894 + 1)

119896 (119896 + 1) (1)

The algorithm may be summarized as follows

Step 0 Construct the ldquoalphabet tablerdquo based on the reducedcost matrix Repeat the following steps for the fixed popula-tion size (119875

119904)

Step 1 Since ldquovertex 1rdquo is the starting vertex so initialize 119901 =

1 and go to Step 2

Step 2 Go to the 119901th row of the ldquoalphabet tablerdquo and visitprobabilistically by using (1) any unvisited vertex of the row(say vertex 119902) in the present cluster and go to Step 3

Step 3 Rename the ldquovertex 119902rdquo as ldquovertex 119901rdquo and go to Step 4

Step 4 If all vertices of the present cluster are visited then goto the next cluster in the order (if any) and go to Step 5 elsego to Step 2

Step 5 If all vertices of the network are visited then go toStep 1 for generating another chromosome in the populationelse go to Step 2

Let us illustrate the algorithm through the example givenin Table 1 with 119881

1= 2 3 4 119881

2= 5 6 7 and 119881

1is followed

by 1198812 We start from 1st row of the ldquoalphabet tablerdquo The

number of unvisited vertices of the 1st cluster in the row is3 which are 4 2 and 3 with cumulative probabilities 05000833 and 1000 respectively Suppose the vertex 4 is selectedrandomly then the partial chromosome will be (1 4) Nextwe go to 4th row of the ldquoalphabet tablerdquo and probabilisticallyselect another node and so on Proceeding in this way it maylead to a complete chromosome (1 4 2 3 6 5 7)

A preliminary study shows the effectiveness of the sam-pling algorithm for initial population However instead ofconsidering all unvisited vertices if we consider at most firstten vertices in a cluster then the algorithm generates verygood population A similar observation has been reportedfor the bottleneck TSP also [22] Hence we consider thisrestricted domain of unvisited vertices of a cluster for ourstudy Further to start with better population we apply 2-optsearch to each chromosome The 2-opt search removes twoedges and then replaces them by a different set of edges insuch a way so as to maintain the feasibility of the tour Let 120572

119894

120572119894+1

120572119895 and 120572

119895+1be four vertices in a cluster then if the edges

(120572119894 120572119894+1

) and (120572119895 120572119895+1

) are removed the only way to form anew valid tour is to connect 120572

119894to 120572119895and 120572

119894+1to 120572119895+1

35 Fitness Function and Selection Method The objectivefunction of each chromosome in the population is the cost ofthe tour represented by the chromosomeThe fitness functionof a chromosome is defined as multiplicative inverse of theobjective functionThere are various selectionmethods in theliteratureThe selection operation considered for our study isthe stochastic remainder selection method [23]

36 The Sequential Constructive Crossover Operation Sincecrossover operation plays main role in GAs hence severalcrossover operators have been proposed for the usual TSPwhich are then used for the variant TSPs also Out of themthe sequential constructive crossover (SCX) [7] is found to

The Scientific World Journal 5

1

3

5

4

2

7

6

1

3

5

4

2

7

6

1

3

5

4

2

7

6

(a) P1 (1 2 4 3 6 7 5) (b) P2 (1 2 4 3 6 7 5) (c) Offspring (1 2 4 3 6 5 7)

Figure 2 Example of sequential constructive crossover operation

be one of the best crossover operators for the usual TSP Amultiparent extension of the SCX has been applied to theusual TSP and found good results [24]The SCXhas also beensuccessfully applied to the TSP with some other variations[22 25] In general it produces an offspring using betteredges of the parents However it does not depend only onthe parentsrsquo structure it sometimes introduces new but goodedges to the offspring which are not even available in thepresent population We modify the SCX operator for theOCTSP as follows

Step 1 Start from ldquovertex 1rdquo (ie current vertex 119901 = 1)

Step 2 Sequentially search both of the parent chromosomesand consider the first unvisited vertex of the present clusterappearing after ldquovertex 119901rdquo in each parent If no unvisitedvertex after ldquovertex 119901rdquo is available in any (or both) of theparents search sequentially from the starting of that parentand consider the first unvisited vertex of the cluster and goto Step 3

Step 3 Suppose the ldquovertex 120572rdquo and the ldquovertex 120573rdquo are foundin the 1st and 2nd parents respectively then for selecting thenext vertex in the offspring chromosome go to Step 4

Step 4 If 119888119901120572

lt 119888119901120573 then select ldquovertex 120572rdquo otherwise select

ldquovertex120573rdquo as the next vertex and concatenate it to the partiallyconstructed offspring chromosome and go to Step 5

Step 5 If there is not any vertex left in that cluster thengo to the next cluster if any If the offspring is a completechromosome then stop otherwise rename the present vertexas ldquovertex 119901rdquo and go to Step 2

Let a pair of parent chromosomes be 1198751 (1 2 4 3 6

7 5) and 1198752 (1 3 2 4 6 5 7) with costs 357 and 354

respectively with respect to the original cost matrix givenin Table 1 By applying above SCX we obtain the offspringchromosome (1 2 4 3 6 5 7) with cost 318 which is lessthan both parentsTheparent and the offspring chromosomesare shown in Figure 2 In general crossover operator inheritsparentsrsquo characteristics and the operator that preserves goodcharacteristics of parents in the offspring is said to be goodoperatorThe SCX is found to be excellent in this regard Boldedges in Figure 2(c) are the edges which are available either in

the first parent or in second parent For this given example alledges are selected from either of the parents

For the crossover operation a pair of parent chromo-somes is selected sequentially from the mating pool It isreported that the SCX gets stuck in local minimums quicklyfor the TSP [7] which is very often due to the identicalpopulation So to overcome this situation the selectedparents are checked for duplication If the selected parentsare found to be identical then the second parent is modifiedtemporarily by swapping some randomly chosen pair of genesin the chromosome and then the crossover operation isperformed To improve quality of the solution as well as havea mixture of parents and offspring in a population the firstparent is replaced by the offspring only if the offspring valueis better than the average value of the present populationIn this way the mixed population retains diversity alsoTo further improve the quality of the solution obtained bycrossover many researchers applied 2-opt search operator Toimprove solution quality we are going to use a local searchmethod that combines three mutation operators that will bediscussed in Section 38 However we are not applying thislocal search method to all of the offspring rather it is appliedonly to the offspring if its value is better than the averagepopulation valueNow since our crossover operator producesonly one offspring to keep population size fixed throughoutthe generations while pairing with the next chromosome inorder the present second original parent chromosomewill beconsidered as the first parent and so on

37 Mutation Operation The mutation operator randomlyselects a position in the chromosome and changes the cor-responding gene thereby modifying information The needfor mutation comes from the fact that as the less fit chromo-somes of successive generations are discarded some aspectsof genetic material could be lost forever By performingoccasional random changes in the chromosomes GAs ensurethat newparts of the search space are reachedwhich selectionand crossover could not fully guarantee In doing somutationensures that no important features are prematurely lost thusmaintaining the mating pool diversity For this investigationwe have considered reciprocal exchange mutation operatorthat selects two genes randomly of a chromosome in everycluster and swaps them The probability of mutation isusually chosen to be considerably less than the probability

6 The Scientific World Journal

1

3

5

4

2

7

6

1

3

5

4

2

7

6

(a) (1 2 4 3 6 7 5) (b) (1 4 2 3 6 5 7)

Figure 3 Example of reciprocal exchange mutation operation

of crossover So mutation plays a secondary role in the GAsearch For example let the chromosome (1 2 4 3 6 7 5)be selected for mutation and vertices 2 and 4 are swapped incluster 1 and vertices 7 and 5 are swapped in cluster 2 thenthe mutated chromosome becomes (1 4 2 3 6 5 7) whichis shown in Figure 3 Bold edges in Figure 3(b) are the newedges in the mutated chromosome

38 A Local Search Method We have considered the com-bined mutation operation as a local search method whichhas been successfully applied to the bottleneck TSP [822] and maximum TSP [25] It combines three mutationoperators insertion inversion and reciprocal exchange withcent percentage of probabilities Insertion operator selects avertex (gene) in a chromosome and inserts it in a randomplace and inversion operator selects two points along thelength of a chromosome and reverses the subchromosomesbetween these points This local search a modification ofthe hybrid mutation operator [26] is applied to a chro-mosome Recall that sizes of the clusters 119881

1 1198812 119881

119898are

1198991 1198992 119899

119898 respectively Suppose (1 = 120572

1 1205722 1205723 120572

119899) is

a chromosome then the local search for the OCTSP can bedeveloped as follows

Step 0 Set 119909 = 2 and 119910 = 1

Step 1 For 119894 = 1 to 119898 perform Step 2

Step 2 Set 119910 = 119899119894+ 119910 and go to Step 3

Step 3 For 119895 = 119909 to (119910 minus 1) perform Step 4

Step 4 For 119896 = (119895 + 1) to 119910 perform Step 5

Step 5 If inserting vertex 120572119895after vertex 120572

119896reduces the

present tour cost then insert the vertex 120572119895after vertex 120572

119896 In

either case go to Step 6

Step 6 If inverting subchromosome between the vertices120572119895and 120572

119896reduces the present tour cost then invert the

subchromosome In either case go to Step 7

Step 7 If swapping the vertices 120572119895and 120572

119896reduces the present

tour cost then swap them In either case go to Step 8

Step 8 Set 119909 = 119910 + 1 and go to Step 1

39 Immigration It is seen that sometimes GAs get stuckin local minimums for the combinatorial optimization prob-lems which is very often due to the identical populationSo to improve capability of GAs the population should bediversified To diversify the population immigrationmethodis also adopted where some randomly selected chromosomesare replaced by new chromosomes after some generations[22] We are also considering an immigration method Forour investigation 20 of the population is replaced ran-domly using sequential sampling algorithm as discussed inSection 34 if no improvement is found within the last 20generations Once the immigration is applied we wait for thenext 20 generations for any improvement Our hybrid GA(HGA) for the OCTSP may be summarized as in Figure 4[22]

4 Results and Discussions

We encoded our HGA in Visual C++ executed on a PC with340GHz Intel(R) Core (TM) i7-3770CPU and 800GBRAMunderMSWindows 7 operating system and testedwith someTSPLIB [10] instances

41 Parameter Setting GAs are well suited for the com-binatorial optimization problems They find near optimalsolution in reasonable time However they are guided bysuitable choice of parameters namely crossover probability(119875119888) mutation probability (119875

119898) population size (119875

119904) and

termination condition Successful working of GAs dependson a proper selection of these parameters [23] But there isnot any intelligent rule to set these parameters In generalvarious sets of the parameters are tested and then the bestone is selected We are also following a similar methodSo we set the parameters as follows a maximum of 20000generations as termination condition 20 as population size100 (100) as crossover probability and 20 independentruns for each setting However we are not reporting ourexperiments except for the mutation probability

To set mutation probability six mutation probabilities000 001 002 003 004 and 005 are considered andtested on five asymmetric TSPLIB instances with four clusters(1198991 1198992 1198993 and 119899

4) for each of the instances ftv110 ftv120

ftv130 ftv140 and ftv150 For example the 7-vertex instancewith two clusters (3 3)means119881

1= 2 3 4119881

2= 5 6 7 and

1198811is followed by 119881

2

Table 4 reports the mean and standard deviation (inparenthesis) of the best solution values over 20 trials on fiveinstances ftv110ndashftv150 for different mutation probabilitiesThe boldface denotes the best average solution value It is seenthat there is significant improvement of the solutions usingnonzero mutation probabilities over using zero mutationprobability It shows that mutation operation also plays animportant role in GAs Mutation probabilities 003 and 004

The Scientific World Journal 7

Table 4 Mean and standard deviation of best solution values on five asymmetric TSPLIB instances

Instance Clusters 119875119898

= 000 119875119898

= 001 119875119898

= 002 119875119898

= 003 119875119898

= 004 119875119898

= 005

ftv110 (29 27 27 27) 270978 (6750) 252634 (3184) 250800 (2521) 249080 (2351) 252835 (3456) 252867 (2479)ftv120 (30 30 30 30) 283967 (8294) 263634 (4194) 262878 (4089) 259623 (3020) 259568 (3536) 259579 (2722)ftv130 (34 32 32 32) 304411 (9377) 284489 (5532) 286056 (4003) 281756 (2442) 283578 (3423) 284166 (1967)ftv140 (35 35 35 35) 320445 (15065) 306831 (5807) 308629 (5450) 307126 (4318) 307002 (4943) 307741 (5558)ftv150 (39 37 37 37) 357625 (15455) 327566 (5772) 327682 (2714) 326526 (3560) 326555 (2964) 326939 (5968)

Start

Improved initial population

Evaluate the population and assign best chromosome cost as the best solution value

Is termination condition satisfied

Print the best solution value and

the best tour

Yes Selection operation

No

Sequential constructive crossover operation

StopMutation operation with

mutation probability

Evaluate the population

Is best population value better than

best solution value

No

Update best solution value

Yes

Local search to the best chromosome

Immigration

Yes

No

Is number ofgeneration gt20

till last update

Figure 4 Flowchart of our hybrid genetic algorithm

are competing Using 119875119898

= 003 the algorithm obtainsthe best average solution for the instances ftv110 ftv120 andftv150 For the remaining two instances the algorithmobtainsthe best average solution at 119875

119898= 004 However if we look

at the standard deviation solutions are relatively stable at119875119898

= 003Figure 5 plots the average best solution values for the five

instances obtained by the HGA using mutation probabilitiesfrom 000 to 005 The figure shows clearly the effectivenessof mutation operator It is seen that as mutation probabilityincreases solution quality also increases However after119875

119898=

004 solution quality is not found to be good From the tableand the figure we can conclude that 119875

119898= 003 is suitable for

our algorithm Hence we are going to use 119875119898

= 003 for ourfurther study

42 Comparative Study on Asymmetric Instances We presenta comparative study between HGA and LBDCOMP [9] forsome asymmetric TSPLIB instances of sizes from 34 to 171

with various clusters and different cluster sizes It is to bementioned that LBDCOMP [9] is claimed to find exactoptimal solution of the OCTSP instances which has beendisproved by showing results of some small sized instances[11] Anyway since no other literature reports the exactsolution for large size instances we are going to comparewith the LBDCOMP algorithm to see solution quality by ourHGA Table 5 shows this comparative study between HGAand LBDCOMPThe table reports results by LBDCOMP andbest solution value (BestSol) average solution value (AvgSol)in 20 runs average complete computational time (CTime)average computational time when final best solution is seenfor the first time (FTime) in twenty runs and percentage oferror (Error()) of the best solution obtained by our HGAThe percentage of error is calculated by the formula

Error () =119861119890119904119905119878119900119897 minus 119874119901119905119878119900119897

119874119901119905119878119900119897times 100 (2)

8 The Scientific World Journal

Table 5 A comparative study between LBDCOMP and HGA for asymmetric TSPLIB instances

Instance Clusters LBDCOMP HGASolution Time BestSol Error () AvgSol FTime CTime

ftv33(16 17) 1584 511 1501 minus524 150216 012 101(9 24) 1509 587 1501 minus053 150324 013 125(3 30) 1356 500 1356 000 135915 018 163

ftv35(17 18) 1747 1132 1731 minus092 173929 041 113(10 25) 1660 346 1660 000 166334 032 126(3 32) 1527 1389 1527 000 153313 042 141

ftv38(19 19) 1681 413 1681 000 168606 096 135(11 27) 1689 765 1689 000 169225 022 153(3 35) 1573 1766 1573 000 158600 016 211

ftv44(22 22) 1935 2492 1935 000 194023 017 175(13 31) 1830 790 1830 000 185212 047 201(4 40) 1670 4871 1670 000 168900 040 272

ftv47(23 24) 2470 1006 2470 000 252624 078 193(13 34) 2349 1049 2257 minus392 228127 115 242(4 43) 1957 593 1957 000 200631 152 317

ftv55(27 28) 2299 600 2219 minus348 224824 071 260(16 39) 1937 815 1937 000 198212 207 306(5 50) 1763 3206 1763 000 178824 116 448

ftv64(32 32) 2658 2854 2658 000 268615 215 368(19 45) 2383 6527 2383 000 249411 298 437(6 58) 2006 11093 2006 000 204821 224 638

ftv70(35 35) 2308 13586 2308 000 234130 130 411(21 49) 2244 10269 2244 000 226724 272 479(7 63) 2134 32331 2134 000 216332 460 737

ftv90(45 45) 1756 1771 1756 000 183378 528 693(27 63) 1710 5671 1710 000 178480 555 846(9 81) 1579 6824 1579 000 165322 579 1443

ftv100(50 50) 2008 2413 2008 000 208417 489 890(30 70) 1903 14351 1903 000 196966 759 1188(10 90) 1788 18749 1788 000 190415 1172 1927

ftv110 (29 27 27 27) 2410 28919 2411 004 249080 608 748ftv120 (30 30 30 30) 2571 8396 2571 000 259623 566 904ftv130 (34 32 32 32) 2747 33167 2751 015 281756 595 1120ftv140 (35 35 35 35) 2941 57144 2947 020 307126 1002 1323ftv150 (39 37 37 37) 3119 8142 3120 003 326526 794 1579ftv160 (40 40 40 40) 3561 75454 3580 053 369618 1251 1834ftv170 (44 42 42 42) 3927 29771 3891 minus092 399201 1536 2239Average 10548 minus038 356 635

whereBestSol denotes the best solution obtained byHGA andOptSol denotes the solution obtained by LBDCOMP

It is seen from Table 5 that our HGA finds bestoptimalsolution of thirty-two instances at least once in twenty runswhereas LBDCOMP could not find optimal solution for atleast six instancesmdashftv33 with clusters (16 17) and (9 24)ftv35 with clusters (17 18) ftv47 with clusters (13 34) ftv55with clusters (27 28) and ftv170 with clusters (44 42 42 42)That is for these six instances solution quality by HGA is

found to better On the other hand for five instances namelyftv110 ftv130 ftv140 ftv150 and ftv160 with four clusterseach solution quality by LBDCOMP is better than by ourHGA For these five instances percentage of error by HGA isat most 053 However on average solution quality byHGAis 038 better than that of by LBDCOMP

In terms of computational time we cannot directlycompare the algorithms because they are executed in differentmachines and it was not possible to access the original code

The Scientific World Journal 9

Table 6 A comparative study between LSA and HGA for symmetric TSPLIB instances

Instance Clusters LSA HGASolution Time BestSol Error () AvgSol FTime CTime

burma14 (6 7) 3621 000 3621 000 362100 000 025ulysses16 (7 8) 7303 000 7303 000 730300 000 030gr17 (8 8) 2517 000 2517 000 251700 000 032gr21 (10 10) 3465 000 3465 000 346500 000 048ulysses22 (10 11) 8190 017 8190 000 819000 000 054gr24 (11 12) 1558 014 1558 000 155800 032 063fri26 (12 13) 957 005 957 000 95700 000 062

bayg29 (14 14) 2144 2103 2144 000 214400 007 093(9 9 10) 2408 3522 2408 000 240800 000 065

bays29 (14 14) 2702 2733 2702 000 270200 000 094(9 9 10) 2991 2489 2991 000 299100 000 066

dantzig42(20 21) 699 44656 699 000 69900 000 151

(13 14 14) 699 102 699 000 69900 002 126(10 10 10 11) 699 517 699 000 69900 000 112

swiss42(20 21) 1605 1440000 1605 000 161233 077 275

(13 14 14) 1919 1440000 1919 000 192300 053 159(10 10 10 11) 1944 1440000 1944 000 194542 015 115

gr48(23 24) 6656 1440000 6433 minus335 643300 007 201

(15 16 16) 7466 1440000 7466 000 750472 004 158(11 12 12 12) 8554 1440000 8554 000 855400 038 143

eil51(25 25) 570 1440000 564 minus105 56400 073 227(16 17 17) 689 1440000 681 minus116 68100 015 173

(12 12 13 13) 714 1440000 714 000 71400 012 162Average 565920 minus024 015 115

2400

2600

2800

3000

3200

3400

3600

Aver

age b

est s

olut

ion

valu

e

ftv110ftv120ftv130

ftv140ftv150

Pm=000

Pm=001

Pm=002

Pm=003

Pm=004

Pm=005

Figure 5 Average best solution values on five asymmetric TSPLIBinstances using six mutation probabilities

of LBDCOMP However a large gap between computationaltime by LBDCOMP and HGA is seen in the table and HGAtakes much less time Further if FTime is considered forHGA then definitely it is found to be much better thanLBDCOMP It is interesting to see that for any of these

instances with the same number of clusters but differentcluster sizesHGA takes different computational times and asthe size of clusters becomesmore unbalanced computationaltime increases In an unbalanced clustered instance size ofthe clusters is not equal It is also seen that on averageHGA hits final best solution for the first time within 56of complete computational time This shows that HGA findsbest solution on average in the middle of the generations forthese asymmetric TSPLIB instances

43 Comparative Study on Symmetric Instances Now we aregoing to compare our HGA with lexisearch algorithm (LSA)[11] on some small sized symmetric TSPLIB [10] instanceswith various clusters and different cluster sizes It is to benoted that our HGA does not require any modification forsolving different types and cases of the instances Table 6shows comparative study between LSA and HGA The solu-tion quality by HGA is found to be insensitive to the numberof runs for most of the instances HGA finds bestoptimalsolution of twenty-three instances at least once in twentyruns whereas LSA could not find optimal solution for atleast three instances within four hours of computational timefor example the instances gr48 with clusters (23 24) andeil51 with clusters (25 25) and (16 17 17) Overall for thesesymmetric instances solution quality by HGA is found to

10 The Scientific World Journal

Table 7 Results on some symmetric TSPLIB instances using HGA

Instance Clusters BestSol AvgSol FTime CTime

berlin52 (51) 7542 (000) 754200 070 363(25 26) 10422 1042200 034 250

brazil58 (57) 25395 (000) 2539500 109 450(28 29) 34110 3411000 114 305

st70 (69) 675 (000) 67715 149 615(34 35) 916 91600 205 418

eil76 (75) 538 (000) 53926 242 733(37 38) 721 72312 199 511

pr76 (75) 108159 (000) 10825455 209 746(37 38) 120436 12058313 274 510

gr96 (95) 55209 (000) 5567285 589 1207(47 48) 56757 5676722 344 840

rat99 (98) 1211 (000) 121840 116 1291(49 49) 1346 134825 553 910

kroA100 (99) 21282 (000) 2132180 367 1302(24 25 25 25) 45733 4614795 340 677

kroB100 (99) 22141 (000) 2219315 490 1348(24 25 25 25) 45709 4581385 252 724

kroC100 (99) 20749 (000) 2078945 304 1262(24 25 25 25) 46388 4647535 410 690

kroD100 (99) 21294 (000) 2138911 464 1226(24 25 25 25) 45681 4595220 343 607

kroE100 (99) 22068 (000) 2211639 518 1452(24 25 25 25) 45431 4555925 324 727

rd100 (99) 7910 (000) 793270 475 1346(24 25 25 25) 15501 1552405 384 634

eil101 (100) 629 (000) 63275 579 1607(25 25 25 25) 1080 108000 380 892

lin105 (104) 14379 (000) 1441665 560 1439(26 26 26 26) 17584 1761820 213 825

pr107 (106) 44303 (000) 4440567 177 1456(26 26 27 27) 51487 5153880 214 779

gr120 (119) 6942 (000) 698695 616 2073(29 30 30 30) 13109 1312915 523 1048

pr124 (123) 59030 (000) 5918175 351 2077(30 31 31 31) 71295 7129500 112 1238

bier127 (126) 118282 (000) 11841960 933 2827(30 32 32 32) 174112 17425070 683 1954

ch130 (129) 6110 (000) 615050 1311 3012(32 32 32 33) 12000 1202205 566 2005

pr136 (135) 96772 (000) 9724080 1480 2836(33 34 34 34) 106605 10671840 814 2004

gr137 (136) 69853 (000) 7042950 1200 2816(34 34 34 34) 81628 8171501 422 1499

pr144 (143) 58537 (000) 5867119 618 3083(35 36 36 36) 69093 6912834 258 2022

kroA150 (149) 26524 (000) 2662965 1233 3585(37 37 37 38) 52824 5298840 1274 1880

kroB150 (149) 26130 (000) 2626423 1721 3807(37 37 37 38) 54008 5423775 1315 1916

The Scientific World Journal 11

Table 7 Continued

Instance Clusters BestSol AvgSol FTime CTime

ch150 (149) 6528 (000) 655631 2092 3859(37 37 37 38) 13042 1308525 1020 1904

pr152 (151) 73682 (000) 7401745 1192 3425(37 38 38 38) 79941 7994100 180 2428

u159 (158) 42080 (000) 4233610 1285 3887(39 39 40 40) 42287 4230245 329 2094

si175 (174) 21407 (000) 2141210 5119 9568(43 43 44 44) 22893 2291065 761 3472

brg180 (179) 1950 (000) 201035 1429 4458(44 45 45 45) 19430 2106020 690 1529

rat195 (194) 2323 (000) 236220 2713 7136(48 48 49 49) 2544 255172 2086 3524

d198 (197) 15800 (013) 1585875 2460 7779(49 49 49 50) 17320 1733950 2976 4392

kroA200 (199) 29420 (018) 2961880 2108 7041(49 50 50 50) 62514 6294175 2345 3981

kroB200 (199) 29463 (009) 2980700 2387 7501(49 50 50 50) 62842 6325311 1828 4140

gr202 (201) 40160 (000) 4041385 3696 9725(50 50 50 51) 44176 4424820 2486 4043

ts225 (224) 126643 (000) 12700683 2851 9385(56 56 56 56) 171269 17154330 2573 4877

tsp225 (224) 3923 (018) 396735 3829 10530(56 56 56 56) 5133 517115 2011 5491

pr226 (225) 80467 (012) 8095360 1628 8769(56 56 56 57) 96508 9651010 2838 6322

gr229 (228) 134957 (026) 13618435 3733 11979(57 57 57 57) 143028 14363245 3204 5731

gil262 (261) 2391 (055) 240315 5308 15126(65 65 65 66) 4874 490645 6433 9372

pr264 (263) 49219 (017) 4981445 4875 14999(65 66 66 66) 60161 6029415 2278 9720

a280 (279) 2585 (023) 261405 10525 18703(69 70 70 70) 2740 274375 3275 9428

pr299 (298) 48375 (038) 4885706 3811 20284(74 74 75 75) 55253 5595165 4536 12361

lin318 (317) 42301 (065) 4267965 8568 25392(79 79 79 80) 52578 5281135 6229 13335

rd400 (399) 15370 (058) 1545220 20327 54674(99 100 100 100) 30821 3100660 15140 25978

fl417 (416) 11930 (058) 1200463 22738 54417(104 104 104 104) 20457 2057624 20162 34601

gr431 (430) 173270 (108) 17604720 30035 71664(107 107 108 108) 185162 18666156 29181 40647

Average 2986 7016

be better and on average solution quality by HGA is 024better than that by LSA

In terms of computational time it can be easily concludedthat HGA ismuch better than LSA though LSAwas executedon slower machine (Pentium IV PC with speed 3GHz and

448 MB RAM) Of course the nature of LSA and HGA isnot the same LSA gives exact optimal solution whereas HGAgives heuristic solution It is also seen from the table thaton average HGA hits final best solution for the first timewithin 13 of complete computational time This shows that

12 The Scientific World Journal

HGA finds best solution on average in the beginning of thegenerations for these instances

44 Proposed Solution for Some More Symmetric InstancesTable 7 presents results for some more symmetric TSPLIBinstances of sizes from 52 to 431 with various clusters andcluster sizes Since to the best of our knowledge no literaturepresents solution for these instances hence we could notprovide any comparative study on these instances Howeverwe present the results for future study of the OCTSP on theseinstances For our self-comparison we provide solution valueand percentage of error (in parentheses) by our HGA forthe instances with one cluster which are of course usualTSP instances Out of forty-seven usual TSP instances HGAfinds exact optimal solution to thirty-three instances For theremaining instances maximum percentage of error is 108That means our algorithm can provide near exact solutionif not exact Treating this study as a base for effectiveness ofthe algorithm we can conclude that the reported solutionsare near exact solution if not exact It is also seen fromthe table that for the same instances as the number ofclusters increases solution value also increases On the otherhand as the number of clusters increases computational timedecreases In general computational time for solving a singleclustered instance (ie usual TSP instance) is more thanits corresponding multiclustered instances It seems that thestructures of these multiclustered instances are less complexand hence easier than their corresponding single clusteredinstances For these symmetric instances on average HGAhits final best solution for the first time within 43 ofcomplete computational time This shows that HGA findsbest solution for these instances on average in the middleof the generations

5 Conclusions

We presented a hybrid genetic algorithm using sequentialconstructive crossover 2-opt search a local search and animmigration method to obtain heuristic solution to theOCTSPWe have used a sequential samplingmethod for gen-erating initial population The efficiency of the hybrid GA tothe problemhas been examined against the exact partitioningalgorithm (LBDCOMP) [9] for some asymmetric TSPLIBinstances and the lexisearch algorithm (LSA) [11] for somesmall sized symmetric TSPLIB instances The computationalexperiments show that ourHGA is efficient in producing highquality of solution for the benchmark instances On the basisof solution quality our HGA is found to be better than theLBDCOMP and LSA In terms of computational time alsoour algorithm is found to be the best one Finally we presentsolution to the problem for some more symmetric TSPLIBinstances Since to the best of our knowledge no literaturepresents solution for these instances we could not confirmthe quality of our solutions for the instances However forthe symmetric instances of size up to 51 we found that ourHGA obtains exact optimal solution to the instances It isto be noted that HGA does not require any modification forsolving different types of TSPLIB instances

For any instance as the number of clusters increasesthe solution value also increases Computational time forsolving a single clustered instance (ie usual TSP instance) ismore than that for solving its corresponding multiclusteredinstances For any multiclustered instance as the clustersbecome more unbalanced computational time increases

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The author is very much thankful to the honourable review-ers for their constructive comments and suggestions Thisresearch was supported by the NSTIP Strategic Technologiesprogramno 10 in the Kingdomof Saudi Arabia via Award no11-INF1788-08 The author is also very much thankful to theNSTIP for its financial and technical supports

References

[1] J A Chisman ldquoThe clustered traveling salesman problemrdquoComputers and Operations Research vol 2 no 2 pp 115ndash1191975

[2] MGendreau AHertz andG Laporte ldquoThe traveling salesmanproblem with backhaulsrdquo Computers and Operations Researchvol 23 no 5 pp 501ndash508 1996

[3] N Guttmann-Beck R Hassin S Khuller and B RaghavacharildquoApproximation algorithms with bounded performance guar-antees for the clustered traveling salesman problemrdquo Algorith-mica vol 28 no 4 pp 422ndash437 2000

[4] F C J Lokin ldquoProcedures for travelling salesman problemswith additional constraintsrdquo European Journal of OperationalResearch vol 3 no 2 pp 135ndash141 1979

[5] G Laporte J-Y Potvin and F Quilleret ldquoTabu search heuristicusing genetic diversification for the clustered traveling salesmanproblemrdquo Journal of Heuristics vol 2 no 3 pp 187ndash200 1997

[6] G Laporte and U Palekar ldquoSome applications of the clus-tered travelling salesman problemrdquo Journal of the OperationalResearch Society vol 53 no 9 pp 972ndash976 2002

[7] Z H Ahmed ldquoGenetic algorithm for the traveling salesmanproblem using sequential constructive crossover operatorrdquoInternational Journal of Biometrics amp Bioinformatics vol 3 no6 pp 96ndash105 2010

[8] Z H Ahmed ldquoA hybrid sequential constructive samplingalgorithm for the bottleneck traveling salesman problemrdquoInternational Journal of Computational Intelligence Researchvol 6 no 3 pp 475ndash484 2010

[9] T Aramgiatisiris ldquoAn exact decomposition algorithm for thetraveling salesman problemwith backhaulsrdquo Journal of Researchin Engineering and Technology vol 1 pp 151ndash164 2004

[10] TSPLIB 1995 httpcomoptifiuni-heidelbergdesoftwareTSPLIB95

[11] Z H Ahmed ldquoAn exact algorithm for the clustered travelingsalesman problemrdquo Opsearch vol 50 no 2 pp 215ndash228 2013

[12] J D E Little K G Murthy D W Sweeny and C KarelldquoAn algorithm for the travelling salesman problemrdquo OperationsResearch vol 11 pp 972ndash989 1963

The Scientific World Journal 13

[13] K Jongens and T Volgenant ldquoThe symmetric clustered trav-eling salesman problemrdquo European Journal of OperationalResearch vol 19 no 1 pp 68ndash75 1985

[14] M Gendreau G Laporte and J Y Potvin ldquoHeuristics for theclustered traveling salesman problemrdquo Tech Rep CRT-94-54Centre de Recherche sur les Transports Universite deMontrealMontreal Canada 1994

[15] J-Y Potvin and F Guertin ldquoA genetic algorithm for theclustered traveling salesman problem with an a priori order onthe clustersrdquo Tech Rep CRT-95-06 Centre de recherchesur lestransports Universite de Montreal Montreal Canada 1995

[16] J-Y Potvin and F Guertin ldquoThe clustered traveling salesmanproblem a genetic approachrdquo in Meta-Heuristics Theory ampApplications I H Osman and J Kelly Eds pp 619ndash631 KluwerAcademic Norwell Mass USA 1996

[17] S Anily J Bramel andAHertz ldquo53-Approximation algorithmfor the clustered traveling salesman tour and path problemsrdquoOperations Research Letters vol 24 no 1 pp 29ndash35 1999

[18] N Christofides ldquoWorst-case analysis of a new heuristic for thetraveling salesmanproblemrdquoTech Rep 388Graduate School ofIndustrial Administration Carnegie-Mellon University Pitts-burgh Pa USA 1976

[19] W Sheng N Xi M Song and Y Chen ldquoRobot path planningfor dimensional measurement in automotive manufacturingrdquoJournal of Manufacturing Science and Engineering Transactionsof the ASME vol 127 no 2 pp 420ndash428 2005

[20] C Ding Y Cheng andM He ldquoTwo-level genetic algorithm forclustered traveling salesman problem with application in large-scale TSPsrdquo Tsinghua Science and Technology vol 12 no 4 pp459ndash465 2007

[21] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley New York NY USA 1989

[22] Z H Ahmed ldquoA hybrid genetic algorithm for the bottlenecktraveling salesman problemrdquo ACM Transactions on EmbeddedComputing Systems vol 12 no 1 article 9 2013

[23] K Deb Optimization for Engineering Design Algorithms andExamples Prentice Hall India New Delhi India 1995

[24] Z H Ahmed ldquoMulti-parent extension of sequential construc-tive crossover for the travelling salesman problemrdquo Interna-tional Journal of Operational Research vol 11 no 3 pp 331ndash3422011

[25] Z H Ahmed ldquoAn experimental study of a hybrid geneticalgorithm for the maximum travelling salesman problemrdquoMathematical Sciences vol 7 no 1 pp 1ndash7 2013

[26] C-X Wang D-W Cui Z-R Wang and D Chen ldquoA novel antcolony system based on minimum 1-tree and hybrid mutationfor TSPrdquo in Proceedings of the 1st International Conferenceon Natural Computation (ICNC rsquo05) LNCS pp 1269ndash1278Springer Changsha China August 2005

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Stochastic AnalysisInternational Journal of

Page 3: Research Article The Ordered Clustered Travelling Salesman Problem: A Hybrid Genetic ...downloads.hindawi.com/journals/tswj/2014/258207.pdf · 2019-07-31 · Research Article The

The Scientific World Journal 3

1

3 5

4

2

76

Figure 1 An example of result of the OCTSP using GA

no new chromosome is created Some of the chromo-somes are copied (even more than once) to the nextgeneration probabilistically based on their objectivefunction value whereas some other chromosomes arediscarded

(b) Crossover It is a binary operator that applies to twoparent chromosomes with a large probability whichcreates new offspring chromosome(s) It is a veryimportant operator in GAs Also crossover operatortogether with selection operator is found to be themost powerful process in the GA search

(c) Mutation It is a unary operator that applies to eachof the chromosomes with a small probability It is theoccasional random change of some selected gene(s) ofa chromosome to diversify the GA search space

Starting from a randomly generated or heuristicallygenerated initial population the GAs search repeated theabove three operators until the stopping criterion is satisfiedCrossover operator is a unique feature of GAs that wishesto combine good quality parent chromosomes to createone or more new offspring chromosome(s) However it isseen that the crossover alone cannot generate high qualitychromosomes for the combinatorial optimization problemslike the TSP and its variations Thus powerful local searchmethods are incorporated to improve the quality of offspringchromosomes [5] In hybrid GAs crossover operator gener-ates new starting solutions for the local search methods

GAs are found to be successful heuristic algorithms forsolving the usual TSP and its variations However GAs donot guarantee the optimality of the solution but they can findvery good near optimal solution in very short time We areapplying crossover mutation and local search methods foreach cluster in the prespecified order for the OCTSP Resultof the GA for a 7-vertex problem instance is a complete touras shown in Figure 1

32 Bias Removal Bias removal step is adopted in lexisearchalgorithm [11] and found effective for the CTSP We alsoconsider the bias removal step in ourGAThemain advantageof the bias removal is that a large amount of the solutionvalue is kept fixed and for the remaining small value wehave to search The process for bias removal of the costmatrix is as follows subtract each row-minima from its

Table 1 The cost matrix with row-minima and column-minima

Vertex 1 2 3 4 5 6 7 Row-minima1 999 75 99 9 35 63 8 82 51 999 86 46 88 29 20 203 100 5 999 16 28 35 28 54 20 45 11 999 59 53 49 115 86 63 33 65 999 76 72 336 36 53 89 31 21 999 52 217 58 31 43 67 52 60 999 32Column-minima 9 0 0 1 0 9 0

Table 2 The reduced cost matrix

Vertex 1 2 3 4 5 6 71 999lowast 67 91 0 27 46 02 22 999lowast 66 25 68 0 03 86 0 999lowast 10 23 21 234 0 34 0 999lowast 48 33 385 44 30 0 31 999lowast 34 396 6 32 68 9 0 999lowast 317 18 0 12 35 21 29 999lowast

(lowastElements are left as 999)

corresponding row elements repeat the same column-wiseon the resultant matrix The total of the row-minima and thesubsequent column-minima is called the ldquobiasrdquo of the matrixHowever we have not incorporated clusters precedencerelations in our cost matrix This does not affect the value ofa chromosome since while generating a chromosome theclusters precedence relations are taken care of

The bias of the cost matrix given in Table 1 is (row-minima + column-minima = 129 + 19 =) 148 The reducedcost matrix (ie after removing bias of the matrix) is shownin Table 2 We shall solve the problem with respect to thereduced cost matrix After we find solution value with respectto the reduced matrix we shall add the bias to the value forfinding the solution value with respect to the original costmatrix

33 Alphabet Table Alphabet matrix 119860 = [119886(119894 119895)] is asquare matrix of order 119899 formed by positions of elementsof the reduced cost matrix of order 119899 119862

1015840= [1198881015840

119894119895] when

they are arranged in the nondecreasing order of their costsAlphabet table ldquo[119886(119894 119895) minus 119888

1015840

119894119886(119894119895)]rdquo is the combination of

elements (vertices) of matrix A and their costs in the reducedmatrix [11] The alphabet table for the reduced cost matrixin Table 2 is shown in Table 3 This alphabet table also is nottaking care of the clusters precedence relations

34 Improved Initial Population The path representation fora chromosome is used in our GA In this representation anyvertex is assigned to a unique natural number from 1 to 119899that is genes are natural numbers The path of a salesmanis represented by a chromosome which is a permutation ofnumber genes A gene segment is defined as a permutation

4 The Scientific World Journal

Table 3 The alphabet table

Vertex 119881mdashCost 119881mdashCost 119881mdashCost 119881mdashCost 119881mdashCost 119881mdashCost 119881mdashCost1 4mdash0 7mdash0 5mdash27 6mdash46 2mdash67 3mdash91 1mdash9992 6mdash0 7mdash0 1mdash22 4mdash25 3mdash66 5mdash68 2mdash9993 2mdash0 4mdash10 6mdash21 5mdash23 7mdash23 1mdash86 3mdash9994 1mdash0 3mdash0 6mdash33 2mdash34 7mdash38 5mdash48 4mdash9995 3mdash0 2mdash30 4mdash31 6mdash34 7mdash39 1mdash44 5mdash9996 5mdash0 1mdash6 4mdash9 7mdash31 2mdash32 3mdash68 6mdash9997 2mdash0 3mdash12 1mdash18 5mdash21 6mdash29 4mdash35 7mdash999

of the vertices in a cluster A chromosome is a permutationof all the gene segments with one gene segment per clusterFor example let 1 2 3 4 5 6 7 be the vertices with 119881

1=

2 3 4 1198812

= 5 6 7 and 1198811is followed by 119881

2 in a 7-

vertex instance then starting from vertex 1 a complete tour1rarr 3rarr 4rarr 2rarr 6rarr 7rarr 5rarr 1 may be represented as thechromosome (1 3 4 2 6 7 5) where (3 4 2) and (6 7 5) arethe gene segments for cluster 1 and cluster 2 respectively

It is to be noted that starting from a good initialpopulation can deliver better quality of solutions quicklyand that is why many literatures report generating initialpopulation using heuristics Hence we are going to usesequential sampling algorithm for heuristically generatinginitial population that has been applied successfully to thebottleneck TSP [22]This algorithm is a simple version of thesequential constructive sampling algorithm [8] It is basicallya probabilistic method to generate a tour of the salesmanTheprobability of visiting each unvisited vertex of a cluster in arow of the alphabet table is assigned in such a way that thefirst unvisited vertex gets more probability than the secondone and so on Thereafter cumulative probability of eachunvisited vertex of a cluster is calculated Next a randomnumber 119903 isin [0 1] is generated and the vertex that representsthe chosen random number in the cumulative probabilityrange is accepted The probability of visiting each unvisitedvertex of a cluster is assigned as follows Suppose the numberof unvisited vertices of a cluster in a row of the alphabet tableis 119896 The probability of visiting the 119894th unvisited vertex is

119901119894=

2 (119896 minus 119894 + 1)

119896 (119896 + 1) (1)

The algorithm may be summarized as follows

Step 0 Construct the ldquoalphabet tablerdquo based on the reducedcost matrix Repeat the following steps for the fixed popula-tion size (119875

119904)

Step 1 Since ldquovertex 1rdquo is the starting vertex so initialize 119901 =

1 and go to Step 2

Step 2 Go to the 119901th row of the ldquoalphabet tablerdquo and visitprobabilistically by using (1) any unvisited vertex of the row(say vertex 119902) in the present cluster and go to Step 3

Step 3 Rename the ldquovertex 119902rdquo as ldquovertex 119901rdquo and go to Step 4

Step 4 If all vertices of the present cluster are visited then goto the next cluster in the order (if any) and go to Step 5 elsego to Step 2

Step 5 If all vertices of the network are visited then go toStep 1 for generating another chromosome in the populationelse go to Step 2

Let us illustrate the algorithm through the example givenin Table 1 with 119881

1= 2 3 4 119881

2= 5 6 7 and 119881

1is followed

by 1198812 We start from 1st row of the ldquoalphabet tablerdquo The

number of unvisited vertices of the 1st cluster in the row is3 which are 4 2 and 3 with cumulative probabilities 05000833 and 1000 respectively Suppose the vertex 4 is selectedrandomly then the partial chromosome will be (1 4) Nextwe go to 4th row of the ldquoalphabet tablerdquo and probabilisticallyselect another node and so on Proceeding in this way it maylead to a complete chromosome (1 4 2 3 6 5 7)

A preliminary study shows the effectiveness of the sam-pling algorithm for initial population However instead ofconsidering all unvisited vertices if we consider at most firstten vertices in a cluster then the algorithm generates verygood population A similar observation has been reportedfor the bottleneck TSP also [22] Hence we consider thisrestricted domain of unvisited vertices of a cluster for ourstudy Further to start with better population we apply 2-optsearch to each chromosome The 2-opt search removes twoedges and then replaces them by a different set of edges insuch a way so as to maintain the feasibility of the tour Let 120572

119894

120572119894+1

120572119895 and 120572

119895+1be four vertices in a cluster then if the edges

(120572119894 120572119894+1

) and (120572119895 120572119895+1

) are removed the only way to form anew valid tour is to connect 120572

119894to 120572119895and 120572

119894+1to 120572119895+1

35 Fitness Function and Selection Method The objectivefunction of each chromosome in the population is the cost ofthe tour represented by the chromosomeThe fitness functionof a chromosome is defined as multiplicative inverse of theobjective functionThere are various selectionmethods in theliteratureThe selection operation considered for our study isthe stochastic remainder selection method [23]

36 The Sequential Constructive Crossover Operation Sincecrossover operation plays main role in GAs hence severalcrossover operators have been proposed for the usual TSPwhich are then used for the variant TSPs also Out of themthe sequential constructive crossover (SCX) [7] is found to

The Scientific World Journal 5

1

3

5

4

2

7

6

1

3

5

4

2

7

6

1

3

5

4

2

7

6

(a) P1 (1 2 4 3 6 7 5) (b) P2 (1 2 4 3 6 7 5) (c) Offspring (1 2 4 3 6 5 7)

Figure 2 Example of sequential constructive crossover operation

be one of the best crossover operators for the usual TSP Amultiparent extension of the SCX has been applied to theusual TSP and found good results [24]The SCXhas also beensuccessfully applied to the TSP with some other variations[22 25] In general it produces an offspring using betteredges of the parents However it does not depend only onthe parentsrsquo structure it sometimes introduces new but goodedges to the offspring which are not even available in thepresent population We modify the SCX operator for theOCTSP as follows

Step 1 Start from ldquovertex 1rdquo (ie current vertex 119901 = 1)

Step 2 Sequentially search both of the parent chromosomesand consider the first unvisited vertex of the present clusterappearing after ldquovertex 119901rdquo in each parent If no unvisitedvertex after ldquovertex 119901rdquo is available in any (or both) of theparents search sequentially from the starting of that parentand consider the first unvisited vertex of the cluster and goto Step 3

Step 3 Suppose the ldquovertex 120572rdquo and the ldquovertex 120573rdquo are foundin the 1st and 2nd parents respectively then for selecting thenext vertex in the offspring chromosome go to Step 4

Step 4 If 119888119901120572

lt 119888119901120573 then select ldquovertex 120572rdquo otherwise select

ldquovertex120573rdquo as the next vertex and concatenate it to the partiallyconstructed offspring chromosome and go to Step 5

Step 5 If there is not any vertex left in that cluster thengo to the next cluster if any If the offspring is a completechromosome then stop otherwise rename the present vertexas ldquovertex 119901rdquo and go to Step 2

Let a pair of parent chromosomes be 1198751 (1 2 4 3 6

7 5) and 1198752 (1 3 2 4 6 5 7) with costs 357 and 354

respectively with respect to the original cost matrix givenin Table 1 By applying above SCX we obtain the offspringchromosome (1 2 4 3 6 5 7) with cost 318 which is lessthan both parentsTheparent and the offspring chromosomesare shown in Figure 2 In general crossover operator inheritsparentsrsquo characteristics and the operator that preserves goodcharacteristics of parents in the offspring is said to be goodoperatorThe SCX is found to be excellent in this regard Boldedges in Figure 2(c) are the edges which are available either in

the first parent or in second parent For this given example alledges are selected from either of the parents

For the crossover operation a pair of parent chromo-somes is selected sequentially from the mating pool It isreported that the SCX gets stuck in local minimums quicklyfor the TSP [7] which is very often due to the identicalpopulation So to overcome this situation the selectedparents are checked for duplication If the selected parentsare found to be identical then the second parent is modifiedtemporarily by swapping some randomly chosen pair of genesin the chromosome and then the crossover operation isperformed To improve quality of the solution as well as havea mixture of parents and offspring in a population the firstparent is replaced by the offspring only if the offspring valueis better than the average value of the present populationIn this way the mixed population retains diversity alsoTo further improve the quality of the solution obtained bycrossover many researchers applied 2-opt search operator Toimprove solution quality we are going to use a local searchmethod that combines three mutation operators that will bediscussed in Section 38 However we are not applying thislocal search method to all of the offspring rather it is appliedonly to the offspring if its value is better than the averagepopulation valueNow since our crossover operator producesonly one offspring to keep population size fixed throughoutthe generations while pairing with the next chromosome inorder the present second original parent chromosomewill beconsidered as the first parent and so on

37 Mutation Operation The mutation operator randomlyselects a position in the chromosome and changes the cor-responding gene thereby modifying information The needfor mutation comes from the fact that as the less fit chromo-somes of successive generations are discarded some aspectsof genetic material could be lost forever By performingoccasional random changes in the chromosomes GAs ensurethat newparts of the search space are reachedwhich selectionand crossover could not fully guarantee In doing somutationensures that no important features are prematurely lost thusmaintaining the mating pool diversity For this investigationwe have considered reciprocal exchange mutation operatorthat selects two genes randomly of a chromosome in everycluster and swaps them The probability of mutation isusually chosen to be considerably less than the probability

6 The Scientific World Journal

1

3

5

4

2

7

6

1

3

5

4

2

7

6

(a) (1 2 4 3 6 7 5) (b) (1 4 2 3 6 5 7)

Figure 3 Example of reciprocal exchange mutation operation

of crossover So mutation plays a secondary role in the GAsearch For example let the chromosome (1 2 4 3 6 7 5)be selected for mutation and vertices 2 and 4 are swapped incluster 1 and vertices 7 and 5 are swapped in cluster 2 thenthe mutated chromosome becomes (1 4 2 3 6 5 7) whichis shown in Figure 3 Bold edges in Figure 3(b) are the newedges in the mutated chromosome

38 A Local Search Method We have considered the com-bined mutation operation as a local search method whichhas been successfully applied to the bottleneck TSP [822] and maximum TSP [25] It combines three mutationoperators insertion inversion and reciprocal exchange withcent percentage of probabilities Insertion operator selects avertex (gene) in a chromosome and inserts it in a randomplace and inversion operator selects two points along thelength of a chromosome and reverses the subchromosomesbetween these points This local search a modification ofthe hybrid mutation operator [26] is applied to a chro-mosome Recall that sizes of the clusters 119881

1 1198812 119881

119898are

1198991 1198992 119899

119898 respectively Suppose (1 = 120572

1 1205722 1205723 120572

119899) is

a chromosome then the local search for the OCTSP can bedeveloped as follows

Step 0 Set 119909 = 2 and 119910 = 1

Step 1 For 119894 = 1 to 119898 perform Step 2

Step 2 Set 119910 = 119899119894+ 119910 and go to Step 3

Step 3 For 119895 = 119909 to (119910 minus 1) perform Step 4

Step 4 For 119896 = (119895 + 1) to 119910 perform Step 5

Step 5 If inserting vertex 120572119895after vertex 120572

119896reduces the

present tour cost then insert the vertex 120572119895after vertex 120572

119896 In

either case go to Step 6

Step 6 If inverting subchromosome between the vertices120572119895and 120572

119896reduces the present tour cost then invert the

subchromosome In either case go to Step 7

Step 7 If swapping the vertices 120572119895and 120572

119896reduces the present

tour cost then swap them In either case go to Step 8

Step 8 Set 119909 = 119910 + 1 and go to Step 1

39 Immigration It is seen that sometimes GAs get stuckin local minimums for the combinatorial optimization prob-lems which is very often due to the identical populationSo to improve capability of GAs the population should bediversified To diversify the population immigrationmethodis also adopted where some randomly selected chromosomesare replaced by new chromosomes after some generations[22] We are also considering an immigration method Forour investigation 20 of the population is replaced ran-domly using sequential sampling algorithm as discussed inSection 34 if no improvement is found within the last 20generations Once the immigration is applied we wait for thenext 20 generations for any improvement Our hybrid GA(HGA) for the OCTSP may be summarized as in Figure 4[22]

4 Results and Discussions

We encoded our HGA in Visual C++ executed on a PC with340GHz Intel(R) Core (TM) i7-3770CPU and 800GBRAMunderMSWindows 7 operating system and testedwith someTSPLIB [10] instances

41 Parameter Setting GAs are well suited for the com-binatorial optimization problems They find near optimalsolution in reasonable time However they are guided bysuitable choice of parameters namely crossover probability(119875119888) mutation probability (119875

119898) population size (119875

119904) and

termination condition Successful working of GAs dependson a proper selection of these parameters [23] But there isnot any intelligent rule to set these parameters In generalvarious sets of the parameters are tested and then the bestone is selected We are also following a similar methodSo we set the parameters as follows a maximum of 20000generations as termination condition 20 as population size100 (100) as crossover probability and 20 independentruns for each setting However we are not reporting ourexperiments except for the mutation probability

To set mutation probability six mutation probabilities000 001 002 003 004 and 005 are considered andtested on five asymmetric TSPLIB instances with four clusters(1198991 1198992 1198993 and 119899

4) for each of the instances ftv110 ftv120

ftv130 ftv140 and ftv150 For example the 7-vertex instancewith two clusters (3 3)means119881

1= 2 3 4119881

2= 5 6 7 and

1198811is followed by 119881

2

Table 4 reports the mean and standard deviation (inparenthesis) of the best solution values over 20 trials on fiveinstances ftv110ndashftv150 for different mutation probabilitiesThe boldface denotes the best average solution value It is seenthat there is significant improvement of the solutions usingnonzero mutation probabilities over using zero mutationprobability It shows that mutation operation also plays animportant role in GAs Mutation probabilities 003 and 004

The Scientific World Journal 7

Table 4 Mean and standard deviation of best solution values on five asymmetric TSPLIB instances

Instance Clusters 119875119898

= 000 119875119898

= 001 119875119898

= 002 119875119898

= 003 119875119898

= 004 119875119898

= 005

ftv110 (29 27 27 27) 270978 (6750) 252634 (3184) 250800 (2521) 249080 (2351) 252835 (3456) 252867 (2479)ftv120 (30 30 30 30) 283967 (8294) 263634 (4194) 262878 (4089) 259623 (3020) 259568 (3536) 259579 (2722)ftv130 (34 32 32 32) 304411 (9377) 284489 (5532) 286056 (4003) 281756 (2442) 283578 (3423) 284166 (1967)ftv140 (35 35 35 35) 320445 (15065) 306831 (5807) 308629 (5450) 307126 (4318) 307002 (4943) 307741 (5558)ftv150 (39 37 37 37) 357625 (15455) 327566 (5772) 327682 (2714) 326526 (3560) 326555 (2964) 326939 (5968)

Start

Improved initial population

Evaluate the population and assign best chromosome cost as the best solution value

Is termination condition satisfied

Print the best solution value and

the best tour

Yes Selection operation

No

Sequential constructive crossover operation

StopMutation operation with

mutation probability

Evaluate the population

Is best population value better than

best solution value

No

Update best solution value

Yes

Local search to the best chromosome

Immigration

Yes

No

Is number ofgeneration gt20

till last update

Figure 4 Flowchart of our hybrid genetic algorithm

are competing Using 119875119898

= 003 the algorithm obtainsthe best average solution for the instances ftv110 ftv120 andftv150 For the remaining two instances the algorithmobtainsthe best average solution at 119875

119898= 004 However if we look

at the standard deviation solutions are relatively stable at119875119898

= 003Figure 5 plots the average best solution values for the five

instances obtained by the HGA using mutation probabilitiesfrom 000 to 005 The figure shows clearly the effectivenessof mutation operator It is seen that as mutation probabilityincreases solution quality also increases However after119875

119898=

004 solution quality is not found to be good From the tableand the figure we can conclude that 119875

119898= 003 is suitable for

our algorithm Hence we are going to use 119875119898

= 003 for ourfurther study

42 Comparative Study on Asymmetric Instances We presenta comparative study between HGA and LBDCOMP [9] forsome asymmetric TSPLIB instances of sizes from 34 to 171

with various clusters and different cluster sizes It is to bementioned that LBDCOMP [9] is claimed to find exactoptimal solution of the OCTSP instances which has beendisproved by showing results of some small sized instances[11] Anyway since no other literature reports the exactsolution for large size instances we are going to comparewith the LBDCOMP algorithm to see solution quality by ourHGA Table 5 shows this comparative study between HGAand LBDCOMPThe table reports results by LBDCOMP andbest solution value (BestSol) average solution value (AvgSol)in 20 runs average complete computational time (CTime)average computational time when final best solution is seenfor the first time (FTime) in twenty runs and percentage oferror (Error()) of the best solution obtained by our HGAThe percentage of error is calculated by the formula

Error () =119861119890119904119905119878119900119897 minus 119874119901119905119878119900119897

119874119901119905119878119900119897times 100 (2)

8 The Scientific World Journal

Table 5 A comparative study between LBDCOMP and HGA for asymmetric TSPLIB instances

Instance Clusters LBDCOMP HGASolution Time BestSol Error () AvgSol FTime CTime

ftv33(16 17) 1584 511 1501 minus524 150216 012 101(9 24) 1509 587 1501 minus053 150324 013 125(3 30) 1356 500 1356 000 135915 018 163

ftv35(17 18) 1747 1132 1731 minus092 173929 041 113(10 25) 1660 346 1660 000 166334 032 126(3 32) 1527 1389 1527 000 153313 042 141

ftv38(19 19) 1681 413 1681 000 168606 096 135(11 27) 1689 765 1689 000 169225 022 153(3 35) 1573 1766 1573 000 158600 016 211

ftv44(22 22) 1935 2492 1935 000 194023 017 175(13 31) 1830 790 1830 000 185212 047 201(4 40) 1670 4871 1670 000 168900 040 272

ftv47(23 24) 2470 1006 2470 000 252624 078 193(13 34) 2349 1049 2257 minus392 228127 115 242(4 43) 1957 593 1957 000 200631 152 317

ftv55(27 28) 2299 600 2219 minus348 224824 071 260(16 39) 1937 815 1937 000 198212 207 306(5 50) 1763 3206 1763 000 178824 116 448

ftv64(32 32) 2658 2854 2658 000 268615 215 368(19 45) 2383 6527 2383 000 249411 298 437(6 58) 2006 11093 2006 000 204821 224 638

ftv70(35 35) 2308 13586 2308 000 234130 130 411(21 49) 2244 10269 2244 000 226724 272 479(7 63) 2134 32331 2134 000 216332 460 737

ftv90(45 45) 1756 1771 1756 000 183378 528 693(27 63) 1710 5671 1710 000 178480 555 846(9 81) 1579 6824 1579 000 165322 579 1443

ftv100(50 50) 2008 2413 2008 000 208417 489 890(30 70) 1903 14351 1903 000 196966 759 1188(10 90) 1788 18749 1788 000 190415 1172 1927

ftv110 (29 27 27 27) 2410 28919 2411 004 249080 608 748ftv120 (30 30 30 30) 2571 8396 2571 000 259623 566 904ftv130 (34 32 32 32) 2747 33167 2751 015 281756 595 1120ftv140 (35 35 35 35) 2941 57144 2947 020 307126 1002 1323ftv150 (39 37 37 37) 3119 8142 3120 003 326526 794 1579ftv160 (40 40 40 40) 3561 75454 3580 053 369618 1251 1834ftv170 (44 42 42 42) 3927 29771 3891 minus092 399201 1536 2239Average 10548 minus038 356 635

whereBestSol denotes the best solution obtained byHGA andOptSol denotes the solution obtained by LBDCOMP

It is seen from Table 5 that our HGA finds bestoptimalsolution of thirty-two instances at least once in twenty runswhereas LBDCOMP could not find optimal solution for atleast six instancesmdashftv33 with clusters (16 17) and (9 24)ftv35 with clusters (17 18) ftv47 with clusters (13 34) ftv55with clusters (27 28) and ftv170 with clusters (44 42 42 42)That is for these six instances solution quality by HGA is

found to better On the other hand for five instances namelyftv110 ftv130 ftv140 ftv150 and ftv160 with four clusterseach solution quality by LBDCOMP is better than by ourHGA For these five instances percentage of error by HGA isat most 053 However on average solution quality byHGAis 038 better than that of by LBDCOMP

In terms of computational time we cannot directlycompare the algorithms because they are executed in differentmachines and it was not possible to access the original code

The Scientific World Journal 9

Table 6 A comparative study between LSA and HGA for symmetric TSPLIB instances

Instance Clusters LSA HGASolution Time BestSol Error () AvgSol FTime CTime

burma14 (6 7) 3621 000 3621 000 362100 000 025ulysses16 (7 8) 7303 000 7303 000 730300 000 030gr17 (8 8) 2517 000 2517 000 251700 000 032gr21 (10 10) 3465 000 3465 000 346500 000 048ulysses22 (10 11) 8190 017 8190 000 819000 000 054gr24 (11 12) 1558 014 1558 000 155800 032 063fri26 (12 13) 957 005 957 000 95700 000 062

bayg29 (14 14) 2144 2103 2144 000 214400 007 093(9 9 10) 2408 3522 2408 000 240800 000 065

bays29 (14 14) 2702 2733 2702 000 270200 000 094(9 9 10) 2991 2489 2991 000 299100 000 066

dantzig42(20 21) 699 44656 699 000 69900 000 151

(13 14 14) 699 102 699 000 69900 002 126(10 10 10 11) 699 517 699 000 69900 000 112

swiss42(20 21) 1605 1440000 1605 000 161233 077 275

(13 14 14) 1919 1440000 1919 000 192300 053 159(10 10 10 11) 1944 1440000 1944 000 194542 015 115

gr48(23 24) 6656 1440000 6433 minus335 643300 007 201

(15 16 16) 7466 1440000 7466 000 750472 004 158(11 12 12 12) 8554 1440000 8554 000 855400 038 143

eil51(25 25) 570 1440000 564 minus105 56400 073 227(16 17 17) 689 1440000 681 minus116 68100 015 173

(12 12 13 13) 714 1440000 714 000 71400 012 162Average 565920 minus024 015 115

2400

2600

2800

3000

3200

3400

3600

Aver

age b

est s

olut

ion

valu

e

ftv110ftv120ftv130

ftv140ftv150

Pm=000

Pm=001

Pm=002

Pm=003

Pm=004

Pm=005

Figure 5 Average best solution values on five asymmetric TSPLIBinstances using six mutation probabilities

of LBDCOMP However a large gap between computationaltime by LBDCOMP and HGA is seen in the table and HGAtakes much less time Further if FTime is considered forHGA then definitely it is found to be much better thanLBDCOMP It is interesting to see that for any of these

instances with the same number of clusters but differentcluster sizesHGA takes different computational times and asthe size of clusters becomesmore unbalanced computationaltime increases In an unbalanced clustered instance size ofthe clusters is not equal It is also seen that on averageHGA hits final best solution for the first time within 56of complete computational time This shows that HGA findsbest solution on average in the middle of the generations forthese asymmetric TSPLIB instances

43 Comparative Study on Symmetric Instances Now we aregoing to compare our HGA with lexisearch algorithm (LSA)[11] on some small sized symmetric TSPLIB [10] instanceswith various clusters and different cluster sizes It is to benoted that our HGA does not require any modification forsolving different types and cases of the instances Table 6shows comparative study between LSA and HGA The solu-tion quality by HGA is found to be insensitive to the numberof runs for most of the instances HGA finds bestoptimalsolution of twenty-three instances at least once in twentyruns whereas LSA could not find optimal solution for atleast three instances within four hours of computational timefor example the instances gr48 with clusters (23 24) andeil51 with clusters (25 25) and (16 17 17) Overall for thesesymmetric instances solution quality by HGA is found to

10 The Scientific World Journal

Table 7 Results on some symmetric TSPLIB instances using HGA

Instance Clusters BestSol AvgSol FTime CTime

berlin52 (51) 7542 (000) 754200 070 363(25 26) 10422 1042200 034 250

brazil58 (57) 25395 (000) 2539500 109 450(28 29) 34110 3411000 114 305

st70 (69) 675 (000) 67715 149 615(34 35) 916 91600 205 418

eil76 (75) 538 (000) 53926 242 733(37 38) 721 72312 199 511

pr76 (75) 108159 (000) 10825455 209 746(37 38) 120436 12058313 274 510

gr96 (95) 55209 (000) 5567285 589 1207(47 48) 56757 5676722 344 840

rat99 (98) 1211 (000) 121840 116 1291(49 49) 1346 134825 553 910

kroA100 (99) 21282 (000) 2132180 367 1302(24 25 25 25) 45733 4614795 340 677

kroB100 (99) 22141 (000) 2219315 490 1348(24 25 25 25) 45709 4581385 252 724

kroC100 (99) 20749 (000) 2078945 304 1262(24 25 25 25) 46388 4647535 410 690

kroD100 (99) 21294 (000) 2138911 464 1226(24 25 25 25) 45681 4595220 343 607

kroE100 (99) 22068 (000) 2211639 518 1452(24 25 25 25) 45431 4555925 324 727

rd100 (99) 7910 (000) 793270 475 1346(24 25 25 25) 15501 1552405 384 634

eil101 (100) 629 (000) 63275 579 1607(25 25 25 25) 1080 108000 380 892

lin105 (104) 14379 (000) 1441665 560 1439(26 26 26 26) 17584 1761820 213 825

pr107 (106) 44303 (000) 4440567 177 1456(26 26 27 27) 51487 5153880 214 779

gr120 (119) 6942 (000) 698695 616 2073(29 30 30 30) 13109 1312915 523 1048

pr124 (123) 59030 (000) 5918175 351 2077(30 31 31 31) 71295 7129500 112 1238

bier127 (126) 118282 (000) 11841960 933 2827(30 32 32 32) 174112 17425070 683 1954

ch130 (129) 6110 (000) 615050 1311 3012(32 32 32 33) 12000 1202205 566 2005

pr136 (135) 96772 (000) 9724080 1480 2836(33 34 34 34) 106605 10671840 814 2004

gr137 (136) 69853 (000) 7042950 1200 2816(34 34 34 34) 81628 8171501 422 1499

pr144 (143) 58537 (000) 5867119 618 3083(35 36 36 36) 69093 6912834 258 2022

kroA150 (149) 26524 (000) 2662965 1233 3585(37 37 37 38) 52824 5298840 1274 1880

kroB150 (149) 26130 (000) 2626423 1721 3807(37 37 37 38) 54008 5423775 1315 1916

The Scientific World Journal 11

Table 7 Continued

Instance Clusters BestSol AvgSol FTime CTime

ch150 (149) 6528 (000) 655631 2092 3859(37 37 37 38) 13042 1308525 1020 1904

pr152 (151) 73682 (000) 7401745 1192 3425(37 38 38 38) 79941 7994100 180 2428

u159 (158) 42080 (000) 4233610 1285 3887(39 39 40 40) 42287 4230245 329 2094

si175 (174) 21407 (000) 2141210 5119 9568(43 43 44 44) 22893 2291065 761 3472

brg180 (179) 1950 (000) 201035 1429 4458(44 45 45 45) 19430 2106020 690 1529

rat195 (194) 2323 (000) 236220 2713 7136(48 48 49 49) 2544 255172 2086 3524

d198 (197) 15800 (013) 1585875 2460 7779(49 49 49 50) 17320 1733950 2976 4392

kroA200 (199) 29420 (018) 2961880 2108 7041(49 50 50 50) 62514 6294175 2345 3981

kroB200 (199) 29463 (009) 2980700 2387 7501(49 50 50 50) 62842 6325311 1828 4140

gr202 (201) 40160 (000) 4041385 3696 9725(50 50 50 51) 44176 4424820 2486 4043

ts225 (224) 126643 (000) 12700683 2851 9385(56 56 56 56) 171269 17154330 2573 4877

tsp225 (224) 3923 (018) 396735 3829 10530(56 56 56 56) 5133 517115 2011 5491

pr226 (225) 80467 (012) 8095360 1628 8769(56 56 56 57) 96508 9651010 2838 6322

gr229 (228) 134957 (026) 13618435 3733 11979(57 57 57 57) 143028 14363245 3204 5731

gil262 (261) 2391 (055) 240315 5308 15126(65 65 65 66) 4874 490645 6433 9372

pr264 (263) 49219 (017) 4981445 4875 14999(65 66 66 66) 60161 6029415 2278 9720

a280 (279) 2585 (023) 261405 10525 18703(69 70 70 70) 2740 274375 3275 9428

pr299 (298) 48375 (038) 4885706 3811 20284(74 74 75 75) 55253 5595165 4536 12361

lin318 (317) 42301 (065) 4267965 8568 25392(79 79 79 80) 52578 5281135 6229 13335

rd400 (399) 15370 (058) 1545220 20327 54674(99 100 100 100) 30821 3100660 15140 25978

fl417 (416) 11930 (058) 1200463 22738 54417(104 104 104 104) 20457 2057624 20162 34601

gr431 (430) 173270 (108) 17604720 30035 71664(107 107 108 108) 185162 18666156 29181 40647

Average 2986 7016

be better and on average solution quality by HGA is 024better than that by LSA

In terms of computational time it can be easily concludedthat HGA ismuch better than LSA though LSAwas executedon slower machine (Pentium IV PC with speed 3GHz and

448 MB RAM) Of course the nature of LSA and HGA isnot the same LSA gives exact optimal solution whereas HGAgives heuristic solution It is also seen from the table thaton average HGA hits final best solution for the first timewithin 13 of complete computational time This shows that

12 The Scientific World Journal

HGA finds best solution on average in the beginning of thegenerations for these instances

44 Proposed Solution for Some More Symmetric InstancesTable 7 presents results for some more symmetric TSPLIBinstances of sizes from 52 to 431 with various clusters andcluster sizes Since to the best of our knowledge no literaturepresents solution for these instances hence we could notprovide any comparative study on these instances Howeverwe present the results for future study of the OCTSP on theseinstances For our self-comparison we provide solution valueand percentage of error (in parentheses) by our HGA forthe instances with one cluster which are of course usualTSP instances Out of forty-seven usual TSP instances HGAfinds exact optimal solution to thirty-three instances For theremaining instances maximum percentage of error is 108That means our algorithm can provide near exact solutionif not exact Treating this study as a base for effectiveness ofthe algorithm we can conclude that the reported solutionsare near exact solution if not exact It is also seen fromthe table that for the same instances as the number ofclusters increases solution value also increases On the otherhand as the number of clusters increases computational timedecreases In general computational time for solving a singleclustered instance (ie usual TSP instance) is more thanits corresponding multiclustered instances It seems that thestructures of these multiclustered instances are less complexand hence easier than their corresponding single clusteredinstances For these symmetric instances on average HGAhits final best solution for the first time within 43 ofcomplete computational time This shows that HGA findsbest solution for these instances on average in the middleof the generations

5 Conclusions

We presented a hybrid genetic algorithm using sequentialconstructive crossover 2-opt search a local search and animmigration method to obtain heuristic solution to theOCTSPWe have used a sequential samplingmethod for gen-erating initial population The efficiency of the hybrid GA tothe problemhas been examined against the exact partitioningalgorithm (LBDCOMP) [9] for some asymmetric TSPLIBinstances and the lexisearch algorithm (LSA) [11] for somesmall sized symmetric TSPLIB instances The computationalexperiments show that ourHGA is efficient in producing highquality of solution for the benchmark instances On the basisof solution quality our HGA is found to be better than theLBDCOMP and LSA In terms of computational time alsoour algorithm is found to be the best one Finally we presentsolution to the problem for some more symmetric TSPLIBinstances Since to the best of our knowledge no literaturepresents solution for these instances we could not confirmthe quality of our solutions for the instances However forthe symmetric instances of size up to 51 we found that ourHGA obtains exact optimal solution to the instances It isto be noted that HGA does not require any modification forsolving different types of TSPLIB instances

For any instance as the number of clusters increasesthe solution value also increases Computational time forsolving a single clustered instance (ie usual TSP instance) ismore than that for solving its corresponding multiclusteredinstances For any multiclustered instance as the clustersbecome more unbalanced computational time increases

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The author is very much thankful to the honourable review-ers for their constructive comments and suggestions Thisresearch was supported by the NSTIP Strategic Technologiesprogramno 10 in the Kingdomof Saudi Arabia via Award no11-INF1788-08 The author is also very much thankful to theNSTIP for its financial and technical supports

References

[1] J A Chisman ldquoThe clustered traveling salesman problemrdquoComputers and Operations Research vol 2 no 2 pp 115ndash1191975

[2] MGendreau AHertz andG Laporte ldquoThe traveling salesmanproblem with backhaulsrdquo Computers and Operations Researchvol 23 no 5 pp 501ndash508 1996

[3] N Guttmann-Beck R Hassin S Khuller and B RaghavacharildquoApproximation algorithms with bounded performance guar-antees for the clustered traveling salesman problemrdquo Algorith-mica vol 28 no 4 pp 422ndash437 2000

[4] F C J Lokin ldquoProcedures for travelling salesman problemswith additional constraintsrdquo European Journal of OperationalResearch vol 3 no 2 pp 135ndash141 1979

[5] G Laporte J-Y Potvin and F Quilleret ldquoTabu search heuristicusing genetic diversification for the clustered traveling salesmanproblemrdquo Journal of Heuristics vol 2 no 3 pp 187ndash200 1997

[6] G Laporte and U Palekar ldquoSome applications of the clus-tered travelling salesman problemrdquo Journal of the OperationalResearch Society vol 53 no 9 pp 972ndash976 2002

[7] Z H Ahmed ldquoGenetic algorithm for the traveling salesmanproblem using sequential constructive crossover operatorrdquoInternational Journal of Biometrics amp Bioinformatics vol 3 no6 pp 96ndash105 2010

[8] Z H Ahmed ldquoA hybrid sequential constructive samplingalgorithm for the bottleneck traveling salesman problemrdquoInternational Journal of Computational Intelligence Researchvol 6 no 3 pp 475ndash484 2010

[9] T Aramgiatisiris ldquoAn exact decomposition algorithm for thetraveling salesman problemwith backhaulsrdquo Journal of Researchin Engineering and Technology vol 1 pp 151ndash164 2004

[10] TSPLIB 1995 httpcomoptifiuni-heidelbergdesoftwareTSPLIB95

[11] Z H Ahmed ldquoAn exact algorithm for the clustered travelingsalesman problemrdquo Opsearch vol 50 no 2 pp 215ndash228 2013

[12] J D E Little K G Murthy D W Sweeny and C KarelldquoAn algorithm for the travelling salesman problemrdquo OperationsResearch vol 11 pp 972ndash989 1963

The Scientific World Journal 13

[13] K Jongens and T Volgenant ldquoThe symmetric clustered trav-eling salesman problemrdquo European Journal of OperationalResearch vol 19 no 1 pp 68ndash75 1985

[14] M Gendreau G Laporte and J Y Potvin ldquoHeuristics for theclustered traveling salesman problemrdquo Tech Rep CRT-94-54Centre de Recherche sur les Transports Universite deMontrealMontreal Canada 1994

[15] J-Y Potvin and F Guertin ldquoA genetic algorithm for theclustered traveling salesman problem with an a priori order onthe clustersrdquo Tech Rep CRT-95-06 Centre de recherchesur lestransports Universite de Montreal Montreal Canada 1995

[16] J-Y Potvin and F Guertin ldquoThe clustered traveling salesmanproblem a genetic approachrdquo in Meta-Heuristics Theory ampApplications I H Osman and J Kelly Eds pp 619ndash631 KluwerAcademic Norwell Mass USA 1996

[17] S Anily J Bramel andAHertz ldquo53-Approximation algorithmfor the clustered traveling salesman tour and path problemsrdquoOperations Research Letters vol 24 no 1 pp 29ndash35 1999

[18] N Christofides ldquoWorst-case analysis of a new heuristic for thetraveling salesmanproblemrdquoTech Rep 388Graduate School ofIndustrial Administration Carnegie-Mellon University Pitts-burgh Pa USA 1976

[19] W Sheng N Xi M Song and Y Chen ldquoRobot path planningfor dimensional measurement in automotive manufacturingrdquoJournal of Manufacturing Science and Engineering Transactionsof the ASME vol 127 no 2 pp 420ndash428 2005

[20] C Ding Y Cheng andM He ldquoTwo-level genetic algorithm forclustered traveling salesman problem with application in large-scale TSPsrdquo Tsinghua Science and Technology vol 12 no 4 pp459ndash465 2007

[21] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley New York NY USA 1989

[22] Z H Ahmed ldquoA hybrid genetic algorithm for the bottlenecktraveling salesman problemrdquo ACM Transactions on EmbeddedComputing Systems vol 12 no 1 article 9 2013

[23] K Deb Optimization for Engineering Design Algorithms andExamples Prentice Hall India New Delhi India 1995

[24] Z H Ahmed ldquoMulti-parent extension of sequential construc-tive crossover for the travelling salesman problemrdquo Interna-tional Journal of Operational Research vol 11 no 3 pp 331ndash3422011

[25] Z H Ahmed ldquoAn experimental study of a hybrid geneticalgorithm for the maximum travelling salesman problemrdquoMathematical Sciences vol 7 no 1 pp 1ndash7 2013

[26] C-X Wang D-W Cui Z-R Wang and D Chen ldquoA novel antcolony system based on minimum 1-tree and hybrid mutationfor TSPrdquo in Proceedings of the 1st International Conferenceon Natural Computation (ICNC rsquo05) LNCS pp 1269ndash1278Springer Changsha China August 2005

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Page 4: Research Article The Ordered Clustered Travelling Salesman Problem: A Hybrid Genetic ...downloads.hindawi.com/journals/tswj/2014/258207.pdf · 2019-07-31 · Research Article The

4 The Scientific World Journal

Table 3 The alphabet table

Vertex 119881mdashCost 119881mdashCost 119881mdashCost 119881mdashCost 119881mdashCost 119881mdashCost 119881mdashCost1 4mdash0 7mdash0 5mdash27 6mdash46 2mdash67 3mdash91 1mdash9992 6mdash0 7mdash0 1mdash22 4mdash25 3mdash66 5mdash68 2mdash9993 2mdash0 4mdash10 6mdash21 5mdash23 7mdash23 1mdash86 3mdash9994 1mdash0 3mdash0 6mdash33 2mdash34 7mdash38 5mdash48 4mdash9995 3mdash0 2mdash30 4mdash31 6mdash34 7mdash39 1mdash44 5mdash9996 5mdash0 1mdash6 4mdash9 7mdash31 2mdash32 3mdash68 6mdash9997 2mdash0 3mdash12 1mdash18 5mdash21 6mdash29 4mdash35 7mdash999

of the vertices in a cluster A chromosome is a permutationof all the gene segments with one gene segment per clusterFor example let 1 2 3 4 5 6 7 be the vertices with 119881

1=

2 3 4 1198812

= 5 6 7 and 1198811is followed by 119881

2 in a 7-

vertex instance then starting from vertex 1 a complete tour1rarr 3rarr 4rarr 2rarr 6rarr 7rarr 5rarr 1 may be represented as thechromosome (1 3 4 2 6 7 5) where (3 4 2) and (6 7 5) arethe gene segments for cluster 1 and cluster 2 respectively

It is to be noted that starting from a good initialpopulation can deliver better quality of solutions quicklyand that is why many literatures report generating initialpopulation using heuristics Hence we are going to usesequential sampling algorithm for heuristically generatinginitial population that has been applied successfully to thebottleneck TSP [22]This algorithm is a simple version of thesequential constructive sampling algorithm [8] It is basicallya probabilistic method to generate a tour of the salesmanTheprobability of visiting each unvisited vertex of a cluster in arow of the alphabet table is assigned in such a way that thefirst unvisited vertex gets more probability than the secondone and so on Thereafter cumulative probability of eachunvisited vertex of a cluster is calculated Next a randomnumber 119903 isin [0 1] is generated and the vertex that representsthe chosen random number in the cumulative probabilityrange is accepted The probability of visiting each unvisitedvertex of a cluster is assigned as follows Suppose the numberof unvisited vertices of a cluster in a row of the alphabet tableis 119896 The probability of visiting the 119894th unvisited vertex is

119901119894=

2 (119896 minus 119894 + 1)

119896 (119896 + 1) (1)

The algorithm may be summarized as follows

Step 0 Construct the ldquoalphabet tablerdquo based on the reducedcost matrix Repeat the following steps for the fixed popula-tion size (119875

119904)

Step 1 Since ldquovertex 1rdquo is the starting vertex so initialize 119901 =

1 and go to Step 2

Step 2 Go to the 119901th row of the ldquoalphabet tablerdquo and visitprobabilistically by using (1) any unvisited vertex of the row(say vertex 119902) in the present cluster and go to Step 3

Step 3 Rename the ldquovertex 119902rdquo as ldquovertex 119901rdquo and go to Step 4

Step 4 If all vertices of the present cluster are visited then goto the next cluster in the order (if any) and go to Step 5 elsego to Step 2

Step 5 If all vertices of the network are visited then go toStep 1 for generating another chromosome in the populationelse go to Step 2

Let us illustrate the algorithm through the example givenin Table 1 with 119881

1= 2 3 4 119881

2= 5 6 7 and 119881

1is followed

by 1198812 We start from 1st row of the ldquoalphabet tablerdquo The

number of unvisited vertices of the 1st cluster in the row is3 which are 4 2 and 3 with cumulative probabilities 05000833 and 1000 respectively Suppose the vertex 4 is selectedrandomly then the partial chromosome will be (1 4) Nextwe go to 4th row of the ldquoalphabet tablerdquo and probabilisticallyselect another node and so on Proceeding in this way it maylead to a complete chromosome (1 4 2 3 6 5 7)

A preliminary study shows the effectiveness of the sam-pling algorithm for initial population However instead ofconsidering all unvisited vertices if we consider at most firstten vertices in a cluster then the algorithm generates verygood population A similar observation has been reportedfor the bottleneck TSP also [22] Hence we consider thisrestricted domain of unvisited vertices of a cluster for ourstudy Further to start with better population we apply 2-optsearch to each chromosome The 2-opt search removes twoedges and then replaces them by a different set of edges insuch a way so as to maintain the feasibility of the tour Let 120572

119894

120572119894+1

120572119895 and 120572

119895+1be four vertices in a cluster then if the edges

(120572119894 120572119894+1

) and (120572119895 120572119895+1

) are removed the only way to form anew valid tour is to connect 120572

119894to 120572119895and 120572

119894+1to 120572119895+1

35 Fitness Function and Selection Method The objectivefunction of each chromosome in the population is the cost ofthe tour represented by the chromosomeThe fitness functionof a chromosome is defined as multiplicative inverse of theobjective functionThere are various selectionmethods in theliteratureThe selection operation considered for our study isthe stochastic remainder selection method [23]

36 The Sequential Constructive Crossover Operation Sincecrossover operation plays main role in GAs hence severalcrossover operators have been proposed for the usual TSPwhich are then used for the variant TSPs also Out of themthe sequential constructive crossover (SCX) [7] is found to

The Scientific World Journal 5

1

3

5

4

2

7

6

1

3

5

4

2

7

6

1

3

5

4

2

7

6

(a) P1 (1 2 4 3 6 7 5) (b) P2 (1 2 4 3 6 7 5) (c) Offspring (1 2 4 3 6 5 7)

Figure 2 Example of sequential constructive crossover operation

be one of the best crossover operators for the usual TSP Amultiparent extension of the SCX has been applied to theusual TSP and found good results [24]The SCXhas also beensuccessfully applied to the TSP with some other variations[22 25] In general it produces an offspring using betteredges of the parents However it does not depend only onthe parentsrsquo structure it sometimes introduces new but goodedges to the offspring which are not even available in thepresent population We modify the SCX operator for theOCTSP as follows

Step 1 Start from ldquovertex 1rdquo (ie current vertex 119901 = 1)

Step 2 Sequentially search both of the parent chromosomesand consider the first unvisited vertex of the present clusterappearing after ldquovertex 119901rdquo in each parent If no unvisitedvertex after ldquovertex 119901rdquo is available in any (or both) of theparents search sequentially from the starting of that parentand consider the first unvisited vertex of the cluster and goto Step 3

Step 3 Suppose the ldquovertex 120572rdquo and the ldquovertex 120573rdquo are foundin the 1st and 2nd parents respectively then for selecting thenext vertex in the offspring chromosome go to Step 4

Step 4 If 119888119901120572

lt 119888119901120573 then select ldquovertex 120572rdquo otherwise select

ldquovertex120573rdquo as the next vertex and concatenate it to the partiallyconstructed offspring chromosome and go to Step 5

Step 5 If there is not any vertex left in that cluster thengo to the next cluster if any If the offspring is a completechromosome then stop otherwise rename the present vertexas ldquovertex 119901rdquo and go to Step 2

Let a pair of parent chromosomes be 1198751 (1 2 4 3 6

7 5) and 1198752 (1 3 2 4 6 5 7) with costs 357 and 354

respectively with respect to the original cost matrix givenin Table 1 By applying above SCX we obtain the offspringchromosome (1 2 4 3 6 5 7) with cost 318 which is lessthan both parentsTheparent and the offspring chromosomesare shown in Figure 2 In general crossover operator inheritsparentsrsquo characteristics and the operator that preserves goodcharacteristics of parents in the offspring is said to be goodoperatorThe SCX is found to be excellent in this regard Boldedges in Figure 2(c) are the edges which are available either in

the first parent or in second parent For this given example alledges are selected from either of the parents

For the crossover operation a pair of parent chromo-somes is selected sequentially from the mating pool It isreported that the SCX gets stuck in local minimums quicklyfor the TSP [7] which is very often due to the identicalpopulation So to overcome this situation the selectedparents are checked for duplication If the selected parentsare found to be identical then the second parent is modifiedtemporarily by swapping some randomly chosen pair of genesin the chromosome and then the crossover operation isperformed To improve quality of the solution as well as havea mixture of parents and offspring in a population the firstparent is replaced by the offspring only if the offspring valueis better than the average value of the present populationIn this way the mixed population retains diversity alsoTo further improve the quality of the solution obtained bycrossover many researchers applied 2-opt search operator Toimprove solution quality we are going to use a local searchmethod that combines three mutation operators that will bediscussed in Section 38 However we are not applying thislocal search method to all of the offspring rather it is appliedonly to the offspring if its value is better than the averagepopulation valueNow since our crossover operator producesonly one offspring to keep population size fixed throughoutthe generations while pairing with the next chromosome inorder the present second original parent chromosomewill beconsidered as the first parent and so on

37 Mutation Operation The mutation operator randomlyselects a position in the chromosome and changes the cor-responding gene thereby modifying information The needfor mutation comes from the fact that as the less fit chromo-somes of successive generations are discarded some aspectsof genetic material could be lost forever By performingoccasional random changes in the chromosomes GAs ensurethat newparts of the search space are reachedwhich selectionand crossover could not fully guarantee In doing somutationensures that no important features are prematurely lost thusmaintaining the mating pool diversity For this investigationwe have considered reciprocal exchange mutation operatorthat selects two genes randomly of a chromosome in everycluster and swaps them The probability of mutation isusually chosen to be considerably less than the probability

6 The Scientific World Journal

1

3

5

4

2

7

6

1

3

5

4

2

7

6

(a) (1 2 4 3 6 7 5) (b) (1 4 2 3 6 5 7)

Figure 3 Example of reciprocal exchange mutation operation

of crossover So mutation plays a secondary role in the GAsearch For example let the chromosome (1 2 4 3 6 7 5)be selected for mutation and vertices 2 and 4 are swapped incluster 1 and vertices 7 and 5 are swapped in cluster 2 thenthe mutated chromosome becomes (1 4 2 3 6 5 7) whichis shown in Figure 3 Bold edges in Figure 3(b) are the newedges in the mutated chromosome

38 A Local Search Method We have considered the com-bined mutation operation as a local search method whichhas been successfully applied to the bottleneck TSP [822] and maximum TSP [25] It combines three mutationoperators insertion inversion and reciprocal exchange withcent percentage of probabilities Insertion operator selects avertex (gene) in a chromosome and inserts it in a randomplace and inversion operator selects two points along thelength of a chromosome and reverses the subchromosomesbetween these points This local search a modification ofthe hybrid mutation operator [26] is applied to a chro-mosome Recall that sizes of the clusters 119881

1 1198812 119881

119898are

1198991 1198992 119899

119898 respectively Suppose (1 = 120572

1 1205722 1205723 120572

119899) is

a chromosome then the local search for the OCTSP can bedeveloped as follows

Step 0 Set 119909 = 2 and 119910 = 1

Step 1 For 119894 = 1 to 119898 perform Step 2

Step 2 Set 119910 = 119899119894+ 119910 and go to Step 3

Step 3 For 119895 = 119909 to (119910 minus 1) perform Step 4

Step 4 For 119896 = (119895 + 1) to 119910 perform Step 5

Step 5 If inserting vertex 120572119895after vertex 120572

119896reduces the

present tour cost then insert the vertex 120572119895after vertex 120572

119896 In

either case go to Step 6

Step 6 If inverting subchromosome between the vertices120572119895and 120572

119896reduces the present tour cost then invert the

subchromosome In either case go to Step 7

Step 7 If swapping the vertices 120572119895and 120572

119896reduces the present

tour cost then swap them In either case go to Step 8

Step 8 Set 119909 = 119910 + 1 and go to Step 1

39 Immigration It is seen that sometimes GAs get stuckin local minimums for the combinatorial optimization prob-lems which is very often due to the identical populationSo to improve capability of GAs the population should bediversified To diversify the population immigrationmethodis also adopted where some randomly selected chromosomesare replaced by new chromosomes after some generations[22] We are also considering an immigration method Forour investigation 20 of the population is replaced ran-domly using sequential sampling algorithm as discussed inSection 34 if no improvement is found within the last 20generations Once the immigration is applied we wait for thenext 20 generations for any improvement Our hybrid GA(HGA) for the OCTSP may be summarized as in Figure 4[22]

4 Results and Discussions

We encoded our HGA in Visual C++ executed on a PC with340GHz Intel(R) Core (TM) i7-3770CPU and 800GBRAMunderMSWindows 7 operating system and testedwith someTSPLIB [10] instances

41 Parameter Setting GAs are well suited for the com-binatorial optimization problems They find near optimalsolution in reasonable time However they are guided bysuitable choice of parameters namely crossover probability(119875119888) mutation probability (119875

119898) population size (119875

119904) and

termination condition Successful working of GAs dependson a proper selection of these parameters [23] But there isnot any intelligent rule to set these parameters In generalvarious sets of the parameters are tested and then the bestone is selected We are also following a similar methodSo we set the parameters as follows a maximum of 20000generations as termination condition 20 as population size100 (100) as crossover probability and 20 independentruns for each setting However we are not reporting ourexperiments except for the mutation probability

To set mutation probability six mutation probabilities000 001 002 003 004 and 005 are considered andtested on five asymmetric TSPLIB instances with four clusters(1198991 1198992 1198993 and 119899

4) for each of the instances ftv110 ftv120

ftv130 ftv140 and ftv150 For example the 7-vertex instancewith two clusters (3 3)means119881

1= 2 3 4119881

2= 5 6 7 and

1198811is followed by 119881

2

Table 4 reports the mean and standard deviation (inparenthesis) of the best solution values over 20 trials on fiveinstances ftv110ndashftv150 for different mutation probabilitiesThe boldface denotes the best average solution value It is seenthat there is significant improvement of the solutions usingnonzero mutation probabilities over using zero mutationprobability It shows that mutation operation also plays animportant role in GAs Mutation probabilities 003 and 004

The Scientific World Journal 7

Table 4 Mean and standard deviation of best solution values on five asymmetric TSPLIB instances

Instance Clusters 119875119898

= 000 119875119898

= 001 119875119898

= 002 119875119898

= 003 119875119898

= 004 119875119898

= 005

ftv110 (29 27 27 27) 270978 (6750) 252634 (3184) 250800 (2521) 249080 (2351) 252835 (3456) 252867 (2479)ftv120 (30 30 30 30) 283967 (8294) 263634 (4194) 262878 (4089) 259623 (3020) 259568 (3536) 259579 (2722)ftv130 (34 32 32 32) 304411 (9377) 284489 (5532) 286056 (4003) 281756 (2442) 283578 (3423) 284166 (1967)ftv140 (35 35 35 35) 320445 (15065) 306831 (5807) 308629 (5450) 307126 (4318) 307002 (4943) 307741 (5558)ftv150 (39 37 37 37) 357625 (15455) 327566 (5772) 327682 (2714) 326526 (3560) 326555 (2964) 326939 (5968)

Start

Improved initial population

Evaluate the population and assign best chromosome cost as the best solution value

Is termination condition satisfied

Print the best solution value and

the best tour

Yes Selection operation

No

Sequential constructive crossover operation

StopMutation operation with

mutation probability

Evaluate the population

Is best population value better than

best solution value

No

Update best solution value

Yes

Local search to the best chromosome

Immigration

Yes

No

Is number ofgeneration gt20

till last update

Figure 4 Flowchart of our hybrid genetic algorithm

are competing Using 119875119898

= 003 the algorithm obtainsthe best average solution for the instances ftv110 ftv120 andftv150 For the remaining two instances the algorithmobtainsthe best average solution at 119875

119898= 004 However if we look

at the standard deviation solutions are relatively stable at119875119898

= 003Figure 5 plots the average best solution values for the five

instances obtained by the HGA using mutation probabilitiesfrom 000 to 005 The figure shows clearly the effectivenessof mutation operator It is seen that as mutation probabilityincreases solution quality also increases However after119875

119898=

004 solution quality is not found to be good From the tableand the figure we can conclude that 119875

119898= 003 is suitable for

our algorithm Hence we are going to use 119875119898

= 003 for ourfurther study

42 Comparative Study on Asymmetric Instances We presenta comparative study between HGA and LBDCOMP [9] forsome asymmetric TSPLIB instances of sizes from 34 to 171

with various clusters and different cluster sizes It is to bementioned that LBDCOMP [9] is claimed to find exactoptimal solution of the OCTSP instances which has beendisproved by showing results of some small sized instances[11] Anyway since no other literature reports the exactsolution for large size instances we are going to comparewith the LBDCOMP algorithm to see solution quality by ourHGA Table 5 shows this comparative study between HGAand LBDCOMPThe table reports results by LBDCOMP andbest solution value (BestSol) average solution value (AvgSol)in 20 runs average complete computational time (CTime)average computational time when final best solution is seenfor the first time (FTime) in twenty runs and percentage oferror (Error()) of the best solution obtained by our HGAThe percentage of error is calculated by the formula

Error () =119861119890119904119905119878119900119897 minus 119874119901119905119878119900119897

119874119901119905119878119900119897times 100 (2)

8 The Scientific World Journal

Table 5 A comparative study between LBDCOMP and HGA for asymmetric TSPLIB instances

Instance Clusters LBDCOMP HGASolution Time BestSol Error () AvgSol FTime CTime

ftv33(16 17) 1584 511 1501 minus524 150216 012 101(9 24) 1509 587 1501 minus053 150324 013 125(3 30) 1356 500 1356 000 135915 018 163

ftv35(17 18) 1747 1132 1731 minus092 173929 041 113(10 25) 1660 346 1660 000 166334 032 126(3 32) 1527 1389 1527 000 153313 042 141

ftv38(19 19) 1681 413 1681 000 168606 096 135(11 27) 1689 765 1689 000 169225 022 153(3 35) 1573 1766 1573 000 158600 016 211

ftv44(22 22) 1935 2492 1935 000 194023 017 175(13 31) 1830 790 1830 000 185212 047 201(4 40) 1670 4871 1670 000 168900 040 272

ftv47(23 24) 2470 1006 2470 000 252624 078 193(13 34) 2349 1049 2257 minus392 228127 115 242(4 43) 1957 593 1957 000 200631 152 317

ftv55(27 28) 2299 600 2219 minus348 224824 071 260(16 39) 1937 815 1937 000 198212 207 306(5 50) 1763 3206 1763 000 178824 116 448

ftv64(32 32) 2658 2854 2658 000 268615 215 368(19 45) 2383 6527 2383 000 249411 298 437(6 58) 2006 11093 2006 000 204821 224 638

ftv70(35 35) 2308 13586 2308 000 234130 130 411(21 49) 2244 10269 2244 000 226724 272 479(7 63) 2134 32331 2134 000 216332 460 737

ftv90(45 45) 1756 1771 1756 000 183378 528 693(27 63) 1710 5671 1710 000 178480 555 846(9 81) 1579 6824 1579 000 165322 579 1443

ftv100(50 50) 2008 2413 2008 000 208417 489 890(30 70) 1903 14351 1903 000 196966 759 1188(10 90) 1788 18749 1788 000 190415 1172 1927

ftv110 (29 27 27 27) 2410 28919 2411 004 249080 608 748ftv120 (30 30 30 30) 2571 8396 2571 000 259623 566 904ftv130 (34 32 32 32) 2747 33167 2751 015 281756 595 1120ftv140 (35 35 35 35) 2941 57144 2947 020 307126 1002 1323ftv150 (39 37 37 37) 3119 8142 3120 003 326526 794 1579ftv160 (40 40 40 40) 3561 75454 3580 053 369618 1251 1834ftv170 (44 42 42 42) 3927 29771 3891 minus092 399201 1536 2239Average 10548 minus038 356 635

whereBestSol denotes the best solution obtained byHGA andOptSol denotes the solution obtained by LBDCOMP

It is seen from Table 5 that our HGA finds bestoptimalsolution of thirty-two instances at least once in twenty runswhereas LBDCOMP could not find optimal solution for atleast six instancesmdashftv33 with clusters (16 17) and (9 24)ftv35 with clusters (17 18) ftv47 with clusters (13 34) ftv55with clusters (27 28) and ftv170 with clusters (44 42 42 42)That is for these six instances solution quality by HGA is

found to better On the other hand for five instances namelyftv110 ftv130 ftv140 ftv150 and ftv160 with four clusterseach solution quality by LBDCOMP is better than by ourHGA For these five instances percentage of error by HGA isat most 053 However on average solution quality byHGAis 038 better than that of by LBDCOMP

In terms of computational time we cannot directlycompare the algorithms because they are executed in differentmachines and it was not possible to access the original code

The Scientific World Journal 9

Table 6 A comparative study between LSA and HGA for symmetric TSPLIB instances

Instance Clusters LSA HGASolution Time BestSol Error () AvgSol FTime CTime

burma14 (6 7) 3621 000 3621 000 362100 000 025ulysses16 (7 8) 7303 000 7303 000 730300 000 030gr17 (8 8) 2517 000 2517 000 251700 000 032gr21 (10 10) 3465 000 3465 000 346500 000 048ulysses22 (10 11) 8190 017 8190 000 819000 000 054gr24 (11 12) 1558 014 1558 000 155800 032 063fri26 (12 13) 957 005 957 000 95700 000 062

bayg29 (14 14) 2144 2103 2144 000 214400 007 093(9 9 10) 2408 3522 2408 000 240800 000 065

bays29 (14 14) 2702 2733 2702 000 270200 000 094(9 9 10) 2991 2489 2991 000 299100 000 066

dantzig42(20 21) 699 44656 699 000 69900 000 151

(13 14 14) 699 102 699 000 69900 002 126(10 10 10 11) 699 517 699 000 69900 000 112

swiss42(20 21) 1605 1440000 1605 000 161233 077 275

(13 14 14) 1919 1440000 1919 000 192300 053 159(10 10 10 11) 1944 1440000 1944 000 194542 015 115

gr48(23 24) 6656 1440000 6433 minus335 643300 007 201

(15 16 16) 7466 1440000 7466 000 750472 004 158(11 12 12 12) 8554 1440000 8554 000 855400 038 143

eil51(25 25) 570 1440000 564 minus105 56400 073 227(16 17 17) 689 1440000 681 minus116 68100 015 173

(12 12 13 13) 714 1440000 714 000 71400 012 162Average 565920 minus024 015 115

2400

2600

2800

3000

3200

3400

3600

Aver

age b

est s

olut

ion

valu

e

ftv110ftv120ftv130

ftv140ftv150

Pm=000

Pm=001

Pm=002

Pm=003

Pm=004

Pm=005

Figure 5 Average best solution values on five asymmetric TSPLIBinstances using six mutation probabilities

of LBDCOMP However a large gap between computationaltime by LBDCOMP and HGA is seen in the table and HGAtakes much less time Further if FTime is considered forHGA then definitely it is found to be much better thanLBDCOMP It is interesting to see that for any of these

instances with the same number of clusters but differentcluster sizesHGA takes different computational times and asthe size of clusters becomesmore unbalanced computationaltime increases In an unbalanced clustered instance size ofthe clusters is not equal It is also seen that on averageHGA hits final best solution for the first time within 56of complete computational time This shows that HGA findsbest solution on average in the middle of the generations forthese asymmetric TSPLIB instances

43 Comparative Study on Symmetric Instances Now we aregoing to compare our HGA with lexisearch algorithm (LSA)[11] on some small sized symmetric TSPLIB [10] instanceswith various clusters and different cluster sizes It is to benoted that our HGA does not require any modification forsolving different types and cases of the instances Table 6shows comparative study between LSA and HGA The solu-tion quality by HGA is found to be insensitive to the numberof runs for most of the instances HGA finds bestoptimalsolution of twenty-three instances at least once in twentyruns whereas LSA could not find optimal solution for atleast three instances within four hours of computational timefor example the instances gr48 with clusters (23 24) andeil51 with clusters (25 25) and (16 17 17) Overall for thesesymmetric instances solution quality by HGA is found to

10 The Scientific World Journal

Table 7 Results on some symmetric TSPLIB instances using HGA

Instance Clusters BestSol AvgSol FTime CTime

berlin52 (51) 7542 (000) 754200 070 363(25 26) 10422 1042200 034 250

brazil58 (57) 25395 (000) 2539500 109 450(28 29) 34110 3411000 114 305

st70 (69) 675 (000) 67715 149 615(34 35) 916 91600 205 418

eil76 (75) 538 (000) 53926 242 733(37 38) 721 72312 199 511

pr76 (75) 108159 (000) 10825455 209 746(37 38) 120436 12058313 274 510

gr96 (95) 55209 (000) 5567285 589 1207(47 48) 56757 5676722 344 840

rat99 (98) 1211 (000) 121840 116 1291(49 49) 1346 134825 553 910

kroA100 (99) 21282 (000) 2132180 367 1302(24 25 25 25) 45733 4614795 340 677

kroB100 (99) 22141 (000) 2219315 490 1348(24 25 25 25) 45709 4581385 252 724

kroC100 (99) 20749 (000) 2078945 304 1262(24 25 25 25) 46388 4647535 410 690

kroD100 (99) 21294 (000) 2138911 464 1226(24 25 25 25) 45681 4595220 343 607

kroE100 (99) 22068 (000) 2211639 518 1452(24 25 25 25) 45431 4555925 324 727

rd100 (99) 7910 (000) 793270 475 1346(24 25 25 25) 15501 1552405 384 634

eil101 (100) 629 (000) 63275 579 1607(25 25 25 25) 1080 108000 380 892

lin105 (104) 14379 (000) 1441665 560 1439(26 26 26 26) 17584 1761820 213 825

pr107 (106) 44303 (000) 4440567 177 1456(26 26 27 27) 51487 5153880 214 779

gr120 (119) 6942 (000) 698695 616 2073(29 30 30 30) 13109 1312915 523 1048

pr124 (123) 59030 (000) 5918175 351 2077(30 31 31 31) 71295 7129500 112 1238

bier127 (126) 118282 (000) 11841960 933 2827(30 32 32 32) 174112 17425070 683 1954

ch130 (129) 6110 (000) 615050 1311 3012(32 32 32 33) 12000 1202205 566 2005

pr136 (135) 96772 (000) 9724080 1480 2836(33 34 34 34) 106605 10671840 814 2004

gr137 (136) 69853 (000) 7042950 1200 2816(34 34 34 34) 81628 8171501 422 1499

pr144 (143) 58537 (000) 5867119 618 3083(35 36 36 36) 69093 6912834 258 2022

kroA150 (149) 26524 (000) 2662965 1233 3585(37 37 37 38) 52824 5298840 1274 1880

kroB150 (149) 26130 (000) 2626423 1721 3807(37 37 37 38) 54008 5423775 1315 1916

The Scientific World Journal 11

Table 7 Continued

Instance Clusters BestSol AvgSol FTime CTime

ch150 (149) 6528 (000) 655631 2092 3859(37 37 37 38) 13042 1308525 1020 1904

pr152 (151) 73682 (000) 7401745 1192 3425(37 38 38 38) 79941 7994100 180 2428

u159 (158) 42080 (000) 4233610 1285 3887(39 39 40 40) 42287 4230245 329 2094

si175 (174) 21407 (000) 2141210 5119 9568(43 43 44 44) 22893 2291065 761 3472

brg180 (179) 1950 (000) 201035 1429 4458(44 45 45 45) 19430 2106020 690 1529

rat195 (194) 2323 (000) 236220 2713 7136(48 48 49 49) 2544 255172 2086 3524

d198 (197) 15800 (013) 1585875 2460 7779(49 49 49 50) 17320 1733950 2976 4392

kroA200 (199) 29420 (018) 2961880 2108 7041(49 50 50 50) 62514 6294175 2345 3981

kroB200 (199) 29463 (009) 2980700 2387 7501(49 50 50 50) 62842 6325311 1828 4140

gr202 (201) 40160 (000) 4041385 3696 9725(50 50 50 51) 44176 4424820 2486 4043

ts225 (224) 126643 (000) 12700683 2851 9385(56 56 56 56) 171269 17154330 2573 4877

tsp225 (224) 3923 (018) 396735 3829 10530(56 56 56 56) 5133 517115 2011 5491

pr226 (225) 80467 (012) 8095360 1628 8769(56 56 56 57) 96508 9651010 2838 6322

gr229 (228) 134957 (026) 13618435 3733 11979(57 57 57 57) 143028 14363245 3204 5731

gil262 (261) 2391 (055) 240315 5308 15126(65 65 65 66) 4874 490645 6433 9372

pr264 (263) 49219 (017) 4981445 4875 14999(65 66 66 66) 60161 6029415 2278 9720

a280 (279) 2585 (023) 261405 10525 18703(69 70 70 70) 2740 274375 3275 9428

pr299 (298) 48375 (038) 4885706 3811 20284(74 74 75 75) 55253 5595165 4536 12361

lin318 (317) 42301 (065) 4267965 8568 25392(79 79 79 80) 52578 5281135 6229 13335

rd400 (399) 15370 (058) 1545220 20327 54674(99 100 100 100) 30821 3100660 15140 25978

fl417 (416) 11930 (058) 1200463 22738 54417(104 104 104 104) 20457 2057624 20162 34601

gr431 (430) 173270 (108) 17604720 30035 71664(107 107 108 108) 185162 18666156 29181 40647

Average 2986 7016

be better and on average solution quality by HGA is 024better than that by LSA

In terms of computational time it can be easily concludedthat HGA ismuch better than LSA though LSAwas executedon slower machine (Pentium IV PC with speed 3GHz and

448 MB RAM) Of course the nature of LSA and HGA isnot the same LSA gives exact optimal solution whereas HGAgives heuristic solution It is also seen from the table thaton average HGA hits final best solution for the first timewithin 13 of complete computational time This shows that

12 The Scientific World Journal

HGA finds best solution on average in the beginning of thegenerations for these instances

44 Proposed Solution for Some More Symmetric InstancesTable 7 presents results for some more symmetric TSPLIBinstances of sizes from 52 to 431 with various clusters andcluster sizes Since to the best of our knowledge no literaturepresents solution for these instances hence we could notprovide any comparative study on these instances Howeverwe present the results for future study of the OCTSP on theseinstances For our self-comparison we provide solution valueand percentage of error (in parentheses) by our HGA forthe instances with one cluster which are of course usualTSP instances Out of forty-seven usual TSP instances HGAfinds exact optimal solution to thirty-three instances For theremaining instances maximum percentage of error is 108That means our algorithm can provide near exact solutionif not exact Treating this study as a base for effectiveness ofthe algorithm we can conclude that the reported solutionsare near exact solution if not exact It is also seen fromthe table that for the same instances as the number ofclusters increases solution value also increases On the otherhand as the number of clusters increases computational timedecreases In general computational time for solving a singleclustered instance (ie usual TSP instance) is more thanits corresponding multiclustered instances It seems that thestructures of these multiclustered instances are less complexand hence easier than their corresponding single clusteredinstances For these symmetric instances on average HGAhits final best solution for the first time within 43 ofcomplete computational time This shows that HGA findsbest solution for these instances on average in the middleof the generations

5 Conclusions

We presented a hybrid genetic algorithm using sequentialconstructive crossover 2-opt search a local search and animmigration method to obtain heuristic solution to theOCTSPWe have used a sequential samplingmethod for gen-erating initial population The efficiency of the hybrid GA tothe problemhas been examined against the exact partitioningalgorithm (LBDCOMP) [9] for some asymmetric TSPLIBinstances and the lexisearch algorithm (LSA) [11] for somesmall sized symmetric TSPLIB instances The computationalexperiments show that ourHGA is efficient in producing highquality of solution for the benchmark instances On the basisof solution quality our HGA is found to be better than theLBDCOMP and LSA In terms of computational time alsoour algorithm is found to be the best one Finally we presentsolution to the problem for some more symmetric TSPLIBinstances Since to the best of our knowledge no literaturepresents solution for these instances we could not confirmthe quality of our solutions for the instances However forthe symmetric instances of size up to 51 we found that ourHGA obtains exact optimal solution to the instances It isto be noted that HGA does not require any modification forsolving different types of TSPLIB instances

For any instance as the number of clusters increasesthe solution value also increases Computational time forsolving a single clustered instance (ie usual TSP instance) ismore than that for solving its corresponding multiclusteredinstances For any multiclustered instance as the clustersbecome more unbalanced computational time increases

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The author is very much thankful to the honourable review-ers for their constructive comments and suggestions Thisresearch was supported by the NSTIP Strategic Technologiesprogramno 10 in the Kingdomof Saudi Arabia via Award no11-INF1788-08 The author is also very much thankful to theNSTIP for its financial and technical supports

References

[1] J A Chisman ldquoThe clustered traveling salesman problemrdquoComputers and Operations Research vol 2 no 2 pp 115ndash1191975

[2] MGendreau AHertz andG Laporte ldquoThe traveling salesmanproblem with backhaulsrdquo Computers and Operations Researchvol 23 no 5 pp 501ndash508 1996

[3] N Guttmann-Beck R Hassin S Khuller and B RaghavacharildquoApproximation algorithms with bounded performance guar-antees for the clustered traveling salesman problemrdquo Algorith-mica vol 28 no 4 pp 422ndash437 2000

[4] F C J Lokin ldquoProcedures for travelling salesman problemswith additional constraintsrdquo European Journal of OperationalResearch vol 3 no 2 pp 135ndash141 1979

[5] G Laporte J-Y Potvin and F Quilleret ldquoTabu search heuristicusing genetic diversification for the clustered traveling salesmanproblemrdquo Journal of Heuristics vol 2 no 3 pp 187ndash200 1997

[6] G Laporte and U Palekar ldquoSome applications of the clus-tered travelling salesman problemrdquo Journal of the OperationalResearch Society vol 53 no 9 pp 972ndash976 2002

[7] Z H Ahmed ldquoGenetic algorithm for the traveling salesmanproblem using sequential constructive crossover operatorrdquoInternational Journal of Biometrics amp Bioinformatics vol 3 no6 pp 96ndash105 2010

[8] Z H Ahmed ldquoA hybrid sequential constructive samplingalgorithm for the bottleneck traveling salesman problemrdquoInternational Journal of Computational Intelligence Researchvol 6 no 3 pp 475ndash484 2010

[9] T Aramgiatisiris ldquoAn exact decomposition algorithm for thetraveling salesman problemwith backhaulsrdquo Journal of Researchin Engineering and Technology vol 1 pp 151ndash164 2004

[10] TSPLIB 1995 httpcomoptifiuni-heidelbergdesoftwareTSPLIB95

[11] Z H Ahmed ldquoAn exact algorithm for the clustered travelingsalesman problemrdquo Opsearch vol 50 no 2 pp 215ndash228 2013

[12] J D E Little K G Murthy D W Sweeny and C KarelldquoAn algorithm for the travelling salesman problemrdquo OperationsResearch vol 11 pp 972ndash989 1963

The Scientific World Journal 13

[13] K Jongens and T Volgenant ldquoThe symmetric clustered trav-eling salesman problemrdquo European Journal of OperationalResearch vol 19 no 1 pp 68ndash75 1985

[14] M Gendreau G Laporte and J Y Potvin ldquoHeuristics for theclustered traveling salesman problemrdquo Tech Rep CRT-94-54Centre de Recherche sur les Transports Universite deMontrealMontreal Canada 1994

[15] J-Y Potvin and F Guertin ldquoA genetic algorithm for theclustered traveling salesman problem with an a priori order onthe clustersrdquo Tech Rep CRT-95-06 Centre de recherchesur lestransports Universite de Montreal Montreal Canada 1995

[16] J-Y Potvin and F Guertin ldquoThe clustered traveling salesmanproblem a genetic approachrdquo in Meta-Heuristics Theory ampApplications I H Osman and J Kelly Eds pp 619ndash631 KluwerAcademic Norwell Mass USA 1996

[17] S Anily J Bramel andAHertz ldquo53-Approximation algorithmfor the clustered traveling salesman tour and path problemsrdquoOperations Research Letters vol 24 no 1 pp 29ndash35 1999

[18] N Christofides ldquoWorst-case analysis of a new heuristic for thetraveling salesmanproblemrdquoTech Rep 388Graduate School ofIndustrial Administration Carnegie-Mellon University Pitts-burgh Pa USA 1976

[19] W Sheng N Xi M Song and Y Chen ldquoRobot path planningfor dimensional measurement in automotive manufacturingrdquoJournal of Manufacturing Science and Engineering Transactionsof the ASME vol 127 no 2 pp 420ndash428 2005

[20] C Ding Y Cheng andM He ldquoTwo-level genetic algorithm forclustered traveling salesman problem with application in large-scale TSPsrdquo Tsinghua Science and Technology vol 12 no 4 pp459ndash465 2007

[21] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley New York NY USA 1989

[22] Z H Ahmed ldquoA hybrid genetic algorithm for the bottlenecktraveling salesman problemrdquo ACM Transactions on EmbeddedComputing Systems vol 12 no 1 article 9 2013

[23] K Deb Optimization for Engineering Design Algorithms andExamples Prentice Hall India New Delhi India 1995

[24] Z H Ahmed ldquoMulti-parent extension of sequential construc-tive crossover for the travelling salesman problemrdquo Interna-tional Journal of Operational Research vol 11 no 3 pp 331ndash3422011

[25] Z H Ahmed ldquoAn experimental study of a hybrid geneticalgorithm for the maximum travelling salesman problemrdquoMathematical Sciences vol 7 no 1 pp 1ndash7 2013

[26] C-X Wang D-W Cui Z-R Wang and D Chen ldquoA novel antcolony system based on minimum 1-tree and hybrid mutationfor TSPrdquo in Proceedings of the 1st International Conferenceon Natural Computation (ICNC rsquo05) LNCS pp 1269ndash1278Springer Changsha China August 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 5: Research Article The Ordered Clustered Travelling Salesman Problem: A Hybrid Genetic ...downloads.hindawi.com/journals/tswj/2014/258207.pdf · 2019-07-31 · Research Article The

The Scientific World Journal 5

1

3

5

4

2

7

6

1

3

5

4

2

7

6

1

3

5

4

2

7

6

(a) P1 (1 2 4 3 6 7 5) (b) P2 (1 2 4 3 6 7 5) (c) Offspring (1 2 4 3 6 5 7)

Figure 2 Example of sequential constructive crossover operation

be one of the best crossover operators for the usual TSP Amultiparent extension of the SCX has been applied to theusual TSP and found good results [24]The SCXhas also beensuccessfully applied to the TSP with some other variations[22 25] In general it produces an offspring using betteredges of the parents However it does not depend only onthe parentsrsquo structure it sometimes introduces new but goodedges to the offspring which are not even available in thepresent population We modify the SCX operator for theOCTSP as follows

Step 1 Start from ldquovertex 1rdquo (ie current vertex 119901 = 1)

Step 2 Sequentially search both of the parent chromosomesand consider the first unvisited vertex of the present clusterappearing after ldquovertex 119901rdquo in each parent If no unvisitedvertex after ldquovertex 119901rdquo is available in any (or both) of theparents search sequentially from the starting of that parentand consider the first unvisited vertex of the cluster and goto Step 3

Step 3 Suppose the ldquovertex 120572rdquo and the ldquovertex 120573rdquo are foundin the 1st and 2nd parents respectively then for selecting thenext vertex in the offspring chromosome go to Step 4

Step 4 If 119888119901120572

lt 119888119901120573 then select ldquovertex 120572rdquo otherwise select

ldquovertex120573rdquo as the next vertex and concatenate it to the partiallyconstructed offspring chromosome and go to Step 5

Step 5 If there is not any vertex left in that cluster thengo to the next cluster if any If the offspring is a completechromosome then stop otherwise rename the present vertexas ldquovertex 119901rdquo and go to Step 2

Let a pair of parent chromosomes be 1198751 (1 2 4 3 6

7 5) and 1198752 (1 3 2 4 6 5 7) with costs 357 and 354

respectively with respect to the original cost matrix givenin Table 1 By applying above SCX we obtain the offspringchromosome (1 2 4 3 6 5 7) with cost 318 which is lessthan both parentsTheparent and the offspring chromosomesare shown in Figure 2 In general crossover operator inheritsparentsrsquo characteristics and the operator that preserves goodcharacteristics of parents in the offspring is said to be goodoperatorThe SCX is found to be excellent in this regard Boldedges in Figure 2(c) are the edges which are available either in

the first parent or in second parent For this given example alledges are selected from either of the parents

For the crossover operation a pair of parent chromo-somes is selected sequentially from the mating pool It isreported that the SCX gets stuck in local minimums quicklyfor the TSP [7] which is very often due to the identicalpopulation So to overcome this situation the selectedparents are checked for duplication If the selected parentsare found to be identical then the second parent is modifiedtemporarily by swapping some randomly chosen pair of genesin the chromosome and then the crossover operation isperformed To improve quality of the solution as well as havea mixture of parents and offspring in a population the firstparent is replaced by the offspring only if the offspring valueis better than the average value of the present populationIn this way the mixed population retains diversity alsoTo further improve the quality of the solution obtained bycrossover many researchers applied 2-opt search operator Toimprove solution quality we are going to use a local searchmethod that combines three mutation operators that will bediscussed in Section 38 However we are not applying thislocal search method to all of the offspring rather it is appliedonly to the offspring if its value is better than the averagepopulation valueNow since our crossover operator producesonly one offspring to keep population size fixed throughoutthe generations while pairing with the next chromosome inorder the present second original parent chromosomewill beconsidered as the first parent and so on

37 Mutation Operation The mutation operator randomlyselects a position in the chromosome and changes the cor-responding gene thereby modifying information The needfor mutation comes from the fact that as the less fit chromo-somes of successive generations are discarded some aspectsof genetic material could be lost forever By performingoccasional random changes in the chromosomes GAs ensurethat newparts of the search space are reachedwhich selectionand crossover could not fully guarantee In doing somutationensures that no important features are prematurely lost thusmaintaining the mating pool diversity For this investigationwe have considered reciprocal exchange mutation operatorthat selects two genes randomly of a chromosome in everycluster and swaps them The probability of mutation isusually chosen to be considerably less than the probability

6 The Scientific World Journal

1

3

5

4

2

7

6

1

3

5

4

2

7

6

(a) (1 2 4 3 6 7 5) (b) (1 4 2 3 6 5 7)

Figure 3 Example of reciprocal exchange mutation operation

of crossover So mutation plays a secondary role in the GAsearch For example let the chromosome (1 2 4 3 6 7 5)be selected for mutation and vertices 2 and 4 are swapped incluster 1 and vertices 7 and 5 are swapped in cluster 2 thenthe mutated chromosome becomes (1 4 2 3 6 5 7) whichis shown in Figure 3 Bold edges in Figure 3(b) are the newedges in the mutated chromosome

38 A Local Search Method We have considered the com-bined mutation operation as a local search method whichhas been successfully applied to the bottleneck TSP [822] and maximum TSP [25] It combines three mutationoperators insertion inversion and reciprocal exchange withcent percentage of probabilities Insertion operator selects avertex (gene) in a chromosome and inserts it in a randomplace and inversion operator selects two points along thelength of a chromosome and reverses the subchromosomesbetween these points This local search a modification ofthe hybrid mutation operator [26] is applied to a chro-mosome Recall that sizes of the clusters 119881

1 1198812 119881

119898are

1198991 1198992 119899

119898 respectively Suppose (1 = 120572

1 1205722 1205723 120572

119899) is

a chromosome then the local search for the OCTSP can bedeveloped as follows

Step 0 Set 119909 = 2 and 119910 = 1

Step 1 For 119894 = 1 to 119898 perform Step 2

Step 2 Set 119910 = 119899119894+ 119910 and go to Step 3

Step 3 For 119895 = 119909 to (119910 minus 1) perform Step 4

Step 4 For 119896 = (119895 + 1) to 119910 perform Step 5

Step 5 If inserting vertex 120572119895after vertex 120572

119896reduces the

present tour cost then insert the vertex 120572119895after vertex 120572

119896 In

either case go to Step 6

Step 6 If inverting subchromosome between the vertices120572119895and 120572

119896reduces the present tour cost then invert the

subchromosome In either case go to Step 7

Step 7 If swapping the vertices 120572119895and 120572

119896reduces the present

tour cost then swap them In either case go to Step 8

Step 8 Set 119909 = 119910 + 1 and go to Step 1

39 Immigration It is seen that sometimes GAs get stuckin local minimums for the combinatorial optimization prob-lems which is very often due to the identical populationSo to improve capability of GAs the population should bediversified To diversify the population immigrationmethodis also adopted where some randomly selected chromosomesare replaced by new chromosomes after some generations[22] We are also considering an immigration method Forour investigation 20 of the population is replaced ran-domly using sequential sampling algorithm as discussed inSection 34 if no improvement is found within the last 20generations Once the immigration is applied we wait for thenext 20 generations for any improvement Our hybrid GA(HGA) for the OCTSP may be summarized as in Figure 4[22]

4 Results and Discussions

We encoded our HGA in Visual C++ executed on a PC with340GHz Intel(R) Core (TM) i7-3770CPU and 800GBRAMunderMSWindows 7 operating system and testedwith someTSPLIB [10] instances

41 Parameter Setting GAs are well suited for the com-binatorial optimization problems They find near optimalsolution in reasonable time However they are guided bysuitable choice of parameters namely crossover probability(119875119888) mutation probability (119875

119898) population size (119875

119904) and

termination condition Successful working of GAs dependson a proper selection of these parameters [23] But there isnot any intelligent rule to set these parameters In generalvarious sets of the parameters are tested and then the bestone is selected We are also following a similar methodSo we set the parameters as follows a maximum of 20000generations as termination condition 20 as population size100 (100) as crossover probability and 20 independentruns for each setting However we are not reporting ourexperiments except for the mutation probability

To set mutation probability six mutation probabilities000 001 002 003 004 and 005 are considered andtested on five asymmetric TSPLIB instances with four clusters(1198991 1198992 1198993 and 119899

4) for each of the instances ftv110 ftv120

ftv130 ftv140 and ftv150 For example the 7-vertex instancewith two clusters (3 3)means119881

1= 2 3 4119881

2= 5 6 7 and

1198811is followed by 119881

2

Table 4 reports the mean and standard deviation (inparenthesis) of the best solution values over 20 trials on fiveinstances ftv110ndashftv150 for different mutation probabilitiesThe boldface denotes the best average solution value It is seenthat there is significant improvement of the solutions usingnonzero mutation probabilities over using zero mutationprobability It shows that mutation operation also plays animportant role in GAs Mutation probabilities 003 and 004

The Scientific World Journal 7

Table 4 Mean and standard deviation of best solution values on five asymmetric TSPLIB instances

Instance Clusters 119875119898

= 000 119875119898

= 001 119875119898

= 002 119875119898

= 003 119875119898

= 004 119875119898

= 005

ftv110 (29 27 27 27) 270978 (6750) 252634 (3184) 250800 (2521) 249080 (2351) 252835 (3456) 252867 (2479)ftv120 (30 30 30 30) 283967 (8294) 263634 (4194) 262878 (4089) 259623 (3020) 259568 (3536) 259579 (2722)ftv130 (34 32 32 32) 304411 (9377) 284489 (5532) 286056 (4003) 281756 (2442) 283578 (3423) 284166 (1967)ftv140 (35 35 35 35) 320445 (15065) 306831 (5807) 308629 (5450) 307126 (4318) 307002 (4943) 307741 (5558)ftv150 (39 37 37 37) 357625 (15455) 327566 (5772) 327682 (2714) 326526 (3560) 326555 (2964) 326939 (5968)

Start

Improved initial population

Evaluate the population and assign best chromosome cost as the best solution value

Is termination condition satisfied

Print the best solution value and

the best tour

Yes Selection operation

No

Sequential constructive crossover operation

StopMutation operation with

mutation probability

Evaluate the population

Is best population value better than

best solution value

No

Update best solution value

Yes

Local search to the best chromosome

Immigration

Yes

No

Is number ofgeneration gt20

till last update

Figure 4 Flowchart of our hybrid genetic algorithm

are competing Using 119875119898

= 003 the algorithm obtainsthe best average solution for the instances ftv110 ftv120 andftv150 For the remaining two instances the algorithmobtainsthe best average solution at 119875

119898= 004 However if we look

at the standard deviation solutions are relatively stable at119875119898

= 003Figure 5 plots the average best solution values for the five

instances obtained by the HGA using mutation probabilitiesfrom 000 to 005 The figure shows clearly the effectivenessof mutation operator It is seen that as mutation probabilityincreases solution quality also increases However after119875

119898=

004 solution quality is not found to be good From the tableand the figure we can conclude that 119875

119898= 003 is suitable for

our algorithm Hence we are going to use 119875119898

= 003 for ourfurther study

42 Comparative Study on Asymmetric Instances We presenta comparative study between HGA and LBDCOMP [9] forsome asymmetric TSPLIB instances of sizes from 34 to 171

with various clusters and different cluster sizes It is to bementioned that LBDCOMP [9] is claimed to find exactoptimal solution of the OCTSP instances which has beendisproved by showing results of some small sized instances[11] Anyway since no other literature reports the exactsolution for large size instances we are going to comparewith the LBDCOMP algorithm to see solution quality by ourHGA Table 5 shows this comparative study between HGAand LBDCOMPThe table reports results by LBDCOMP andbest solution value (BestSol) average solution value (AvgSol)in 20 runs average complete computational time (CTime)average computational time when final best solution is seenfor the first time (FTime) in twenty runs and percentage oferror (Error()) of the best solution obtained by our HGAThe percentage of error is calculated by the formula

Error () =119861119890119904119905119878119900119897 minus 119874119901119905119878119900119897

119874119901119905119878119900119897times 100 (2)

8 The Scientific World Journal

Table 5 A comparative study between LBDCOMP and HGA for asymmetric TSPLIB instances

Instance Clusters LBDCOMP HGASolution Time BestSol Error () AvgSol FTime CTime

ftv33(16 17) 1584 511 1501 minus524 150216 012 101(9 24) 1509 587 1501 minus053 150324 013 125(3 30) 1356 500 1356 000 135915 018 163

ftv35(17 18) 1747 1132 1731 minus092 173929 041 113(10 25) 1660 346 1660 000 166334 032 126(3 32) 1527 1389 1527 000 153313 042 141

ftv38(19 19) 1681 413 1681 000 168606 096 135(11 27) 1689 765 1689 000 169225 022 153(3 35) 1573 1766 1573 000 158600 016 211

ftv44(22 22) 1935 2492 1935 000 194023 017 175(13 31) 1830 790 1830 000 185212 047 201(4 40) 1670 4871 1670 000 168900 040 272

ftv47(23 24) 2470 1006 2470 000 252624 078 193(13 34) 2349 1049 2257 minus392 228127 115 242(4 43) 1957 593 1957 000 200631 152 317

ftv55(27 28) 2299 600 2219 minus348 224824 071 260(16 39) 1937 815 1937 000 198212 207 306(5 50) 1763 3206 1763 000 178824 116 448

ftv64(32 32) 2658 2854 2658 000 268615 215 368(19 45) 2383 6527 2383 000 249411 298 437(6 58) 2006 11093 2006 000 204821 224 638

ftv70(35 35) 2308 13586 2308 000 234130 130 411(21 49) 2244 10269 2244 000 226724 272 479(7 63) 2134 32331 2134 000 216332 460 737

ftv90(45 45) 1756 1771 1756 000 183378 528 693(27 63) 1710 5671 1710 000 178480 555 846(9 81) 1579 6824 1579 000 165322 579 1443

ftv100(50 50) 2008 2413 2008 000 208417 489 890(30 70) 1903 14351 1903 000 196966 759 1188(10 90) 1788 18749 1788 000 190415 1172 1927

ftv110 (29 27 27 27) 2410 28919 2411 004 249080 608 748ftv120 (30 30 30 30) 2571 8396 2571 000 259623 566 904ftv130 (34 32 32 32) 2747 33167 2751 015 281756 595 1120ftv140 (35 35 35 35) 2941 57144 2947 020 307126 1002 1323ftv150 (39 37 37 37) 3119 8142 3120 003 326526 794 1579ftv160 (40 40 40 40) 3561 75454 3580 053 369618 1251 1834ftv170 (44 42 42 42) 3927 29771 3891 minus092 399201 1536 2239Average 10548 minus038 356 635

whereBestSol denotes the best solution obtained byHGA andOptSol denotes the solution obtained by LBDCOMP

It is seen from Table 5 that our HGA finds bestoptimalsolution of thirty-two instances at least once in twenty runswhereas LBDCOMP could not find optimal solution for atleast six instancesmdashftv33 with clusters (16 17) and (9 24)ftv35 with clusters (17 18) ftv47 with clusters (13 34) ftv55with clusters (27 28) and ftv170 with clusters (44 42 42 42)That is for these six instances solution quality by HGA is

found to better On the other hand for five instances namelyftv110 ftv130 ftv140 ftv150 and ftv160 with four clusterseach solution quality by LBDCOMP is better than by ourHGA For these five instances percentage of error by HGA isat most 053 However on average solution quality byHGAis 038 better than that of by LBDCOMP

In terms of computational time we cannot directlycompare the algorithms because they are executed in differentmachines and it was not possible to access the original code

The Scientific World Journal 9

Table 6 A comparative study between LSA and HGA for symmetric TSPLIB instances

Instance Clusters LSA HGASolution Time BestSol Error () AvgSol FTime CTime

burma14 (6 7) 3621 000 3621 000 362100 000 025ulysses16 (7 8) 7303 000 7303 000 730300 000 030gr17 (8 8) 2517 000 2517 000 251700 000 032gr21 (10 10) 3465 000 3465 000 346500 000 048ulysses22 (10 11) 8190 017 8190 000 819000 000 054gr24 (11 12) 1558 014 1558 000 155800 032 063fri26 (12 13) 957 005 957 000 95700 000 062

bayg29 (14 14) 2144 2103 2144 000 214400 007 093(9 9 10) 2408 3522 2408 000 240800 000 065

bays29 (14 14) 2702 2733 2702 000 270200 000 094(9 9 10) 2991 2489 2991 000 299100 000 066

dantzig42(20 21) 699 44656 699 000 69900 000 151

(13 14 14) 699 102 699 000 69900 002 126(10 10 10 11) 699 517 699 000 69900 000 112

swiss42(20 21) 1605 1440000 1605 000 161233 077 275

(13 14 14) 1919 1440000 1919 000 192300 053 159(10 10 10 11) 1944 1440000 1944 000 194542 015 115

gr48(23 24) 6656 1440000 6433 minus335 643300 007 201

(15 16 16) 7466 1440000 7466 000 750472 004 158(11 12 12 12) 8554 1440000 8554 000 855400 038 143

eil51(25 25) 570 1440000 564 minus105 56400 073 227(16 17 17) 689 1440000 681 minus116 68100 015 173

(12 12 13 13) 714 1440000 714 000 71400 012 162Average 565920 minus024 015 115

2400

2600

2800

3000

3200

3400

3600

Aver

age b

est s

olut

ion

valu

e

ftv110ftv120ftv130

ftv140ftv150

Pm=000

Pm=001

Pm=002

Pm=003

Pm=004

Pm=005

Figure 5 Average best solution values on five asymmetric TSPLIBinstances using six mutation probabilities

of LBDCOMP However a large gap between computationaltime by LBDCOMP and HGA is seen in the table and HGAtakes much less time Further if FTime is considered forHGA then definitely it is found to be much better thanLBDCOMP It is interesting to see that for any of these

instances with the same number of clusters but differentcluster sizesHGA takes different computational times and asthe size of clusters becomesmore unbalanced computationaltime increases In an unbalanced clustered instance size ofthe clusters is not equal It is also seen that on averageHGA hits final best solution for the first time within 56of complete computational time This shows that HGA findsbest solution on average in the middle of the generations forthese asymmetric TSPLIB instances

43 Comparative Study on Symmetric Instances Now we aregoing to compare our HGA with lexisearch algorithm (LSA)[11] on some small sized symmetric TSPLIB [10] instanceswith various clusters and different cluster sizes It is to benoted that our HGA does not require any modification forsolving different types and cases of the instances Table 6shows comparative study between LSA and HGA The solu-tion quality by HGA is found to be insensitive to the numberof runs for most of the instances HGA finds bestoptimalsolution of twenty-three instances at least once in twentyruns whereas LSA could not find optimal solution for atleast three instances within four hours of computational timefor example the instances gr48 with clusters (23 24) andeil51 with clusters (25 25) and (16 17 17) Overall for thesesymmetric instances solution quality by HGA is found to

10 The Scientific World Journal

Table 7 Results on some symmetric TSPLIB instances using HGA

Instance Clusters BestSol AvgSol FTime CTime

berlin52 (51) 7542 (000) 754200 070 363(25 26) 10422 1042200 034 250

brazil58 (57) 25395 (000) 2539500 109 450(28 29) 34110 3411000 114 305

st70 (69) 675 (000) 67715 149 615(34 35) 916 91600 205 418

eil76 (75) 538 (000) 53926 242 733(37 38) 721 72312 199 511

pr76 (75) 108159 (000) 10825455 209 746(37 38) 120436 12058313 274 510

gr96 (95) 55209 (000) 5567285 589 1207(47 48) 56757 5676722 344 840

rat99 (98) 1211 (000) 121840 116 1291(49 49) 1346 134825 553 910

kroA100 (99) 21282 (000) 2132180 367 1302(24 25 25 25) 45733 4614795 340 677

kroB100 (99) 22141 (000) 2219315 490 1348(24 25 25 25) 45709 4581385 252 724

kroC100 (99) 20749 (000) 2078945 304 1262(24 25 25 25) 46388 4647535 410 690

kroD100 (99) 21294 (000) 2138911 464 1226(24 25 25 25) 45681 4595220 343 607

kroE100 (99) 22068 (000) 2211639 518 1452(24 25 25 25) 45431 4555925 324 727

rd100 (99) 7910 (000) 793270 475 1346(24 25 25 25) 15501 1552405 384 634

eil101 (100) 629 (000) 63275 579 1607(25 25 25 25) 1080 108000 380 892

lin105 (104) 14379 (000) 1441665 560 1439(26 26 26 26) 17584 1761820 213 825

pr107 (106) 44303 (000) 4440567 177 1456(26 26 27 27) 51487 5153880 214 779

gr120 (119) 6942 (000) 698695 616 2073(29 30 30 30) 13109 1312915 523 1048

pr124 (123) 59030 (000) 5918175 351 2077(30 31 31 31) 71295 7129500 112 1238

bier127 (126) 118282 (000) 11841960 933 2827(30 32 32 32) 174112 17425070 683 1954

ch130 (129) 6110 (000) 615050 1311 3012(32 32 32 33) 12000 1202205 566 2005

pr136 (135) 96772 (000) 9724080 1480 2836(33 34 34 34) 106605 10671840 814 2004

gr137 (136) 69853 (000) 7042950 1200 2816(34 34 34 34) 81628 8171501 422 1499

pr144 (143) 58537 (000) 5867119 618 3083(35 36 36 36) 69093 6912834 258 2022

kroA150 (149) 26524 (000) 2662965 1233 3585(37 37 37 38) 52824 5298840 1274 1880

kroB150 (149) 26130 (000) 2626423 1721 3807(37 37 37 38) 54008 5423775 1315 1916

The Scientific World Journal 11

Table 7 Continued

Instance Clusters BestSol AvgSol FTime CTime

ch150 (149) 6528 (000) 655631 2092 3859(37 37 37 38) 13042 1308525 1020 1904

pr152 (151) 73682 (000) 7401745 1192 3425(37 38 38 38) 79941 7994100 180 2428

u159 (158) 42080 (000) 4233610 1285 3887(39 39 40 40) 42287 4230245 329 2094

si175 (174) 21407 (000) 2141210 5119 9568(43 43 44 44) 22893 2291065 761 3472

brg180 (179) 1950 (000) 201035 1429 4458(44 45 45 45) 19430 2106020 690 1529

rat195 (194) 2323 (000) 236220 2713 7136(48 48 49 49) 2544 255172 2086 3524

d198 (197) 15800 (013) 1585875 2460 7779(49 49 49 50) 17320 1733950 2976 4392

kroA200 (199) 29420 (018) 2961880 2108 7041(49 50 50 50) 62514 6294175 2345 3981

kroB200 (199) 29463 (009) 2980700 2387 7501(49 50 50 50) 62842 6325311 1828 4140

gr202 (201) 40160 (000) 4041385 3696 9725(50 50 50 51) 44176 4424820 2486 4043

ts225 (224) 126643 (000) 12700683 2851 9385(56 56 56 56) 171269 17154330 2573 4877

tsp225 (224) 3923 (018) 396735 3829 10530(56 56 56 56) 5133 517115 2011 5491

pr226 (225) 80467 (012) 8095360 1628 8769(56 56 56 57) 96508 9651010 2838 6322

gr229 (228) 134957 (026) 13618435 3733 11979(57 57 57 57) 143028 14363245 3204 5731

gil262 (261) 2391 (055) 240315 5308 15126(65 65 65 66) 4874 490645 6433 9372

pr264 (263) 49219 (017) 4981445 4875 14999(65 66 66 66) 60161 6029415 2278 9720

a280 (279) 2585 (023) 261405 10525 18703(69 70 70 70) 2740 274375 3275 9428

pr299 (298) 48375 (038) 4885706 3811 20284(74 74 75 75) 55253 5595165 4536 12361

lin318 (317) 42301 (065) 4267965 8568 25392(79 79 79 80) 52578 5281135 6229 13335

rd400 (399) 15370 (058) 1545220 20327 54674(99 100 100 100) 30821 3100660 15140 25978

fl417 (416) 11930 (058) 1200463 22738 54417(104 104 104 104) 20457 2057624 20162 34601

gr431 (430) 173270 (108) 17604720 30035 71664(107 107 108 108) 185162 18666156 29181 40647

Average 2986 7016

be better and on average solution quality by HGA is 024better than that by LSA

In terms of computational time it can be easily concludedthat HGA ismuch better than LSA though LSAwas executedon slower machine (Pentium IV PC with speed 3GHz and

448 MB RAM) Of course the nature of LSA and HGA isnot the same LSA gives exact optimal solution whereas HGAgives heuristic solution It is also seen from the table thaton average HGA hits final best solution for the first timewithin 13 of complete computational time This shows that

12 The Scientific World Journal

HGA finds best solution on average in the beginning of thegenerations for these instances

44 Proposed Solution for Some More Symmetric InstancesTable 7 presents results for some more symmetric TSPLIBinstances of sizes from 52 to 431 with various clusters andcluster sizes Since to the best of our knowledge no literaturepresents solution for these instances hence we could notprovide any comparative study on these instances Howeverwe present the results for future study of the OCTSP on theseinstances For our self-comparison we provide solution valueand percentage of error (in parentheses) by our HGA forthe instances with one cluster which are of course usualTSP instances Out of forty-seven usual TSP instances HGAfinds exact optimal solution to thirty-three instances For theremaining instances maximum percentage of error is 108That means our algorithm can provide near exact solutionif not exact Treating this study as a base for effectiveness ofthe algorithm we can conclude that the reported solutionsare near exact solution if not exact It is also seen fromthe table that for the same instances as the number ofclusters increases solution value also increases On the otherhand as the number of clusters increases computational timedecreases In general computational time for solving a singleclustered instance (ie usual TSP instance) is more thanits corresponding multiclustered instances It seems that thestructures of these multiclustered instances are less complexand hence easier than their corresponding single clusteredinstances For these symmetric instances on average HGAhits final best solution for the first time within 43 ofcomplete computational time This shows that HGA findsbest solution for these instances on average in the middleof the generations

5 Conclusions

We presented a hybrid genetic algorithm using sequentialconstructive crossover 2-opt search a local search and animmigration method to obtain heuristic solution to theOCTSPWe have used a sequential samplingmethod for gen-erating initial population The efficiency of the hybrid GA tothe problemhas been examined against the exact partitioningalgorithm (LBDCOMP) [9] for some asymmetric TSPLIBinstances and the lexisearch algorithm (LSA) [11] for somesmall sized symmetric TSPLIB instances The computationalexperiments show that ourHGA is efficient in producing highquality of solution for the benchmark instances On the basisof solution quality our HGA is found to be better than theLBDCOMP and LSA In terms of computational time alsoour algorithm is found to be the best one Finally we presentsolution to the problem for some more symmetric TSPLIBinstances Since to the best of our knowledge no literaturepresents solution for these instances we could not confirmthe quality of our solutions for the instances However forthe symmetric instances of size up to 51 we found that ourHGA obtains exact optimal solution to the instances It isto be noted that HGA does not require any modification forsolving different types of TSPLIB instances

For any instance as the number of clusters increasesthe solution value also increases Computational time forsolving a single clustered instance (ie usual TSP instance) ismore than that for solving its corresponding multiclusteredinstances For any multiclustered instance as the clustersbecome more unbalanced computational time increases

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The author is very much thankful to the honourable review-ers for their constructive comments and suggestions Thisresearch was supported by the NSTIP Strategic Technologiesprogramno 10 in the Kingdomof Saudi Arabia via Award no11-INF1788-08 The author is also very much thankful to theNSTIP for its financial and technical supports

References

[1] J A Chisman ldquoThe clustered traveling salesman problemrdquoComputers and Operations Research vol 2 no 2 pp 115ndash1191975

[2] MGendreau AHertz andG Laporte ldquoThe traveling salesmanproblem with backhaulsrdquo Computers and Operations Researchvol 23 no 5 pp 501ndash508 1996

[3] N Guttmann-Beck R Hassin S Khuller and B RaghavacharildquoApproximation algorithms with bounded performance guar-antees for the clustered traveling salesman problemrdquo Algorith-mica vol 28 no 4 pp 422ndash437 2000

[4] F C J Lokin ldquoProcedures for travelling salesman problemswith additional constraintsrdquo European Journal of OperationalResearch vol 3 no 2 pp 135ndash141 1979

[5] G Laporte J-Y Potvin and F Quilleret ldquoTabu search heuristicusing genetic diversification for the clustered traveling salesmanproblemrdquo Journal of Heuristics vol 2 no 3 pp 187ndash200 1997

[6] G Laporte and U Palekar ldquoSome applications of the clus-tered travelling salesman problemrdquo Journal of the OperationalResearch Society vol 53 no 9 pp 972ndash976 2002

[7] Z H Ahmed ldquoGenetic algorithm for the traveling salesmanproblem using sequential constructive crossover operatorrdquoInternational Journal of Biometrics amp Bioinformatics vol 3 no6 pp 96ndash105 2010

[8] Z H Ahmed ldquoA hybrid sequential constructive samplingalgorithm for the bottleneck traveling salesman problemrdquoInternational Journal of Computational Intelligence Researchvol 6 no 3 pp 475ndash484 2010

[9] T Aramgiatisiris ldquoAn exact decomposition algorithm for thetraveling salesman problemwith backhaulsrdquo Journal of Researchin Engineering and Technology vol 1 pp 151ndash164 2004

[10] TSPLIB 1995 httpcomoptifiuni-heidelbergdesoftwareTSPLIB95

[11] Z H Ahmed ldquoAn exact algorithm for the clustered travelingsalesman problemrdquo Opsearch vol 50 no 2 pp 215ndash228 2013

[12] J D E Little K G Murthy D W Sweeny and C KarelldquoAn algorithm for the travelling salesman problemrdquo OperationsResearch vol 11 pp 972ndash989 1963

The Scientific World Journal 13

[13] K Jongens and T Volgenant ldquoThe symmetric clustered trav-eling salesman problemrdquo European Journal of OperationalResearch vol 19 no 1 pp 68ndash75 1985

[14] M Gendreau G Laporte and J Y Potvin ldquoHeuristics for theclustered traveling salesman problemrdquo Tech Rep CRT-94-54Centre de Recherche sur les Transports Universite deMontrealMontreal Canada 1994

[15] J-Y Potvin and F Guertin ldquoA genetic algorithm for theclustered traveling salesman problem with an a priori order onthe clustersrdquo Tech Rep CRT-95-06 Centre de recherchesur lestransports Universite de Montreal Montreal Canada 1995

[16] J-Y Potvin and F Guertin ldquoThe clustered traveling salesmanproblem a genetic approachrdquo in Meta-Heuristics Theory ampApplications I H Osman and J Kelly Eds pp 619ndash631 KluwerAcademic Norwell Mass USA 1996

[17] S Anily J Bramel andAHertz ldquo53-Approximation algorithmfor the clustered traveling salesman tour and path problemsrdquoOperations Research Letters vol 24 no 1 pp 29ndash35 1999

[18] N Christofides ldquoWorst-case analysis of a new heuristic for thetraveling salesmanproblemrdquoTech Rep 388Graduate School ofIndustrial Administration Carnegie-Mellon University Pitts-burgh Pa USA 1976

[19] W Sheng N Xi M Song and Y Chen ldquoRobot path planningfor dimensional measurement in automotive manufacturingrdquoJournal of Manufacturing Science and Engineering Transactionsof the ASME vol 127 no 2 pp 420ndash428 2005

[20] C Ding Y Cheng andM He ldquoTwo-level genetic algorithm forclustered traveling salesman problem with application in large-scale TSPsrdquo Tsinghua Science and Technology vol 12 no 4 pp459ndash465 2007

[21] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley New York NY USA 1989

[22] Z H Ahmed ldquoA hybrid genetic algorithm for the bottlenecktraveling salesman problemrdquo ACM Transactions on EmbeddedComputing Systems vol 12 no 1 article 9 2013

[23] K Deb Optimization for Engineering Design Algorithms andExamples Prentice Hall India New Delhi India 1995

[24] Z H Ahmed ldquoMulti-parent extension of sequential construc-tive crossover for the travelling salesman problemrdquo Interna-tional Journal of Operational Research vol 11 no 3 pp 331ndash3422011

[25] Z H Ahmed ldquoAn experimental study of a hybrid geneticalgorithm for the maximum travelling salesman problemrdquoMathematical Sciences vol 7 no 1 pp 1ndash7 2013

[26] C-X Wang D-W Cui Z-R Wang and D Chen ldquoA novel antcolony system based on minimum 1-tree and hybrid mutationfor TSPrdquo in Proceedings of the 1st International Conferenceon Natural Computation (ICNC rsquo05) LNCS pp 1269ndash1278Springer Changsha China August 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Differential EquationsInternational Journal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 6: Research Article The Ordered Clustered Travelling Salesman Problem: A Hybrid Genetic ...downloads.hindawi.com/journals/tswj/2014/258207.pdf · 2019-07-31 · Research Article The

6 The Scientific World Journal

1

3

5

4

2

7

6

1

3

5

4

2

7

6

(a) (1 2 4 3 6 7 5) (b) (1 4 2 3 6 5 7)

Figure 3 Example of reciprocal exchange mutation operation

of crossover So mutation plays a secondary role in the GAsearch For example let the chromosome (1 2 4 3 6 7 5)be selected for mutation and vertices 2 and 4 are swapped incluster 1 and vertices 7 and 5 are swapped in cluster 2 thenthe mutated chromosome becomes (1 4 2 3 6 5 7) whichis shown in Figure 3 Bold edges in Figure 3(b) are the newedges in the mutated chromosome

38 A Local Search Method We have considered the com-bined mutation operation as a local search method whichhas been successfully applied to the bottleneck TSP [822] and maximum TSP [25] It combines three mutationoperators insertion inversion and reciprocal exchange withcent percentage of probabilities Insertion operator selects avertex (gene) in a chromosome and inserts it in a randomplace and inversion operator selects two points along thelength of a chromosome and reverses the subchromosomesbetween these points This local search a modification ofthe hybrid mutation operator [26] is applied to a chro-mosome Recall that sizes of the clusters 119881

1 1198812 119881

119898are

1198991 1198992 119899

119898 respectively Suppose (1 = 120572

1 1205722 1205723 120572

119899) is

a chromosome then the local search for the OCTSP can bedeveloped as follows

Step 0 Set 119909 = 2 and 119910 = 1

Step 1 For 119894 = 1 to 119898 perform Step 2

Step 2 Set 119910 = 119899119894+ 119910 and go to Step 3

Step 3 For 119895 = 119909 to (119910 minus 1) perform Step 4

Step 4 For 119896 = (119895 + 1) to 119910 perform Step 5

Step 5 If inserting vertex 120572119895after vertex 120572

119896reduces the

present tour cost then insert the vertex 120572119895after vertex 120572

119896 In

either case go to Step 6

Step 6 If inverting subchromosome between the vertices120572119895and 120572

119896reduces the present tour cost then invert the

subchromosome In either case go to Step 7

Step 7 If swapping the vertices 120572119895and 120572

119896reduces the present

tour cost then swap them In either case go to Step 8

Step 8 Set 119909 = 119910 + 1 and go to Step 1

39 Immigration It is seen that sometimes GAs get stuckin local minimums for the combinatorial optimization prob-lems which is very often due to the identical populationSo to improve capability of GAs the population should bediversified To diversify the population immigrationmethodis also adopted where some randomly selected chromosomesare replaced by new chromosomes after some generations[22] We are also considering an immigration method Forour investigation 20 of the population is replaced ran-domly using sequential sampling algorithm as discussed inSection 34 if no improvement is found within the last 20generations Once the immigration is applied we wait for thenext 20 generations for any improvement Our hybrid GA(HGA) for the OCTSP may be summarized as in Figure 4[22]

4 Results and Discussions

We encoded our HGA in Visual C++ executed on a PC with340GHz Intel(R) Core (TM) i7-3770CPU and 800GBRAMunderMSWindows 7 operating system and testedwith someTSPLIB [10] instances

41 Parameter Setting GAs are well suited for the com-binatorial optimization problems They find near optimalsolution in reasonable time However they are guided bysuitable choice of parameters namely crossover probability(119875119888) mutation probability (119875

119898) population size (119875

119904) and

termination condition Successful working of GAs dependson a proper selection of these parameters [23] But there isnot any intelligent rule to set these parameters In generalvarious sets of the parameters are tested and then the bestone is selected We are also following a similar methodSo we set the parameters as follows a maximum of 20000generations as termination condition 20 as population size100 (100) as crossover probability and 20 independentruns for each setting However we are not reporting ourexperiments except for the mutation probability

To set mutation probability six mutation probabilities000 001 002 003 004 and 005 are considered andtested on five asymmetric TSPLIB instances with four clusters(1198991 1198992 1198993 and 119899

4) for each of the instances ftv110 ftv120

ftv130 ftv140 and ftv150 For example the 7-vertex instancewith two clusters (3 3)means119881

1= 2 3 4119881

2= 5 6 7 and

1198811is followed by 119881

2

Table 4 reports the mean and standard deviation (inparenthesis) of the best solution values over 20 trials on fiveinstances ftv110ndashftv150 for different mutation probabilitiesThe boldface denotes the best average solution value It is seenthat there is significant improvement of the solutions usingnonzero mutation probabilities over using zero mutationprobability It shows that mutation operation also plays animportant role in GAs Mutation probabilities 003 and 004

The Scientific World Journal 7

Table 4 Mean and standard deviation of best solution values on five asymmetric TSPLIB instances

Instance Clusters 119875119898

= 000 119875119898

= 001 119875119898

= 002 119875119898

= 003 119875119898

= 004 119875119898

= 005

ftv110 (29 27 27 27) 270978 (6750) 252634 (3184) 250800 (2521) 249080 (2351) 252835 (3456) 252867 (2479)ftv120 (30 30 30 30) 283967 (8294) 263634 (4194) 262878 (4089) 259623 (3020) 259568 (3536) 259579 (2722)ftv130 (34 32 32 32) 304411 (9377) 284489 (5532) 286056 (4003) 281756 (2442) 283578 (3423) 284166 (1967)ftv140 (35 35 35 35) 320445 (15065) 306831 (5807) 308629 (5450) 307126 (4318) 307002 (4943) 307741 (5558)ftv150 (39 37 37 37) 357625 (15455) 327566 (5772) 327682 (2714) 326526 (3560) 326555 (2964) 326939 (5968)

Start

Improved initial population

Evaluate the population and assign best chromosome cost as the best solution value

Is termination condition satisfied

Print the best solution value and

the best tour

Yes Selection operation

No

Sequential constructive crossover operation

StopMutation operation with

mutation probability

Evaluate the population

Is best population value better than

best solution value

No

Update best solution value

Yes

Local search to the best chromosome

Immigration

Yes

No

Is number ofgeneration gt20

till last update

Figure 4 Flowchart of our hybrid genetic algorithm

are competing Using 119875119898

= 003 the algorithm obtainsthe best average solution for the instances ftv110 ftv120 andftv150 For the remaining two instances the algorithmobtainsthe best average solution at 119875

119898= 004 However if we look

at the standard deviation solutions are relatively stable at119875119898

= 003Figure 5 plots the average best solution values for the five

instances obtained by the HGA using mutation probabilitiesfrom 000 to 005 The figure shows clearly the effectivenessof mutation operator It is seen that as mutation probabilityincreases solution quality also increases However after119875

119898=

004 solution quality is not found to be good From the tableand the figure we can conclude that 119875

119898= 003 is suitable for

our algorithm Hence we are going to use 119875119898

= 003 for ourfurther study

42 Comparative Study on Asymmetric Instances We presenta comparative study between HGA and LBDCOMP [9] forsome asymmetric TSPLIB instances of sizes from 34 to 171

with various clusters and different cluster sizes It is to bementioned that LBDCOMP [9] is claimed to find exactoptimal solution of the OCTSP instances which has beendisproved by showing results of some small sized instances[11] Anyway since no other literature reports the exactsolution for large size instances we are going to comparewith the LBDCOMP algorithm to see solution quality by ourHGA Table 5 shows this comparative study between HGAand LBDCOMPThe table reports results by LBDCOMP andbest solution value (BestSol) average solution value (AvgSol)in 20 runs average complete computational time (CTime)average computational time when final best solution is seenfor the first time (FTime) in twenty runs and percentage oferror (Error()) of the best solution obtained by our HGAThe percentage of error is calculated by the formula

Error () =119861119890119904119905119878119900119897 minus 119874119901119905119878119900119897

119874119901119905119878119900119897times 100 (2)

8 The Scientific World Journal

Table 5 A comparative study between LBDCOMP and HGA for asymmetric TSPLIB instances

Instance Clusters LBDCOMP HGASolution Time BestSol Error () AvgSol FTime CTime

ftv33(16 17) 1584 511 1501 minus524 150216 012 101(9 24) 1509 587 1501 minus053 150324 013 125(3 30) 1356 500 1356 000 135915 018 163

ftv35(17 18) 1747 1132 1731 minus092 173929 041 113(10 25) 1660 346 1660 000 166334 032 126(3 32) 1527 1389 1527 000 153313 042 141

ftv38(19 19) 1681 413 1681 000 168606 096 135(11 27) 1689 765 1689 000 169225 022 153(3 35) 1573 1766 1573 000 158600 016 211

ftv44(22 22) 1935 2492 1935 000 194023 017 175(13 31) 1830 790 1830 000 185212 047 201(4 40) 1670 4871 1670 000 168900 040 272

ftv47(23 24) 2470 1006 2470 000 252624 078 193(13 34) 2349 1049 2257 minus392 228127 115 242(4 43) 1957 593 1957 000 200631 152 317

ftv55(27 28) 2299 600 2219 minus348 224824 071 260(16 39) 1937 815 1937 000 198212 207 306(5 50) 1763 3206 1763 000 178824 116 448

ftv64(32 32) 2658 2854 2658 000 268615 215 368(19 45) 2383 6527 2383 000 249411 298 437(6 58) 2006 11093 2006 000 204821 224 638

ftv70(35 35) 2308 13586 2308 000 234130 130 411(21 49) 2244 10269 2244 000 226724 272 479(7 63) 2134 32331 2134 000 216332 460 737

ftv90(45 45) 1756 1771 1756 000 183378 528 693(27 63) 1710 5671 1710 000 178480 555 846(9 81) 1579 6824 1579 000 165322 579 1443

ftv100(50 50) 2008 2413 2008 000 208417 489 890(30 70) 1903 14351 1903 000 196966 759 1188(10 90) 1788 18749 1788 000 190415 1172 1927

ftv110 (29 27 27 27) 2410 28919 2411 004 249080 608 748ftv120 (30 30 30 30) 2571 8396 2571 000 259623 566 904ftv130 (34 32 32 32) 2747 33167 2751 015 281756 595 1120ftv140 (35 35 35 35) 2941 57144 2947 020 307126 1002 1323ftv150 (39 37 37 37) 3119 8142 3120 003 326526 794 1579ftv160 (40 40 40 40) 3561 75454 3580 053 369618 1251 1834ftv170 (44 42 42 42) 3927 29771 3891 minus092 399201 1536 2239Average 10548 minus038 356 635

whereBestSol denotes the best solution obtained byHGA andOptSol denotes the solution obtained by LBDCOMP

It is seen from Table 5 that our HGA finds bestoptimalsolution of thirty-two instances at least once in twenty runswhereas LBDCOMP could not find optimal solution for atleast six instancesmdashftv33 with clusters (16 17) and (9 24)ftv35 with clusters (17 18) ftv47 with clusters (13 34) ftv55with clusters (27 28) and ftv170 with clusters (44 42 42 42)That is for these six instances solution quality by HGA is

found to better On the other hand for five instances namelyftv110 ftv130 ftv140 ftv150 and ftv160 with four clusterseach solution quality by LBDCOMP is better than by ourHGA For these five instances percentage of error by HGA isat most 053 However on average solution quality byHGAis 038 better than that of by LBDCOMP

In terms of computational time we cannot directlycompare the algorithms because they are executed in differentmachines and it was not possible to access the original code

The Scientific World Journal 9

Table 6 A comparative study between LSA and HGA for symmetric TSPLIB instances

Instance Clusters LSA HGASolution Time BestSol Error () AvgSol FTime CTime

burma14 (6 7) 3621 000 3621 000 362100 000 025ulysses16 (7 8) 7303 000 7303 000 730300 000 030gr17 (8 8) 2517 000 2517 000 251700 000 032gr21 (10 10) 3465 000 3465 000 346500 000 048ulysses22 (10 11) 8190 017 8190 000 819000 000 054gr24 (11 12) 1558 014 1558 000 155800 032 063fri26 (12 13) 957 005 957 000 95700 000 062

bayg29 (14 14) 2144 2103 2144 000 214400 007 093(9 9 10) 2408 3522 2408 000 240800 000 065

bays29 (14 14) 2702 2733 2702 000 270200 000 094(9 9 10) 2991 2489 2991 000 299100 000 066

dantzig42(20 21) 699 44656 699 000 69900 000 151

(13 14 14) 699 102 699 000 69900 002 126(10 10 10 11) 699 517 699 000 69900 000 112

swiss42(20 21) 1605 1440000 1605 000 161233 077 275

(13 14 14) 1919 1440000 1919 000 192300 053 159(10 10 10 11) 1944 1440000 1944 000 194542 015 115

gr48(23 24) 6656 1440000 6433 minus335 643300 007 201

(15 16 16) 7466 1440000 7466 000 750472 004 158(11 12 12 12) 8554 1440000 8554 000 855400 038 143

eil51(25 25) 570 1440000 564 minus105 56400 073 227(16 17 17) 689 1440000 681 minus116 68100 015 173

(12 12 13 13) 714 1440000 714 000 71400 012 162Average 565920 minus024 015 115

2400

2600

2800

3000

3200

3400

3600

Aver

age b

est s

olut

ion

valu

e

ftv110ftv120ftv130

ftv140ftv150

Pm=000

Pm=001

Pm=002

Pm=003

Pm=004

Pm=005

Figure 5 Average best solution values on five asymmetric TSPLIBinstances using six mutation probabilities

of LBDCOMP However a large gap between computationaltime by LBDCOMP and HGA is seen in the table and HGAtakes much less time Further if FTime is considered forHGA then definitely it is found to be much better thanLBDCOMP It is interesting to see that for any of these

instances with the same number of clusters but differentcluster sizesHGA takes different computational times and asthe size of clusters becomesmore unbalanced computationaltime increases In an unbalanced clustered instance size ofthe clusters is not equal It is also seen that on averageHGA hits final best solution for the first time within 56of complete computational time This shows that HGA findsbest solution on average in the middle of the generations forthese asymmetric TSPLIB instances

43 Comparative Study on Symmetric Instances Now we aregoing to compare our HGA with lexisearch algorithm (LSA)[11] on some small sized symmetric TSPLIB [10] instanceswith various clusters and different cluster sizes It is to benoted that our HGA does not require any modification forsolving different types and cases of the instances Table 6shows comparative study between LSA and HGA The solu-tion quality by HGA is found to be insensitive to the numberof runs for most of the instances HGA finds bestoptimalsolution of twenty-three instances at least once in twentyruns whereas LSA could not find optimal solution for atleast three instances within four hours of computational timefor example the instances gr48 with clusters (23 24) andeil51 with clusters (25 25) and (16 17 17) Overall for thesesymmetric instances solution quality by HGA is found to

10 The Scientific World Journal

Table 7 Results on some symmetric TSPLIB instances using HGA

Instance Clusters BestSol AvgSol FTime CTime

berlin52 (51) 7542 (000) 754200 070 363(25 26) 10422 1042200 034 250

brazil58 (57) 25395 (000) 2539500 109 450(28 29) 34110 3411000 114 305

st70 (69) 675 (000) 67715 149 615(34 35) 916 91600 205 418

eil76 (75) 538 (000) 53926 242 733(37 38) 721 72312 199 511

pr76 (75) 108159 (000) 10825455 209 746(37 38) 120436 12058313 274 510

gr96 (95) 55209 (000) 5567285 589 1207(47 48) 56757 5676722 344 840

rat99 (98) 1211 (000) 121840 116 1291(49 49) 1346 134825 553 910

kroA100 (99) 21282 (000) 2132180 367 1302(24 25 25 25) 45733 4614795 340 677

kroB100 (99) 22141 (000) 2219315 490 1348(24 25 25 25) 45709 4581385 252 724

kroC100 (99) 20749 (000) 2078945 304 1262(24 25 25 25) 46388 4647535 410 690

kroD100 (99) 21294 (000) 2138911 464 1226(24 25 25 25) 45681 4595220 343 607

kroE100 (99) 22068 (000) 2211639 518 1452(24 25 25 25) 45431 4555925 324 727

rd100 (99) 7910 (000) 793270 475 1346(24 25 25 25) 15501 1552405 384 634

eil101 (100) 629 (000) 63275 579 1607(25 25 25 25) 1080 108000 380 892

lin105 (104) 14379 (000) 1441665 560 1439(26 26 26 26) 17584 1761820 213 825

pr107 (106) 44303 (000) 4440567 177 1456(26 26 27 27) 51487 5153880 214 779

gr120 (119) 6942 (000) 698695 616 2073(29 30 30 30) 13109 1312915 523 1048

pr124 (123) 59030 (000) 5918175 351 2077(30 31 31 31) 71295 7129500 112 1238

bier127 (126) 118282 (000) 11841960 933 2827(30 32 32 32) 174112 17425070 683 1954

ch130 (129) 6110 (000) 615050 1311 3012(32 32 32 33) 12000 1202205 566 2005

pr136 (135) 96772 (000) 9724080 1480 2836(33 34 34 34) 106605 10671840 814 2004

gr137 (136) 69853 (000) 7042950 1200 2816(34 34 34 34) 81628 8171501 422 1499

pr144 (143) 58537 (000) 5867119 618 3083(35 36 36 36) 69093 6912834 258 2022

kroA150 (149) 26524 (000) 2662965 1233 3585(37 37 37 38) 52824 5298840 1274 1880

kroB150 (149) 26130 (000) 2626423 1721 3807(37 37 37 38) 54008 5423775 1315 1916

The Scientific World Journal 11

Table 7 Continued

Instance Clusters BestSol AvgSol FTime CTime

ch150 (149) 6528 (000) 655631 2092 3859(37 37 37 38) 13042 1308525 1020 1904

pr152 (151) 73682 (000) 7401745 1192 3425(37 38 38 38) 79941 7994100 180 2428

u159 (158) 42080 (000) 4233610 1285 3887(39 39 40 40) 42287 4230245 329 2094

si175 (174) 21407 (000) 2141210 5119 9568(43 43 44 44) 22893 2291065 761 3472

brg180 (179) 1950 (000) 201035 1429 4458(44 45 45 45) 19430 2106020 690 1529

rat195 (194) 2323 (000) 236220 2713 7136(48 48 49 49) 2544 255172 2086 3524

d198 (197) 15800 (013) 1585875 2460 7779(49 49 49 50) 17320 1733950 2976 4392

kroA200 (199) 29420 (018) 2961880 2108 7041(49 50 50 50) 62514 6294175 2345 3981

kroB200 (199) 29463 (009) 2980700 2387 7501(49 50 50 50) 62842 6325311 1828 4140

gr202 (201) 40160 (000) 4041385 3696 9725(50 50 50 51) 44176 4424820 2486 4043

ts225 (224) 126643 (000) 12700683 2851 9385(56 56 56 56) 171269 17154330 2573 4877

tsp225 (224) 3923 (018) 396735 3829 10530(56 56 56 56) 5133 517115 2011 5491

pr226 (225) 80467 (012) 8095360 1628 8769(56 56 56 57) 96508 9651010 2838 6322

gr229 (228) 134957 (026) 13618435 3733 11979(57 57 57 57) 143028 14363245 3204 5731

gil262 (261) 2391 (055) 240315 5308 15126(65 65 65 66) 4874 490645 6433 9372

pr264 (263) 49219 (017) 4981445 4875 14999(65 66 66 66) 60161 6029415 2278 9720

a280 (279) 2585 (023) 261405 10525 18703(69 70 70 70) 2740 274375 3275 9428

pr299 (298) 48375 (038) 4885706 3811 20284(74 74 75 75) 55253 5595165 4536 12361

lin318 (317) 42301 (065) 4267965 8568 25392(79 79 79 80) 52578 5281135 6229 13335

rd400 (399) 15370 (058) 1545220 20327 54674(99 100 100 100) 30821 3100660 15140 25978

fl417 (416) 11930 (058) 1200463 22738 54417(104 104 104 104) 20457 2057624 20162 34601

gr431 (430) 173270 (108) 17604720 30035 71664(107 107 108 108) 185162 18666156 29181 40647

Average 2986 7016

be better and on average solution quality by HGA is 024better than that by LSA

In terms of computational time it can be easily concludedthat HGA ismuch better than LSA though LSAwas executedon slower machine (Pentium IV PC with speed 3GHz and

448 MB RAM) Of course the nature of LSA and HGA isnot the same LSA gives exact optimal solution whereas HGAgives heuristic solution It is also seen from the table thaton average HGA hits final best solution for the first timewithin 13 of complete computational time This shows that

12 The Scientific World Journal

HGA finds best solution on average in the beginning of thegenerations for these instances

44 Proposed Solution for Some More Symmetric InstancesTable 7 presents results for some more symmetric TSPLIBinstances of sizes from 52 to 431 with various clusters andcluster sizes Since to the best of our knowledge no literaturepresents solution for these instances hence we could notprovide any comparative study on these instances Howeverwe present the results for future study of the OCTSP on theseinstances For our self-comparison we provide solution valueand percentage of error (in parentheses) by our HGA forthe instances with one cluster which are of course usualTSP instances Out of forty-seven usual TSP instances HGAfinds exact optimal solution to thirty-three instances For theremaining instances maximum percentage of error is 108That means our algorithm can provide near exact solutionif not exact Treating this study as a base for effectiveness ofthe algorithm we can conclude that the reported solutionsare near exact solution if not exact It is also seen fromthe table that for the same instances as the number ofclusters increases solution value also increases On the otherhand as the number of clusters increases computational timedecreases In general computational time for solving a singleclustered instance (ie usual TSP instance) is more thanits corresponding multiclustered instances It seems that thestructures of these multiclustered instances are less complexand hence easier than their corresponding single clusteredinstances For these symmetric instances on average HGAhits final best solution for the first time within 43 ofcomplete computational time This shows that HGA findsbest solution for these instances on average in the middleof the generations

5 Conclusions

We presented a hybrid genetic algorithm using sequentialconstructive crossover 2-opt search a local search and animmigration method to obtain heuristic solution to theOCTSPWe have used a sequential samplingmethod for gen-erating initial population The efficiency of the hybrid GA tothe problemhas been examined against the exact partitioningalgorithm (LBDCOMP) [9] for some asymmetric TSPLIBinstances and the lexisearch algorithm (LSA) [11] for somesmall sized symmetric TSPLIB instances The computationalexperiments show that ourHGA is efficient in producing highquality of solution for the benchmark instances On the basisof solution quality our HGA is found to be better than theLBDCOMP and LSA In terms of computational time alsoour algorithm is found to be the best one Finally we presentsolution to the problem for some more symmetric TSPLIBinstances Since to the best of our knowledge no literaturepresents solution for these instances we could not confirmthe quality of our solutions for the instances However forthe symmetric instances of size up to 51 we found that ourHGA obtains exact optimal solution to the instances It isto be noted that HGA does not require any modification forsolving different types of TSPLIB instances

For any instance as the number of clusters increasesthe solution value also increases Computational time forsolving a single clustered instance (ie usual TSP instance) ismore than that for solving its corresponding multiclusteredinstances For any multiclustered instance as the clustersbecome more unbalanced computational time increases

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The author is very much thankful to the honourable review-ers for their constructive comments and suggestions Thisresearch was supported by the NSTIP Strategic Technologiesprogramno 10 in the Kingdomof Saudi Arabia via Award no11-INF1788-08 The author is also very much thankful to theNSTIP for its financial and technical supports

References

[1] J A Chisman ldquoThe clustered traveling salesman problemrdquoComputers and Operations Research vol 2 no 2 pp 115ndash1191975

[2] MGendreau AHertz andG Laporte ldquoThe traveling salesmanproblem with backhaulsrdquo Computers and Operations Researchvol 23 no 5 pp 501ndash508 1996

[3] N Guttmann-Beck R Hassin S Khuller and B RaghavacharildquoApproximation algorithms with bounded performance guar-antees for the clustered traveling salesman problemrdquo Algorith-mica vol 28 no 4 pp 422ndash437 2000

[4] F C J Lokin ldquoProcedures for travelling salesman problemswith additional constraintsrdquo European Journal of OperationalResearch vol 3 no 2 pp 135ndash141 1979

[5] G Laporte J-Y Potvin and F Quilleret ldquoTabu search heuristicusing genetic diversification for the clustered traveling salesmanproblemrdquo Journal of Heuristics vol 2 no 3 pp 187ndash200 1997

[6] G Laporte and U Palekar ldquoSome applications of the clus-tered travelling salesman problemrdquo Journal of the OperationalResearch Society vol 53 no 9 pp 972ndash976 2002

[7] Z H Ahmed ldquoGenetic algorithm for the traveling salesmanproblem using sequential constructive crossover operatorrdquoInternational Journal of Biometrics amp Bioinformatics vol 3 no6 pp 96ndash105 2010

[8] Z H Ahmed ldquoA hybrid sequential constructive samplingalgorithm for the bottleneck traveling salesman problemrdquoInternational Journal of Computational Intelligence Researchvol 6 no 3 pp 475ndash484 2010

[9] T Aramgiatisiris ldquoAn exact decomposition algorithm for thetraveling salesman problemwith backhaulsrdquo Journal of Researchin Engineering and Technology vol 1 pp 151ndash164 2004

[10] TSPLIB 1995 httpcomoptifiuni-heidelbergdesoftwareTSPLIB95

[11] Z H Ahmed ldquoAn exact algorithm for the clustered travelingsalesman problemrdquo Opsearch vol 50 no 2 pp 215ndash228 2013

[12] J D E Little K G Murthy D W Sweeny and C KarelldquoAn algorithm for the travelling salesman problemrdquo OperationsResearch vol 11 pp 972ndash989 1963

The Scientific World Journal 13

[13] K Jongens and T Volgenant ldquoThe symmetric clustered trav-eling salesman problemrdquo European Journal of OperationalResearch vol 19 no 1 pp 68ndash75 1985

[14] M Gendreau G Laporte and J Y Potvin ldquoHeuristics for theclustered traveling salesman problemrdquo Tech Rep CRT-94-54Centre de Recherche sur les Transports Universite deMontrealMontreal Canada 1994

[15] J-Y Potvin and F Guertin ldquoA genetic algorithm for theclustered traveling salesman problem with an a priori order onthe clustersrdquo Tech Rep CRT-95-06 Centre de recherchesur lestransports Universite de Montreal Montreal Canada 1995

[16] J-Y Potvin and F Guertin ldquoThe clustered traveling salesmanproblem a genetic approachrdquo in Meta-Heuristics Theory ampApplications I H Osman and J Kelly Eds pp 619ndash631 KluwerAcademic Norwell Mass USA 1996

[17] S Anily J Bramel andAHertz ldquo53-Approximation algorithmfor the clustered traveling salesman tour and path problemsrdquoOperations Research Letters vol 24 no 1 pp 29ndash35 1999

[18] N Christofides ldquoWorst-case analysis of a new heuristic for thetraveling salesmanproblemrdquoTech Rep 388Graduate School ofIndustrial Administration Carnegie-Mellon University Pitts-burgh Pa USA 1976

[19] W Sheng N Xi M Song and Y Chen ldquoRobot path planningfor dimensional measurement in automotive manufacturingrdquoJournal of Manufacturing Science and Engineering Transactionsof the ASME vol 127 no 2 pp 420ndash428 2005

[20] C Ding Y Cheng andM He ldquoTwo-level genetic algorithm forclustered traveling salesman problem with application in large-scale TSPsrdquo Tsinghua Science and Technology vol 12 no 4 pp459ndash465 2007

[21] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley New York NY USA 1989

[22] Z H Ahmed ldquoA hybrid genetic algorithm for the bottlenecktraveling salesman problemrdquo ACM Transactions on EmbeddedComputing Systems vol 12 no 1 article 9 2013

[23] K Deb Optimization for Engineering Design Algorithms andExamples Prentice Hall India New Delhi India 1995

[24] Z H Ahmed ldquoMulti-parent extension of sequential construc-tive crossover for the travelling salesman problemrdquo Interna-tional Journal of Operational Research vol 11 no 3 pp 331ndash3422011

[25] Z H Ahmed ldquoAn experimental study of a hybrid geneticalgorithm for the maximum travelling salesman problemrdquoMathematical Sciences vol 7 no 1 pp 1ndash7 2013

[26] C-X Wang D-W Cui Z-R Wang and D Chen ldquoA novel antcolony system based on minimum 1-tree and hybrid mutationfor TSPrdquo in Proceedings of the 1st International Conferenceon Natural Computation (ICNC rsquo05) LNCS pp 1269ndash1278Springer Changsha China August 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Differential EquationsInternational Journal of

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Stochastic AnalysisInternational Journal of

Page 7: Research Article The Ordered Clustered Travelling Salesman Problem: A Hybrid Genetic ...downloads.hindawi.com/journals/tswj/2014/258207.pdf · 2019-07-31 · Research Article The

The Scientific World Journal 7

Table 4 Mean and standard deviation of best solution values on five asymmetric TSPLIB instances

Instance Clusters 119875119898

= 000 119875119898

= 001 119875119898

= 002 119875119898

= 003 119875119898

= 004 119875119898

= 005

ftv110 (29 27 27 27) 270978 (6750) 252634 (3184) 250800 (2521) 249080 (2351) 252835 (3456) 252867 (2479)ftv120 (30 30 30 30) 283967 (8294) 263634 (4194) 262878 (4089) 259623 (3020) 259568 (3536) 259579 (2722)ftv130 (34 32 32 32) 304411 (9377) 284489 (5532) 286056 (4003) 281756 (2442) 283578 (3423) 284166 (1967)ftv140 (35 35 35 35) 320445 (15065) 306831 (5807) 308629 (5450) 307126 (4318) 307002 (4943) 307741 (5558)ftv150 (39 37 37 37) 357625 (15455) 327566 (5772) 327682 (2714) 326526 (3560) 326555 (2964) 326939 (5968)

Start

Improved initial population

Evaluate the population and assign best chromosome cost as the best solution value

Is termination condition satisfied

Print the best solution value and

the best tour

Yes Selection operation

No

Sequential constructive crossover operation

StopMutation operation with

mutation probability

Evaluate the population

Is best population value better than

best solution value

No

Update best solution value

Yes

Local search to the best chromosome

Immigration

Yes

No

Is number ofgeneration gt20

till last update

Figure 4 Flowchart of our hybrid genetic algorithm

are competing Using 119875119898

= 003 the algorithm obtainsthe best average solution for the instances ftv110 ftv120 andftv150 For the remaining two instances the algorithmobtainsthe best average solution at 119875

119898= 004 However if we look

at the standard deviation solutions are relatively stable at119875119898

= 003Figure 5 plots the average best solution values for the five

instances obtained by the HGA using mutation probabilitiesfrom 000 to 005 The figure shows clearly the effectivenessof mutation operator It is seen that as mutation probabilityincreases solution quality also increases However after119875

119898=

004 solution quality is not found to be good From the tableand the figure we can conclude that 119875

119898= 003 is suitable for

our algorithm Hence we are going to use 119875119898

= 003 for ourfurther study

42 Comparative Study on Asymmetric Instances We presenta comparative study between HGA and LBDCOMP [9] forsome asymmetric TSPLIB instances of sizes from 34 to 171

with various clusters and different cluster sizes It is to bementioned that LBDCOMP [9] is claimed to find exactoptimal solution of the OCTSP instances which has beendisproved by showing results of some small sized instances[11] Anyway since no other literature reports the exactsolution for large size instances we are going to comparewith the LBDCOMP algorithm to see solution quality by ourHGA Table 5 shows this comparative study between HGAand LBDCOMPThe table reports results by LBDCOMP andbest solution value (BestSol) average solution value (AvgSol)in 20 runs average complete computational time (CTime)average computational time when final best solution is seenfor the first time (FTime) in twenty runs and percentage oferror (Error()) of the best solution obtained by our HGAThe percentage of error is calculated by the formula

Error () =119861119890119904119905119878119900119897 minus 119874119901119905119878119900119897

119874119901119905119878119900119897times 100 (2)

8 The Scientific World Journal

Table 5 A comparative study between LBDCOMP and HGA for asymmetric TSPLIB instances

Instance Clusters LBDCOMP HGASolution Time BestSol Error () AvgSol FTime CTime

ftv33(16 17) 1584 511 1501 minus524 150216 012 101(9 24) 1509 587 1501 minus053 150324 013 125(3 30) 1356 500 1356 000 135915 018 163

ftv35(17 18) 1747 1132 1731 minus092 173929 041 113(10 25) 1660 346 1660 000 166334 032 126(3 32) 1527 1389 1527 000 153313 042 141

ftv38(19 19) 1681 413 1681 000 168606 096 135(11 27) 1689 765 1689 000 169225 022 153(3 35) 1573 1766 1573 000 158600 016 211

ftv44(22 22) 1935 2492 1935 000 194023 017 175(13 31) 1830 790 1830 000 185212 047 201(4 40) 1670 4871 1670 000 168900 040 272

ftv47(23 24) 2470 1006 2470 000 252624 078 193(13 34) 2349 1049 2257 minus392 228127 115 242(4 43) 1957 593 1957 000 200631 152 317

ftv55(27 28) 2299 600 2219 minus348 224824 071 260(16 39) 1937 815 1937 000 198212 207 306(5 50) 1763 3206 1763 000 178824 116 448

ftv64(32 32) 2658 2854 2658 000 268615 215 368(19 45) 2383 6527 2383 000 249411 298 437(6 58) 2006 11093 2006 000 204821 224 638

ftv70(35 35) 2308 13586 2308 000 234130 130 411(21 49) 2244 10269 2244 000 226724 272 479(7 63) 2134 32331 2134 000 216332 460 737

ftv90(45 45) 1756 1771 1756 000 183378 528 693(27 63) 1710 5671 1710 000 178480 555 846(9 81) 1579 6824 1579 000 165322 579 1443

ftv100(50 50) 2008 2413 2008 000 208417 489 890(30 70) 1903 14351 1903 000 196966 759 1188(10 90) 1788 18749 1788 000 190415 1172 1927

ftv110 (29 27 27 27) 2410 28919 2411 004 249080 608 748ftv120 (30 30 30 30) 2571 8396 2571 000 259623 566 904ftv130 (34 32 32 32) 2747 33167 2751 015 281756 595 1120ftv140 (35 35 35 35) 2941 57144 2947 020 307126 1002 1323ftv150 (39 37 37 37) 3119 8142 3120 003 326526 794 1579ftv160 (40 40 40 40) 3561 75454 3580 053 369618 1251 1834ftv170 (44 42 42 42) 3927 29771 3891 minus092 399201 1536 2239Average 10548 minus038 356 635

whereBestSol denotes the best solution obtained byHGA andOptSol denotes the solution obtained by LBDCOMP

It is seen from Table 5 that our HGA finds bestoptimalsolution of thirty-two instances at least once in twenty runswhereas LBDCOMP could not find optimal solution for atleast six instancesmdashftv33 with clusters (16 17) and (9 24)ftv35 with clusters (17 18) ftv47 with clusters (13 34) ftv55with clusters (27 28) and ftv170 with clusters (44 42 42 42)That is for these six instances solution quality by HGA is

found to better On the other hand for five instances namelyftv110 ftv130 ftv140 ftv150 and ftv160 with four clusterseach solution quality by LBDCOMP is better than by ourHGA For these five instances percentage of error by HGA isat most 053 However on average solution quality byHGAis 038 better than that of by LBDCOMP

In terms of computational time we cannot directlycompare the algorithms because they are executed in differentmachines and it was not possible to access the original code

The Scientific World Journal 9

Table 6 A comparative study between LSA and HGA for symmetric TSPLIB instances

Instance Clusters LSA HGASolution Time BestSol Error () AvgSol FTime CTime

burma14 (6 7) 3621 000 3621 000 362100 000 025ulysses16 (7 8) 7303 000 7303 000 730300 000 030gr17 (8 8) 2517 000 2517 000 251700 000 032gr21 (10 10) 3465 000 3465 000 346500 000 048ulysses22 (10 11) 8190 017 8190 000 819000 000 054gr24 (11 12) 1558 014 1558 000 155800 032 063fri26 (12 13) 957 005 957 000 95700 000 062

bayg29 (14 14) 2144 2103 2144 000 214400 007 093(9 9 10) 2408 3522 2408 000 240800 000 065

bays29 (14 14) 2702 2733 2702 000 270200 000 094(9 9 10) 2991 2489 2991 000 299100 000 066

dantzig42(20 21) 699 44656 699 000 69900 000 151

(13 14 14) 699 102 699 000 69900 002 126(10 10 10 11) 699 517 699 000 69900 000 112

swiss42(20 21) 1605 1440000 1605 000 161233 077 275

(13 14 14) 1919 1440000 1919 000 192300 053 159(10 10 10 11) 1944 1440000 1944 000 194542 015 115

gr48(23 24) 6656 1440000 6433 minus335 643300 007 201

(15 16 16) 7466 1440000 7466 000 750472 004 158(11 12 12 12) 8554 1440000 8554 000 855400 038 143

eil51(25 25) 570 1440000 564 minus105 56400 073 227(16 17 17) 689 1440000 681 minus116 68100 015 173

(12 12 13 13) 714 1440000 714 000 71400 012 162Average 565920 minus024 015 115

2400

2600

2800

3000

3200

3400

3600

Aver

age b

est s

olut

ion

valu

e

ftv110ftv120ftv130

ftv140ftv150

Pm=000

Pm=001

Pm=002

Pm=003

Pm=004

Pm=005

Figure 5 Average best solution values on five asymmetric TSPLIBinstances using six mutation probabilities

of LBDCOMP However a large gap between computationaltime by LBDCOMP and HGA is seen in the table and HGAtakes much less time Further if FTime is considered forHGA then definitely it is found to be much better thanLBDCOMP It is interesting to see that for any of these

instances with the same number of clusters but differentcluster sizesHGA takes different computational times and asthe size of clusters becomesmore unbalanced computationaltime increases In an unbalanced clustered instance size ofthe clusters is not equal It is also seen that on averageHGA hits final best solution for the first time within 56of complete computational time This shows that HGA findsbest solution on average in the middle of the generations forthese asymmetric TSPLIB instances

43 Comparative Study on Symmetric Instances Now we aregoing to compare our HGA with lexisearch algorithm (LSA)[11] on some small sized symmetric TSPLIB [10] instanceswith various clusters and different cluster sizes It is to benoted that our HGA does not require any modification forsolving different types and cases of the instances Table 6shows comparative study between LSA and HGA The solu-tion quality by HGA is found to be insensitive to the numberof runs for most of the instances HGA finds bestoptimalsolution of twenty-three instances at least once in twentyruns whereas LSA could not find optimal solution for atleast three instances within four hours of computational timefor example the instances gr48 with clusters (23 24) andeil51 with clusters (25 25) and (16 17 17) Overall for thesesymmetric instances solution quality by HGA is found to

10 The Scientific World Journal

Table 7 Results on some symmetric TSPLIB instances using HGA

Instance Clusters BestSol AvgSol FTime CTime

berlin52 (51) 7542 (000) 754200 070 363(25 26) 10422 1042200 034 250

brazil58 (57) 25395 (000) 2539500 109 450(28 29) 34110 3411000 114 305

st70 (69) 675 (000) 67715 149 615(34 35) 916 91600 205 418

eil76 (75) 538 (000) 53926 242 733(37 38) 721 72312 199 511

pr76 (75) 108159 (000) 10825455 209 746(37 38) 120436 12058313 274 510

gr96 (95) 55209 (000) 5567285 589 1207(47 48) 56757 5676722 344 840

rat99 (98) 1211 (000) 121840 116 1291(49 49) 1346 134825 553 910

kroA100 (99) 21282 (000) 2132180 367 1302(24 25 25 25) 45733 4614795 340 677

kroB100 (99) 22141 (000) 2219315 490 1348(24 25 25 25) 45709 4581385 252 724

kroC100 (99) 20749 (000) 2078945 304 1262(24 25 25 25) 46388 4647535 410 690

kroD100 (99) 21294 (000) 2138911 464 1226(24 25 25 25) 45681 4595220 343 607

kroE100 (99) 22068 (000) 2211639 518 1452(24 25 25 25) 45431 4555925 324 727

rd100 (99) 7910 (000) 793270 475 1346(24 25 25 25) 15501 1552405 384 634

eil101 (100) 629 (000) 63275 579 1607(25 25 25 25) 1080 108000 380 892

lin105 (104) 14379 (000) 1441665 560 1439(26 26 26 26) 17584 1761820 213 825

pr107 (106) 44303 (000) 4440567 177 1456(26 26 27 27) 51487 5153880 214 779

gr120 (119) 6942 (000) 698695 616 2073(29 30 30 30) 13109 1312915 523 1048

pr124 (123) 59030 (000) 5918175 351 2077(30 31 31 31) 71295 7129500 112 1238

bier127 (126) 118282 (000) 11841960 933 2827(30 32 32 32) 174112 17425070 683 1954

ch130 (129) 6110 (000) 615050 1311 3012(32 32 32 33) 12000 1202205 566 2005

pr136 (135) 96772 (000) 9724080 1480 2836(33 34 34 34) 106605 10671840 814 2004

gr137 (136) 69853 (000) 7042950 1200 2816(34 34 34 34) 81628 8171501 422 1499

pr144 (143) 58537 (000) 5867119 618 3083(35 36 36 36) 69093 6912834 258 2022

kroA150 (149) 26524 (000) 2662965 1233 3585(37 37 37 38) 52824 5298840 1274 1880

kroB150 (149) 26130 (000) 2626423 1721 3807(37 37 37 38) 54008 5423775 1315 1916

The Scientific World Journal 11

Table 7 Continued

Instance Clusters BestSol AvgSol FTime CTime

ch150 (149) 6528 (000) 655631 2092 3859(37 37 37 38) 13042 1308525 1020 1904

pr152 (151) 73682 (000) 7401745 1192 3425(37 38 38 38) 79941 7994100 180 2428

u159 (158) 42080 (000) 4233610 1285 3887(39 39 40 40) 42287 4230245 329 2094

si175 (174) 21407 (000) 2141210 5119 9568(43 43 44 44) 22893 2291065 761 3472

brg180 (179) 1950 (000) 201035 1429 4458(44 45 45 45) 19430 2106020 690 1529

rat195 (194) 2323 (000) 236220 2713 7136(48 48 49 49) 2544 255172 2086 3524

d198 (197) 15800 (013) 1585875 2460 7779(49 49 49 50) 17320 1733950 2976 4392

kroA200 (199) 29420 (018) 2961880 2108 7041(49 50 50 50) 62514 6294175 2345 3981

kroB200 (199) 29463 (009) 2980700 2387 7501(49 50 50 50) 62842 6325311 1828 4140

gr202 (201) 40160 (000) 4041385 3696 9725(50 50 50 51) 44176 4424820 2486 4043

ts225 (224) 126643 (000) 12700683 2851 9385(56 56 56 56) 171269 17154330 2573 4877

tsp225 (224) 3923 (018) 396735 3829 10530(56 56 56 56) 5133 517115 2011 5491

pr226 (225) 80467 (012) 8095360 1628 8769(56 56 56 57) 96508 9651010 2838 6322

gr229 (228) 134957 (026) 13618435 3733 11979(57 57 57 57) 143028 14363245 3204 5731

gil262 (261) 2391 (055) 240315 5308 15126(65 65 65 66) 4874 490645 6433 9372

pr264 (263) 49219 (017) 4981445 4875 14999(65 66 66 66) 60161 6029415 2278 9720

a280 (279) 2585 (023) 261405 10525 18703(69 70 70 70) 2740 274375 3275 9428

pr299 (298) 48375 (038) 4885706 3811 20284(74 74 75 75) 55253 5595165 4536 12361

lin318 (317) 42301 (065) 4267965 8568 25392(79 79 79 80) 52578 5281135 6229 13335

rd400 (399) 15370 (058) 1545220 20327 54674(99 100 100 100) 30821 3100660 15140 25978

fl417 (416) 11930 (058) 1200463 22738 54417(104 104 104 104) 20457 2057624 20162 34601

gr431 (430) 173270 (108) 17604720 30035 71664(107 107 108 108) 185162 18666156 29181 40647

Average 2986 7016

be better and on average solution quality by HGA is 024better than that by LSA

In terms of computational time it can be easily concludedthat HGA ismuch better than LSA though LSAwas executedon slower machine (Pentium IV PC with speed 3GHz and

448 MB RAM) Of course the nature of LSA and HGA isnot the same LSA gives exact optimal solution whereas HGAgives heuristic solution It is also seen from the table thaton average HGA hits final best solution for the first timewithin 13 of complete computational time This shows that

12 The Scientific World Journal

HGA finds best solution on average in the beginning of thegenerations for these instances

44 Proposed Solution for Some More Symmetric InstancesTable 7 presents results for some more symmetric TSPLIBinstances of sizes from 52 to 431 with various clusters andcluster sizes Since to the best of our knowledge no literaturepresents solution for these instances hence we could notprovide any comparative study on these instances Howeverwe present the results for future study of the OCTSP on theseinstances For our self-comparison we provide solution valueand percentage of error (in parentheses) by our HGA forthe instances with one cluster which are of course usualTSP instances Out of forty-seven usual TSP instances HGAfinds exact optimal solution to thirty-three instances For theremaining instances maximum percentage of error is 108That means our algorithm can provide near exact solutionif not exact Treating this study as a base for effectiveness ofthe algorithm we can conclude that the reported solutionsare near exact solution if not exact It is also seen fromthe table that for the same instances as the number ofclusters increases solution value also increases On the otherhand as the number of clusters increases computational timedecreases In general computational time for solving a singleclustered instance (ie usual TSP instance) is more thanits corresponding multiclustered instances It seems that thestructures of these multiclustered instances are less complexand hence easier than their corresponding single clusteredinstances For these symmetric instances on average HGAhits final best solution for the first time within 43 ofcomplete computational time This shows that HGA findsbest solution for these instances on average in the middleof the generations

5 Conclusions

We presented a hybrid genetic algorithm using sequentialconstructive crossover 2-opt search a local search and animmigration method to obtain heuristic solution to theOCTSPWe have used a sequential samplingmethod for gen-erating initial population The efficiency of the hybrid GA tothe problemhas been examined against the exact partitioningalgorithm (LBDCOMP) [9] for some asymmetric TSPLIBinstances and the lexisearch algorithm (LSA) [11] for somesmall sized symmetric TSPLIB instances The computationalexperiments show that ourHGA is efficient in producing highquality of solution for the benchmark instances On the basisof solution quality our HGA is found to be better than theLBDCOMP and LSA In terms of computational time alsoour algorithm is found to be the best one Finally we presentsolution to the problem for some more symmetric TSPLIBinstances Since to the best of our knowledge no literaturepresents solution for these instances we could not confirmthe quality of our solutions for the instances However forthe symmetric instances of size up to 51 we found that ourHGA obtains exact optimal solution to the instances It isto be noted that HGA does not require any modification forsolving different types of TSPLIB instances

For any instance as the number of clusters increasesthe solution value also increases Computational time forsolving a single clustered instance (ie usual TSP instance) ismore than that for solving its corresponding multiclusteredinstances For any multiclustered instance as the clustersbecome more unbalanced computational time increases

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The author is very much thankful to the honourable review-ers for their constructive comments and suggestions Thisresearch was supported by the NSTIP Strategic Technologiesprogramno 10 in the Kingdomof Saudi Arabia via Award no11-INF1788-08 The author is also very much thankful to theNSTIP for its financial and technical supports

References

[1] J A Chisman ldquoThe clustered traveling salesman problemrdquoComputers and Operations Research vol 2 no 2 pp 115ndash1191975

[2] MGendreau AHertz andG Laporte ldquoThe traveling salesmanproblem with backhaulsrdquo Computers and Operations Researchvol 23 no 5 pp 501ndash508 1996

[3] N Guttmann-Beck R Hassin S Khuller and B RaghavacharildquoApproximation algorithms with bounded performance guar-antees for the clustered traveling salesman problemrdquo Algorith-mica vol 28 no 4 pp 422ndash437 2000

[4] F C J Lokin ldquoProcedures for travelling salesman problemswith additional constraintsrdquo European Journal of OperationalResearch vol 3 no 2 pp 135ndash141 1979

[5] G Laporte J-Y Potvin and F Quilleret ldquoTabu search heuristicusing genetic diversification for the clustered traveling salesmanproblemrdquo Journal of Heuristics vol 2 no 3 pp 187ndash200 1997

[6] G Laporte and U Palekar ldquoSome applications of the clus-tered travelling salesman problemrdquo Journal of the OperationalResearch Society vol 53 no 9 pp 972ndash976 2002

[7] Z H Ahmed ldquoGenetic algorithm for the traveling salesmanproblem using sequential constructive crossover operatorrdquoInternational Journal of Biometrics amp Bioinformatics vol 3 no6 pp 96ndash105 2010

[8] Z H Ahmed ldquoA hybrid sequential constructive samplingalgorithm for the bottleneck traveling salesman problemrdquoInternational Journal of Computational Intelligence Researchvol 6 no 3 pp 475ndash484 2010

[9] T Aramgiatisiris ldquoAn exact decomposition algorithm for thetraveling salesman problemwith backhaulsrdquo Journal of Researchin Engineering and Technology vol 1 pp 151ndash164 2004

[10] TSPLIB 1995 httpcomoptifiuni-heidelbergdesoftwareTSPLIB95

[11] Z H Ahmed ldquoAn exact algorithm for the clustered travelingsalesman problemrdquo Opsearch vol 50 no 2 pp 215ndash228 2013

[12] J D E Little K G Murthy D W Sweeny and C KarelldquoAn algorithm for the travelling salesman problemrdquo OperationsResearch vol 11 pp 972ndash989 1963

The Scientific World Journal 13

[13] K Jongens and T Volgenant ldquoThe symmetric clustered trav-eling salesman problemrdquo European Journal of OperationalResearch vol 19 no 1 pp 68ndash75 1985

[14] M Gendreau G Laporte and J Y Potvin ldquoHeuristics for theclustered traveling salesman problemrdquo Tech Rep CRT-94-54Centre de Recherche sur les Transports Universite deMontrealMontreal Canada 1994

[15] J-Y Potvin and F Guertin ldquoA genetic algorithm for theclustered traveling salesman problem with an a priori order onthe clustersrdquo Tech Rep CRT-95-06 Centre de recherchesur lestransports Universite de Montreal Montreal Canada 1995

[16] J-Y Potvin and F Guertin ldquoThe clustered traveling salesmanproblem a genetic approachrdquo in Meta-Heuristics Theory ampApplications I H Osman and J Kelly Eds pp 619ndash631 KluwerAcademic Norwell Mass USA 1996

[17] S Anily J Bramel andAHertz ldquo53-Approximation algorithmfor the clustered traveling salesman tour and path problemsrdquoOperations Research Letters vol 24 no 1 pp 29ndash35 1999

[18] N Christofides ldquoWorst-case analysis of a new heuristic for thetraveling salesmanproblemrdquoTech Rep 388Graduate School ofIndustrial Administration Carnegie-Mellon University Pitts-burgh Pa USA 1976

[19] W Sheng N Xi M Song and Y Chen ldquoRobot path planningfor dimensional measurement in automotive manufacturingrdquoJournal of Manufacturing Science and Engineering Transactionsof the ASME vol 127 no 2 pp 420ndash428 2005

[20] C Ding Y Cheng andM He ldquoTwo-level genetic algorithm forclustered traveling salesman problem with application in large-scale TSPsrdquo Tsinghua Science and Technology vol 12 no 4 pp459ndash465 2007

[21] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley New York NY USA 1989

[22] Z H Ahmed ldquoA hybrid genetic algorithm for the bottlenecktraveling salesman problemrdquo ACM Transactions on EmbeddedComputing Systems vol 12 no 1 article 9 2013

[23] K Deb Optimization for Engineering Design Algorithms andExamples Prentice Hall India New Delhi India 1995

[24] Z H Ahmed ldquoMulti-parent extension of sequential construc-tive crossover for the travelling salesman problemrdquo Interna-tional Journal of Operational Research vol 11 no 3 pp 331ndash3422011

[25] Z H Ahmed ldquoAn experimental study of a hybrid geneticalgorithm for the maximum travelling salesman problemrdquoMathematical Sciences vol 7 no 1 pp 1ndash7 2013

[26] C-X Wang D-W Cui Z-R Wang and D Chen ldquoA novel antcolony system based on minimum 1-tree and hybrid mutationfor TSPrdquo in Proceedings of the 1st International Conferenceon Natural Computation (ICNC rsquo05) LNCS pp 1269ndash1278Springer Changsha China August 2005

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Stochastic AnalysisInternational Journal of

Page 8: Research Article The Ordered Clustered Travelling Salesman Problem: A Hybrid Genetic ...downloads.hindawi.com/journals/tswj/2014/258207.pdf · 2019-07-31 · Research Article The

8 The Scientific World Journal

Table 5 A comparative study between LBDCOMP and HGA for asymmetric TSPLIB instances

Instance Clusters LBDCOMP HGASolution Time BestSol Error () AvgSol FTime CTime

ftv33(16 17) 1584 511 1501 minus524 150216 012 101(9 24) 1509 587 1501 minus053 150324 013 125(3 30) 1356 500 1356 000 135915 018 163

ftv35(17 18) 1747 1132 1731 minus092 173929 041 113(10 25) 1660 346 1660 000 166334 032 126(3 32) 1527 1389 1527 000 153313 042 141

ftv38(19 19) 1681 413 1681 000 168606 096 135(11 27) 1689 765 1689 000 169225 022 153(3 35) 1573 1766 1573 000 158600 016 211

ftv44(22 22) 1935 2492 1935 000 194023 017 175(13 31) 1830 790 1830 000 185212 047 201(4 40) 1670 4871 1670 000 168900 040 272

ftv47(23 24) 2470 1006 2470 000 252624 078 193(13 34) 2349 1049 2257 minus392 228127 115 242(4 43) 1957 593 1957 000 200631 152 317

ftv55(27 28) 2299 600 2219 minus348 224824 071 260(16 39) 1937 815 1937 000 198212 207 306(5 50) 1763 3206 1763 000 178824 116 448

ftv64(32 32) 2658 2854 2658 000 268615 215 368(19 45) 2383 6527 2383 000 249411 298 437(6 58) 2006 11093 2006 000 204821 224 638

ftv70(35 35) 2308 13586 2308 000 234130 130 411(21 49) 2244 10269 2244 000 226724 272 479(7 63) 2134 32331 2134 000 216332 460 737

ftv90(45 45) 1756 1771 1756 000 183378 528 693(27 63) 1710 5671 1710 000 178480 555 846(9 81) 1579 6824 1579 000 165322 579 1443

ftv100(50 50) 2008 2413 2008 000 208417 489 890(30 70) 1903 14351 1903 000 196966 759 1188(10 90) 1788 18749 1788 000 190415 1172 1927

ftv110 (29 27 27 27) 2410 28919 2411 004 249080 608 748ftv120 (30 30 30 30) 2571 8396 2571 000 259623 566 904ftv130 (34 32 32 32) 2747 33167 2751 015 281756 595 1120ftv140 (35 35 35 35) 2941 57144 2947 020 307126 1002 1323ftv150 (39 37 37 37) 3119 8142 3120 003 326526 794 1579ftv160 (40 40 40 40) 3561 75454 3580 053 369618 1251 1834ftv170 (44 42 42 42) 3927 29771 3891 minus092 399201 1536 2239Average 10548 minus038 356 635

whereBestSol denotes the best solution obtained byHGA andOptSol denotes the solution obtained by LBDCOMP

It is seen from Table 5 that our HGA finds bestoptimalsolution of thirty-two instances at least once in twenty runswhereas LBDCOMP could not find optimal solution for atleast six instancesmdashftv33 with clusters (16 17) and (9 24)ftv35 with clusters (17 18) ftv47 with clusters (13 34) ftv55with clusters (27 28) and ftv170 with clusters (44 42 42 42)That is for these six instances solution quality by HGA is

found to better On the other hand for five instances namelyftv110 ftv130 ftv140 ftv150 and ftv160 with four clusterseach solution quality by LBDCOMP is better than by ourHGA For these five instances percentage of error by HGA isat most 053 However on average solution quality byHGAis 038 better than that of by LBDCOMP

In terms of computational time we cannot directlycompare the algorithms because they are executed in differentmachines and it was not possible to access the original code

The Scientific World Journal 9

Table 6 A comparative study between LSA and HGA for symmetric TSPLIB instances

Instance Clusters LSA HGASolution Time BestSol Error () AvgSol FTime CTime

burma14 (6 7) 3621 000 3621 000 362100 000 025ulysses16 (7 8) 7303 000 7303 000 730300 000 030gr17 (8 8) 2517 000 2517 000 251700 000 032gr21 (10 10) 3465 000 3465 000 346500 000 048ulysses22 (10 11) 8190 017 8190 000 819000 000 054gr24 (11 12) 1558 014 1558 000 155800 032 063fri26 (12 13) 957 005 957 000 95700 000 062

bayg29 (14 14) 2144 2103 2144 000 214400 007 093(9 9 10) 2408 3522 2408 000 240800 000 065

bays29 (14 14) 2702 2733 2702 000 270200 000 094(9 9 10) 2991 2489 2991 000 299100 000 066

dantzig42(20 21) 699 44656 699 000 69900 000 151

(13 14 14) 699 102 699 000 69900 002 126(10 10 10 11) 699 517 699 000 69900 000 112

swiss42(20 21) 1605 1440000 1605 000 161233 077 275

(13 14 14) 1919 1440000 1919 000 192300 053 159(10 10 10 11) 1944 1440000 1944 000 194542 015 115

gr48(23 24) 6656 1440000 6433 minus335 643300 007 201

(15 16 16) 7466 1440000 7466 000 750472 004 158(11 12 12 12) 8554 1440000 8554 000 855400 038 143

eil51(25 25) 570 1440000 564 minus105 56400 073 227(16 17 17) 689 1440000 681 minus116 68100 015 173

(12 12 13 13) 714 1440000 714 000 71400 012 162Average 565920 minus024 015 115

2400

2600

2800

3000

3200

3400

3600

Aver

age b

est s

olut

ion

valu

e

ftv110ftv120ftv130

ftv140ftv150

Pm=000

Pm=001

Pm=002

Pm=003

Pm=004

Pm=005

Figure 5 Average best solution values on five asymmetric TSPLIBinstances using six mutation probabilities

of LBDCOMP However a large gap between computationaltime by LBDCOMP and HGA is seen in the table and HGAtakes much less time Further if FTime is considered forHGA then definitely it is found to be much better thanLBDCOMP It is interesting to see that for any of these

instances with the same number of clusters but differentcluster sizesHGA takes different computational times and asthe size of clusters becomesmore unbalanced computationaltime increases In an unbalanced clustered instance size ofthe clusters is not equal It is also seen that on averageHGA hits final best solution for the first time within 56of complete computational time This shows that HGA findsbest solution on average in the middle of the generations forthese asymmetric TSPLIB instances

43 Comparative Study on Symmetric Instances Now we aregoing to compare our HGA with lexisearch algorithm (LSA)[11] on some small sized symmetric TSPLIB [10] instanceswith various clusters and different cluster sizes It is to benoted that our HGA does not require any modification forsolving different types and cases of the instances Table 6shows comparative study between LSA and HGA The solu-tion quality by HGA is found to be insensitive to the numberof runs for most of the instances HGA finds bestoptimalsolution of twenty-three instances at least once in twentyruns whereas LSA could not find optimal solution for atleast three instances within four hours of computational timefor example the instances gr48 with clusters (23 24) andeil51 with clusters (25 25) and (16 17 17) Overall for thesesymmetric instances solution quality by HGA is found to

10 The Scientific World Journal

Table 7 Results on some symmetric TSPLIB instances using HGA

Instance Clusters BestSol AvgSol FTime CTime

berlin52 (51) 7542 (000) 754200 070 363(25 26) 10422 1042200 034 250

brazil58 (57) 25395 (000) 2539500 109 450(28 29) 34110 3411000 114 305

st70 (69) 675 (000) 67715 149 615(34 35) 916 91600 205 418

eil76 (75) 538 (000) 53926 242 733(37 38) 721 72312 199 511

pr76 (75) 108159 (000) 10825455 209 746(37 38) 120436 12058313 274 510

gr96 (95) 55209 (000) 5567285 589 1207(47 48) 56757 5676722 344 840

rat99 (98) 1211 (000) 121840 116 1291(49 49) 1346 134825 553 910

kroA100 (99) 21282 (000) 2132180 367 1302(24 25 25 25) 45733 4614795 340 677

kroB100 (99) 22141 (000) 2219315 490 1348(24 25 25 25) 45709 4581385 252 724

kroC100 (99) 20749 (000) 2078945 304 1262(24 25 25 25) 46388 4647535 410 690

kroD100 (99) 21294 (000) 2138911 464 1226(24 25 25 25) 45681 4595220 343 607

kroE100 (99) 22068 (000) 2211639 518 1452(24 25 25 25) 45431 4555925 324 727

rd100 (99) 7910 (000) 793270 475 1346(24 25 25 25) 15501 1552405 384 634

eil101 (100) 629 (000) 63275 579 1607(25 25 25 25) 1080 108000 380 892

lin105 (104) 14379 (000) 1441665 560 1439(26 26 26 26) 17584 1761820 213 825

pr107 (106) 44303 (000) 4440567 177 1456(26 26 27 27) 51487 5153880 214 779

gr120 (119) 6942 (000) 698695 616 2073(29 30 30 30) 13109 1312915 523 1048

pr124 (123) 59030 (000) 5918175 351 2077(30 31 31 31) 71295 7129500 112 1238

bier127 (126) 118282 (000) 11841960 933 2827(30 32 32 32) 174112 17425070 683 1954

ch130 (129) 6110 (000) 615050 1311 3012(32 32 32 33) 12000 1202205 566 2005

pr136 (135) 96772 (000) 9724080 1480 2836(33 34 34 34) 106605 10671840 814 2004

gr137 (136) 69853 (000) 7042950 1200 2816(34 34 34 34) 81628 8171501 422 1499

pr144 (143) 58537 (000) 5867119 618 3083(35 36 36 36) 69093 6912834 258 2022

kroA150 (149) 26524 (000) 2662965 1233 3585(37 37 37 38) 52824 5298840 1274 1880

kroB150 (149) 26130 (000) 2626423 1721 3807(37 37 37 38) 54008 5423775 1315 1916

The Scientific World Journal 11

Table 7 Continued

Instance Clusters BestSol AvgSol FTime CTime

ch150 (149) 6528 (000) 655631 2092 3859(37 37 37 38) 13042 1308525 1020 1904

pr152 (151) 73682 (000) 7401745 1192 3425(37 38 38 38) 79941 7994100 180 2428

u159 (158) 42080 (000) 4233610 1285 3887(39 39 40 40) 42287 4230245 329 2094

si175 (174) 21407 (000) 2141210 5119 9568(43 43 44 44) 22893 2291065 761 3472

brg180 (179) 1950 (000) 201035 1429 4458(44 45 45 45) 19430 2106020 690 1529

rat195 (194) 2323 (000) 236220 2713 7136(48 48 49 49) 2544 255172 2086 3524

d198 (197) 15800 (013) 1585875 2460 7779(49 49 49 50) 17320 1733950 2976 4392

kroA200 (199) 29420 (018) 2961880 2108 7041(49 50 50 50) 62514 6294175 2345 3981

kroB200 (199) 29463 (009) 2980700 2387 7501(49 50 50 50) 62842 6325311 1828 4140

gr202 (201) 40160 (000) 4041385 3696 9725(50 50 50 51) 44176 4424820 2486 4043

ts225 (224) 126643 (000) 12700683 2851 9385(56 56 56 56) 171269 17154330 2573 4877

tsp225 (224) 3923 (018) 396735 3829 10530(56 56 56 56) 5133 517115 2011 5491

pr226 (225) 80467 (012) 8095360 1628 8769(56 56 56 57) 96508 9651010 2838 6322

gr229 (228) 134957 (026) 13618435 3733 11979(57 57 57 57) 143028 14363245 3204 5731

gil262 (261) 2391 (055) 240315 5308 15126(65 65 65 66) 4874 490645 6433 9372

pr264 (263) 49219 (017) 4981445 4875 14999(65 66 66 66) 60161 6029415 2278 9720

a280 (279) 2585 (023) 261405 10525 18703(69 70 70 70) 2740 274375 3275 9428

pr299 (298) 48375 (038) 4885706 3811 20284(74 74 75 75) 55253 5595165 4536 12361

lin318 (317) 42301 (065) 4267965 8568 25392(79 79 79 80) 52578 5281135 6229 13335

rd400 (399) 15370 (058) 1545220 20327 54674(99 100 100 100) 30821 3100660 15140 25978

fl417 (416) 11930 (058) 1200463 22738 54417(104 104 104 104) 20457 2057624 20162 34601

gr431 (430) 173270 (108) 17604720 30035 71664(107 107 108 108) 185162 18666156 29181 40647

Average 2986 7016

be better and on average solution quality by HGA is 024better than that by LSA

In terms of computational time it can be easily concludedthat HGA ismuch better than LSA though LSAwas executedon slower machine (Pentium IV PC with speed 3GHz and

448 MB RAM) Of course the nature of LSA and HGA isnot the same LSA gives exact optimal solution whereas HGAgives heuristic solution It is also seen from the table thaton average HGA hits final best solution for the first timewithin 13 of complete computational time This shows that

12 The Scientific World Journal

HGA finds best solution on average in the beginning of thegenerations for these instances

44 Proposed Solution for Some More Symmetric InstancesTable 7 presents results for some more symmetric TSPLIBinstances of sizes from 52 to 431 with various clusters andcluster sizes Since to the best of our knowledge no literaturepresents solution for these instances hence we could notprovide any comparative study on these instances Howeverwe present the results for future study of the OCTSP on theseinstances For our self-comparison we provide solution valueand percentage of error (in parentheses) by our HGA forthe instances with one cluster which are of course usualTSP instances Out of forty-seven usual TSP instances HGAfinds exact optimal solution to thirty-three instances For theremaining instances maximum percentage of error is 108That means our algorithm can provide near exact solutionif not exact Treating this study as a base for effectiveness ofthe algorithm we can conclude that the reported solutionsare near exact solution if not exact It is also seen fromthe table that for the same instances as the number ofclusters increases solution value also increases On the otherhand as the number of clusters increases computational timedecreases In general computational time for solving a singleclustered instance (ie usual TSP instance) is more thanits corresponding multiclustered instances It seems that thestructures of these multiclustered instances are less complexand hence easier than their corresponding single clusteredinstances For these symmetric instances on average HGAhits final best solution for the first time within 43 ofcomplete computational time This shows that HGA findsbest solution for these instances on average in the middleof the generations

5 Conclusions

We presented a hybrid genetic algorithm using sequentialconstructive crossover 2-opt search a local search and animmigration method to obtain heuristic solution to theOCTSPWe have used a sequential samplingmethod for gen-erating initial population The efficiency of the hybrid GA tothe problemhas been examined against the exact partitioningalgorithm (LBDCOMP) [9] for some asymmetric TSPLIBinstances and the lexisearch algorithm (LSA) [11] for somesmall sized symmetric TSPLIB instances The computationalexperiments show that ourHGA is efficient in producing highquality of solution for the benchmark instances On the basisof solution quality our HGA is found to be better than theLBDCOMP and LSA In terms of computational time alsoour algorithm is found to be the best one Finally we presentsolution to the problem for some more symmetric TSPLIBinstances Since to the best of our knowledge no literaturepresents solution for these instances we could not confirmthe quality of our solutions for the instances However forthe symmetric instances of size up to 51 we found that ourHGA obtains exact optimal solution to the instances It isto be noted that HGA does not require any modification forsolving different types of TSPLIB instances

For any instance as the number of clusters increasesthe solution value also increases Computational time forsolving a single clustered instance (ie usual TSP instance) ismore than that for solving its corresponding multiclusteredinstances For any multiclustered instance as the clustersbecome more unbalanced computational time increases

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The author is very much thankful to the honourable review-ers for their constructive comments and suggestions Thisresearch was supported by the NSTIP Strategic Technologiesprogramno 10 in the Kingdomof Saudi Arabia via Award no11-INF1788-08 The author is also very much thankful to theNSTIP for its financial and technical supports

References

[1] J A Chisman ldquoThe clustered traveling salesman problemrdquoComputers and Operations Research vol 2 no 2 pp 115ndash1191975

[2] MGendreau AHertz andG Laporte ldquoThe traveling salesmanproblem with backhaulsrdquo Computers and Operations Researchvol 23 no 5 pp 501ndash508 1996

[3] N Guttmann-Beck R Hassin S Khuller and B RaghavacharildquoApproximation algorithms with bounded performance guar-antees for the clustered traveling salesman problemrdquo Algorith-mica vol 28 no 4 pp 422ndash437 2000

[4] F C J Lokin ldquoProcedures for travelling salesman problemswith additional constraintsrdquo European Journal of OperationalResearch vol 3 no 2 pp 135ndash141 1979

[5] G Laporte J-Y Potvin and F Quilleret ldquoTabu search heuristicusing genetic diversification for the clustered traveling salesmanproblemrdquo Journal of Heuristics vol 2 no 3 pp 187ndash200 1997

[6] G Laporte and U Palekar ldquoSome applications of the clus-tered travelling salesman problemrdquo Journal of the OperationalResearch Society vol 53 no 9 pp 972ndash976 2002

[7] Z H Ahmed ldquoGenetic algorithm for the traveling salesmanproblem using sequential constructive crossover operatorrdquoInternational Journal of Biometrics amp Bioinformatics vol 3 no6 pp 96ndash105 2010

[8] Z H Ahmed ldquoA hybrid sequential constructive samplingalgorithm for the bottleneck traveling salesman problemrdquoInternational Journal of Computational Intelligence Researchvol 6 no 3 pp 475ndash484 2010

[9] T Aramgiatisiris ldquoAn exact decomposition algorithm for thetraveling salesman problemwith backhaulsrdquo Journal of Researchin Engineering and Technology vol 1 pp 151ndash164 2004

[10] TSPLIB 1995 httpcomoptifiuni-heidelbergdesoftwareTSPLIB95

[11] Z H Ahmed ldquoAn exact algorithm for the clustered travelingsalesman problemrdquo Opsearch vol 50 no 2 pp 215ndash228 2013

[12] J D E Little K G Murthy D W Sweeny and C KarelldquoAn algorithm for the travelling salesman problemrdquo OperationsResearch vol 11 pp 972ndash989 1963

The Scientific World Journal 13

[13] K Jongens and T Volgenant ldquoThe symmetric clustered trav-eling salesman problemrdquo European Journal of OperationalResearch vol 19 no 1 pp 68ndash75 1985

[14] M Gendreau G Laporte and J Y Potvin ldquoHeuristics for theclustered traveling salesman problemrdquo Tech Rep CRT-94-54Centre de Recherche sur les Transports Universite deMontrealMontreal Canada 1994

[15] J-Y Potvin and F Guertin ldquoA genetic algorithm for theclustered traveling salesman problem with an a priori order onthe clustersrdquo Tech Rep CRT-95-06 Centre de recherchesur lestransports Universite de Montreal Montreal Canada 1995

[16] J-Y Potvin and F Guertin ldquoThe clustered traveling salesmanproblem a genetic approachrdquo in Meta-Heuristics Theory ampApplications I H Osman and J Kelly Eds pp 619ndash631 KluwerAcademic Norwell Mass USA 1996

[17] S Anily J Bramel andAHertz ldquo53-Approximation algorithmfor the clustered traveling salesman tour and path problemsrdquoOperations Research Letters vol 24 no 1 pp 29ndash35 1999

[18] N Christofides ldquoWorst-case analysis of a new heuristic for thetraveling salesmanproblemrdquoTech Rep 388Graduate School ofIndustrial Administration Carnegie-Mellon University Pitts-burgh Pa USA 1976

[19] W Sheng N Xi M Song and Y Chen ldquoRobot path planningfor dimensional measurement in automotive manufacturingrdquoJournal of Manufacturing Science and Engineering Transactionsof the ASME vol 127 no 2 pp 420ndash428 2005

[20] C Ding Y Cheng andM He ldquoTwo-level genetic algorithm forclustered traveling salesman problem with application in large-scale TSPsrdquo Tsinghua Science and Technology vol 12 no 4 pp459ndash465 2007

[21] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley New York NY USA 1989

[22] Z H Ahmed ldquoA hybrid genetic algorithm for the bottlenecktraveling salesman problemrdquo ACM Transactions on EmbeddedComputing Systems vol 12 no 1 article 9 2013

[23] K Deb Optimization for Engineering Design Algorithms andExamples Prentice Hall India New Delhi India 1995

[24] Z H Ahmed ldquoMulti-parent extension of sequential construc-tive crossover for the travelling salesman problemrdquo Interna-tional Journal of Operational Research vol 11 no 3 pp 331ndash3422011

[25] Z H Ahmed ldquoAn experimental study of a hybrid geneticalgorithm for the maximum travelling salesman problemrdquoMathematical Sciences vol 7 no 1 pp 1ndash7 2013

[26] C-X Wang D-W Cui Z-R Wang and D Chen ldquoA novel antcolony system based on minimum 1-tree and hybrid mutationfor TSPrdquo in Proceedings of the 1st International Conferenceon Natural Computation (ICNC rsquo05) LNCS pp 1269ndash1278Springer Changsha China August 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 9: Research Article The Ordered Clustered Travelling Salesman Problem: A Hybrid Genetic ...downloads.hindawi.com/journals/tswj/2014/258207.pdf · 2019-07-31 · Research Article The

The Scientific World Journal 9

Table 6 A comparative study between LSA and HGA for symmetric TSPLIB instances

Instance Clusters LSA HGASolution Time BestSol Error () AvgSol FTime CTime

burma14 (6 7) 3621 000 3621 000 362100 000 025ulysses16 (7 8) 7303 000 7303 000 730300 000 030gr17 (8 8) 2517 000 2517 000 251700 000 032gr21 (10 10) 3465 000 3465 000 346500 000 048ulysses22 (10 11) 8190 017 8190 000 819000 000 054gr24 (11 12) 1558 014 1558 000 155800 032 063fri26 (12 13) 957 005 957 000 95700 000 062

bayg29 (14 14) 2144 2103 2144 000 214400 007 093(9 9 10) 2408 3522 2408 000 240800 000 065

bays29 (14 14) 2702 2733 2702 000 270200 000 094(9 9 10) 2991 2489 2991 000 299100 000 066

dantzig42(20 21) 699 44656 699 000 69900 000 151

(13 14 14) 699 102 699 000 69900 002 126(10 10 10 11) 699 517 699 000 69900 000 112

swiss42(20 21) 1605 1440000 1605 000 161233 077 275

(13 14 14) 1919 1440000 1919 000 192300 053 159(10 10 10 11) 1944 1440000 1944 000 194542 015 115

gr48(23 24) 6656 1440000 6433 minus335 643300 007 201

(15 16 16) 7466 1440000 7466 000 750472 004 158(11 12 12 12) 8554 1440000 8554 000 855400 038 143

eil51(25 25) 570 1440000 564 minus105 56400 073 227(16 17 17) 689 1440000 681 minus116 68100 015 173

(12 12 13 13) 714 1440000 714 000 71400 012 162Average 565920 minus024 015 115

2400

2600

2800

3000

3200

3400

3600

Aver

age b

est s

olut

ion

valu

e

ftv110ftv120ftv130

ftv140ftv150

Pm=000

Pm=001

Pm=002

Pm=003

Pm=004

Pm=005

Figure 5 Average best solution values on five asymmetric TSPLIBinstances using six mutation probabilities

of LBDCOMP However a large gap between computationaltime by LBDCOMP and HGA is seen in the table and HGAtakes much less time Further if FTime is considered forHGA then definitely it is found to be much better thanLBDCOMP It is interesting to see that for any of these

instances with the same number of clusters but differentcluster sizesHGA takes different computational times and asthe size of clusters becomesmore unbalanced computationaltime increases In an unbalanced clustered instance size ofthe clusters is not equal It is also seen that on averageHGA hits final best solution for the first time within 56of complete computational time This shows that HGA findsbest solution on average in the middle of the generations forthese asymmetric TSPLIB instances

43 Comparative Study on Symmetric Instances Now we aregoing to compare our HGA with lexisearch algorithm (LSA)[11] on some small sized symmetric TSPLIB [10] instanceswith various clusters and different cluster sizes It is to benoted that our HGA does not require any modification forsolving different types and cases of the instances Table 6shows comparative study between LSA and HGA The solu-tion quality by HGA is found to be insensitive to the numberof runs for most of the instances HGA finds bestoptimalsolution of twenty-three instances at least once in twentyruns whereas LSA could not find optimal solution for atleast three instances within four hours of computational timefor example the instances gr48 with clusters (23 24) andeil51 with clusters (25 25) and (16 17 17) Overall for thesesymmetric instances solution quality by HGA is found to

10 The Scientific World Journal

Table 7 Results on some symmetric TSPLIB instances using HGA

Instance Clusters BestSol AvgSol FTime CTime

berlin52 (51) 7542 (000) 754200 070 363(25 26) 10422 1042200 034 250

brazil58 (57) 25395 (000) 2539500 109 450(28 29) 34110 3411000 114 305

st70 (69) 675 (000) 67715 149 615(34 35) 916 91600 205 418

eil76 (75) 538 (000) 53926 242 733(37 38) 721 72312 199 511

pr76 (75) 108159 (000) 10825455 209 746(37 38) 120436 12058313 274 510

gr96 (95) 55209 (000) 5567285 589 1207(47 48) 56757 5676722 344 840

rat99 (98) 1211 (000) 121840 116 1291(49 49) 1346 134825 553 910

kroA100 (99) 21282 (000) 2132180 367 1302(24 25 25 25) 45733 4614795 340 677

kroB100 (99) 22141 (000) 2219315 490 1348(24 25 25 25) 45709 4581385 252 724

kroC100 (99) 20749 (000) 2078945 304 1262(24 25 25 25) 46388 4647535 410 690

kroD100 (99) 21294 (000) 2138911 464 1226(24 25 25 25) 45681 4595220 343 607

kroE100 (99) 22068 (000) 2211639 518 1452(24 25 25 25) 45431 4555925 324 727

rd100 (99) 7910 (000) 793270 475 1346(24 25 25 25) 15501 1552405 384 634

eil101 (100) 629 (000) 63275 579 1607(25 25 25 25) 1080 108000 380 892

lin105 (104) 14379 (000) 1441665 560 1439(26 26 26 26) 17584 1761820 213 825

pr107 (106) 44303 (000) 4440567 177 1456(26 26 27 27) 51487 5153880 214 779

gr120 (119) 6942 (000) 698695 616 2073(29 30 30 30) 13109 1312915 523 1048

pr124 (123) 59030 (000) 5918175 351 2077(30 31 31 31) 71295 7129500 112 1238

bier127 (126) 118282 (000) 11841960 933 2827(30 32 32 32) 174112 17425070 683 1954

ch130 (129) 6110 (000) 615050 1311 3012(32 32 32 33) 12000 1202205 566 2005

pr136 (135) 96772 (000) 9724080 1480 2836(33 34 34 34) 106605 10671840 814 2004

gr137 (136) 69853 (000) 7042950 1200 2816(34 34 34 34) 81628 8171501 422 1499

pr144 (143) 58537 (000) 5867119 618 3083(35 36 36 36) 69093 6912834 258 2022

kroA150 (149) 26524 (000) 2662965 1233 3585(37 37 37 38) 52824 5298840 1274 1880

kroB150 (149) 26130 (000) 2626423 1721 3807(37 37 37 38) 54008 5423775 1315 1916

The Scientific World Journal 11

Table 7 Continued

Instance Clusters BestSol AvgSol FTime CTime

ch150 (149) 6528 (000) 655631 2092 3859(37 37 37 38) 13042 1308525 1020 1904

pr152 (151) 73682 (000) 7401745 1192 3425(37 38 38 38) 79941 7994100 180 2428

u159 (158) 42080 (000) 4233610 1285 3887(39 39 40 40) 42287 4230245 329 2094

si175 (174) 21407 (000) 2141210 5119 9568(43 43 44 44) 22893 2291065 761 3472

brg180 (179) 1950 (000) 201035 1429 4458(44 45 45 45) 19430 2106020 690 1529

rat195 (194) 2323 (000) 236220 2713 7136(48 48 49 49) 2544 255172 2086 3524

d198 (197) 15800 (013) 1585875 2460 7779(49 49 49 50) 17320 1733950 2976 4392

kroA200 (199) 29420 (018) 2961880 2108 7041(49 50 50 50) 62514 6294175 2345 3981

kroB200 (199) 29463 (009) 2980700 2387 7501(49 50 50 50) 62842 6325311 1828 4140

gr202 (201) 40160 (000) 4041385 3696 9725(50 50 50 51) 44176 4424820 2486 4043

ts225 (224) 126643 (000) 12700683 2851 9385(56 56 56 56) 171269 17154330 2573 4877

tsp225 (224) 3923 (018) 396735 3829 10530(56 56 56 56) 5133 517115 2011 5491

pr226 (225) 80467 (012) 8095360 1628 8769(56 56 56 57) 96508 9651010 2838 6322

gr229 (228) 134957 (026) 13618435 3733 11979(57 57 57 57) 143028 14363245 3204 5731

gil262 (261) 2391 (055) 240315 5308 15126(65 65 65 66) 4874 490645 6433 9372

pr264 (263) 49219 (017) 4981445 4875 14999(65 66 66 66) 60161 6029415 2278 9720

a280 (279) 2585 (023) 261405 10525 18703(69 70 70 70) 2740 274375 3275 9428

pr299 (298) 48375 (038) 4885706 3811 20284(74 74 75 75) 55253 5595165 4536 12361

lin318 (317) 42301 (065) 4267965 8568 25392(79 79 79 80) 52578 5281135 6229 13335

rd400 (399) 15370 (058) 1545220 20327 54674(99 100 100 100) 30821 3100660 15140 25978

fl417 (416) 11930 (058) 1200463 22738 54417(104 104 104 104) 20457 2057624 20162 34601

gr431 (430) 173270 (108) 17604720 30035 71664(107 107 108 108) 185162 18666156 29181 40647

Average 2986 7016

be better and on average solution quality by HGA is 024better than that by LSA

In terms of computational time it can be easily concludedthat HGA ismuch better than LSA though LSAwas executedon slower machine (Pentium IV PC with speed 3GHz and

448 MB RAM) Of course the nature of LSA and HGA isnot the same LSA gives exact optimal solution whereas HGAgives heuristic solution It is also seen from the table thaton average HGA hits final best solution for the first timewithin 13 of complete computational time This shows that

12 The Scientific World Journal

HGA finds best solution on average in the beginning of thegenerations for these instances

44 Proposed Solution for Some More Symmetric InstancesTable 7 presents results for some more symmetric TSPLIBinstances of sizes from 52 to 431 with various clusters andcluster sizes Since to the best of our knowledge no literaturepresents solution for these instances hence we could notprovide any comparative study on these instances Howeverwe present the results for future study of the OCTSP on theseinstances For our self-comparison we provide solution valueand percentage of error (in parentheses) by our HGA forthe instances with one cluster which are of course usualTSP instances Out of forty-seven usual TSP instances HGAfinds exact optimal solution to thirty-three instances For theremaining instances maximum percentage of error is 108That means our algorithm can provide near exact solutionif not exact Treating this study as a base for effectiveness ofthe algorithm we can conclude that the reported solutionsare near exact solution if not exact It is also seen fromthe table that for the same instances as the number ofclusters increases solution value also increases On the otherhand as the number of clusters increases computational timedecreases In general computational time for solving a singleclustered instance (ie usual TSP instance) is more thanits corresponding multiclustered instances It seems that thestructures of these multiclustered instances are less complexand hence easier than their corresponding single clusteredinstances For these symmetric instances on average HGAhits final best solution for the first time within 43 ofcomplete computational time This shows that HGA findsbest solution for these instances on average in the middleof the generations

5 Conclusions

We presented a hybrid genetic algorithm using sequentialconstructive crossover 2-opt search a local search and animmigration method to obtain heuristic solution to theOCTSPWe have used a sequential samplingmethod for gen-erating initial population The efficiency of the hybrid GA tothe problemhas been examined against the exact partitioningalgorithm (LBDCOMP) [9] for some asymmetric TSPLIBinstances and the lexisearch algorithm (LSA) [11] for somesmall sized symmetric TSPLIB instances The computationalexperiments show that ourHGA is efficient in producing highquality of solution for the benchmark instances On the basisof solution quality our HGA is found to be better than theLBDCOMP and LSA In terms of computational time alsoour algorithm is found to be the best one Finally we presentsolution to the problem for some more symmetric TSPLIBinstances Since to the best of our knowledge no literaturepresents solution for these instances we could not confirmthe quality of our solutions for the instances However forthe symmetric instances of size up to 51 we found that ourHGA obtains exact optimal solution to the instances It isto be noted that HGA does not require any modification forsolving different types of TSPLIB instances

For any instance as the number of clusters increasesthe solution value also increases Computational time forsolving a single clustered instance (ie usual TSP instance) ismore than that for solving its corresponding multiclusteredinstances For any multiclustered instance as the clustersbecome more unbalanced computational time increases

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The author is very much thankful to the honourable review-ers for their constructive comments and suggestions Thisresearch was supported by the NSTIP Strategic Technologiesprogramno 10 in the Kingdomof Saudi Arabia via Award no11-INF1788-08 The author is also very much thankful to theNSTIP for its financial and technical supports

References

[1] J A Chisman ldquoThe clustered traveling salesman problemrdquoComputers and Operations Research vol 2 no 2 pp 115ndash1191975

[2] MGendreau AHertz andG Laporte ldquoThe traveling salesmanproblem with backhaulsrdquo Computers and Operations Researchvol 23 no 5 pp 501ndash508 1996

[3] N Guttmann-Beck R Hassin S Khuller and B RaghavacharildquoApproximation algorithms with bounded performance guar-antees for the clustered traveling salesman problemrdquo Algorith-mica vol 28 no 4 pp 422ndash437 2000

[4] F C J Lokin ldquoProcedures for travelling salesman problemswith additional constraintsrdquo European Journal of OperationalResearch vol 3 no 2 pp 135ndash141 1979

[5] G Laporte J-Y Potvin and F Quilleret ldquoTabu search heuristicusing genetic diversification for the clustered traveling salesmanproblemrdquo Journal of Heuristics vol 2 no 3 pp 187ndash200 1997

[6] G Laporte and U Palekar ldquoSome applications of the clus-tered travelling salesman problemrdquo Journal of the OperationalResearch Society vol 53 no 9 pp 972ndash976 2002

[7] Z H Ahmed ldquoGenetic algorithm for the traveling salesmanproblem using sequential constructive crossover operatorrdquoInternational Journal of Biometrics amp Bioinformatics vol 3 no6 pp 96ndash105 2010

[8] Z H Ahmed ldquoA hybrid sequential constructive samplingalgorithm for the bottleneck traveling salesman problemrdquoInternational Journal of Computational Intelligence Researchvol 6 no 3 pp 475ndash484 2010

[9] T Aramgiatisiris ldquoAn exact decomposition algorithm for thetraveling salesman problemwith backhaulsrdquo Journal of Researchin Engineering and Technology vol 1 pp 151ndash164 2004

[10] TSPLIB 1995 httpcomoptifiuni-heidelbergdesoftwareTSPLIB95

[11] Z H Ahmed ldquoAn exact algorithm for the clustered travelingsalesman problemrdquo Opsearch vol 50 no 2 pp 215ndash228 2013

[12] J D E Little K G Murthy D W Sweeny and C KarelldquoAn algorithm for the travelling salesman problemrdquo OperationsResearch vol 11 pp 972ndash989 1963

The Scientific World Journal 13

[13] K Jongens and T Volgenant ldquoThe symmetric clustered trav-eling salesman problemrdquo European Journal of OperationalResearch vol 19 no 1 pp 68ndash75 1985

[14] M Gendreau G Laporte and J Y Potvin ldquoHeuristics for theclustered traveling salesman problemrdquo Tech Rep CRT-94-54Centre de Recherche sur les Transports Universite deMontrealMontreal Canada 1994

[15] J-Y Potvin and F Guertin ldquoA genetic algorithm for theclustered traveling salesman problem with an a priori order onthe clustersrdquo Tech Rep CRT-95-06 Centre de recherchesur lestransports Universite de Montreal Montreal Canada 1995

[16] J-Y Potvin and F Guertin ldquoThe clustered traveling salesmanproblem a genetic approachrdquo in Meta-Heuristics Theory ampApplications I H Osman and J Kelly Eds pp 619ndash631 KluwerAcademic Norwell Mass USA 1996

[17] S Anily J Bramel andAHertz ldquo53-Approximation algorithmfor the clustered traveling salesman tour and path problemsrdquoOperations Research Letters vol 24 no 1 pp 29ndash35 1999

[18] N Christofides ldquoWorst-case analysis of a new heuristic for thetraveling salesmanproblemrdquoTech Rep 388Graduate School ofIndustrial Administration Carnegie-Mellon University Pitts-burgh Pa USA 1976

[19] W Sheng N Xi M Song and Y Chen ldquoRobot path planningfor dimensional measurement in automotive manufacturingrdquoJournal of Manufacturing Science and Engineering Transactionsof the ASME vol 127 no 2 pp 420ndash428 2005

[20] C Ding Y Cheng andM He ldquoTwo-level genetic algorithm forclustered traveling salesman problem with application in large-scale TSPsrdquo Tsinghua Science and Technology vol 12 no 4 pp459ndash465 2007

[21] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley New York NY USA 1989

[22] Z H Ahmed ldquoA hybrid genetic algorithm for the bottlenecktraveling salesman problemrdquo ACM Transactions on EmbeddedComputing Systems vol 12 no 1 article 9 2013

[23] K Deb Optimization for Engineering Design Algorithms andExamples Prentice Hall India New Delhi India 1995

[24] Z H Ahmed ldquoMulti-parent extension of sequential construc-tive crossover for the travelling salesman problemrdquo Interna-tional Journal of Operational Research vol 11 no 3 pp 331ndash3422011

[25] Z H Ahmed ldquoAn experimental study of a hybrid geneticalgorithm for the maximum travelling salesman problemrdquoMathematical Sciences vol 7 no 1 pp 1ndash7 2013

[26] C-X Wang D-W Cui Z-R Wang and D Chen ldquoA novel antcolony system based on minimum 1-tree and hybrid mutationfor TSPrdquo in Proceedings of the 1st International Conferenceon Natural Computation (ICNC rsquo05) LNCS pp 1269ndash1278Springer Changsha China August 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 10: Research Article The Ordered Clustered Travelling Salesman Problem: A Hybrid Genetic ...downloads.hindawi.com/journals/tswj/2014/258207.pdf · 2019-07-31 · Research Article The

10 The Scientific World Journal

Table 7 Results on some symmetric TSPLIB instances using HGA

Instance Clusters BestSol AvgSol FTime CTime

berlin52 (51) 7542 (000) 754200 070 363(25 26) 10422 1042200 034 250

brazil58 (57) 25395 (000) 2539500 109 450(28 29) 34110 3411000 114 305

st70 (69) 675 (000) 67715 149 615(34 35) 916 91600 205 418

eil76 (75) 538 (000) 53926 242 733(37 38) 721 72312 199 511

pr76 (75) 108159 (000) 10825455 209 746(37 38) 120436 12058313 274 510

gr96 (95) 55209 (000) 5567285 589 1207(47 48) 56757 5676722 344 840

rat99 (98) 1211 (000) 121840 116 1291(49 49) 1346 134825 553 910

kroA100 (99) 21282 (000) 2132180 367 1302(24 25 25 25) 45733 4614795 340 677

kroB100 (99) 22141 (000) 2219315 490 1348(24 25 25 25) 45709 4581385 252 724

kroC100 (99) 20749 (000) 2078945 304 1262(24 25 25 25) 46388 4647535 410 690

kroD100 (99) 21294 (000) 2138911 464 1226(24 25 25 25) 45681 4595220 343 607

kroE100 (99) 22068 (000) 2211639 518 1452(24 25 25 25) 45431 4555925 324 727

rd100 (99) 7910 (000) 793270 475 1346(24 25 25 25) 15501 1552405 384 634

eil101 (100) 629 (000) 63275 579 1607(25 25 25 25) 1080 108000 380 892

lin105 (104) 14379 (000) 1441665 560 1439(26 26 26 26) 17584 1761820 213 825

pr107 (106) 44303 (000) 4440567 177 1456(26 26 27 27) 51487 5153880 214 779

gr120 (119) 6942 (000) 698695 616 2073(29 30 30 30) 13109 1312915 523 1048

pr124 (123) 59030 (000) 5918175 351 2077(30 31 31 31) 71295 7129500 112 1238

bier127 (126) 118282 (000) 11841960 933 2827(30 32 32 32) 174112 17425070 683 1954

ch130 (129) 6110 (000) 615050 1311 3012(32 32 32 33) 12000 1202205 566 2005

pr136 (135) 96772 (000) 9724080 1480 2836(33 34 34 34) 106605 10671840 814 2004

gr137 (136) 69853 (000) 7042950 1200 2816(34 34 34 34) 81628 8171501 422 1499

pr144 (143) 58537 (000) 5867119 618 3083(35 36 36 36) 69093 6912834 258 2022

kroA150 (149) 26524 (000) 2662965 1233 3585(37 37 37 38) 52824 5298840 1274 1880

kroB150 (149) 26130 (000) 2626423 1721 3807(37 37 37 38) 54008 5423775 1315 1916

The Scientific World Journal 11

Table 7 Continued

Instance Clusters BestSol AvgSol FTime CTime

ch150 (149) 6528 (000) 655631 2092 3859(37 37 37 38) 13042 1308525 1020 1904

pr152 (151) 73682 (000) 7401745 1192 3425(37 38 38 38) 79941 7994100 180 2428

u159 (158) 42080 (000) 4233610 1285 3887(39 39 40 40) 42287 4230245 329 2094

si175 (174) 21407 (000) 2141210 5119 9568(43 43 44 44) 22893 2291065 761 3472

brg180 (179) 1950 (000) 201035 1429 4458(44 45 45 45) 19430 2106020 690 1529

rat195 (194) 2323 (000) 236220 2713 7136(48 48 49 49) 2544 255172 2086 3524

d198 (197) 15800 (013) 1585875 2460 7779(49 49 49 50) 17320 1733950 2976 4392

kroA200 (199) 29420 (018) 2961880 2108 7041(49 50 50 50) 62514 6294175 2345 3981

kroB200 (199) 29463 (009) 2980700 2387 7501(49 50 50 50) 62842 6325311 1828 4140

gr202 (201) 40160 (000) 4041385 3696 9725(50 50 50 51) 44176 4424820 2486 4043

ts225 (224) 126643 (000) 12700683 2851 9385(56 56 56 56) 171269 17154330 2573 4877

tsp225 (224) 3923 (018) 396735 3829 10530(56 56 56 56) 5133 517115 2011 5491

pr226 (225) 80467 (012) 8095360 1628 8769(56 56 56 57) 96508 9651010 2838 6322

gr229 (228) 134957 (026) 13618435 3733 11979(57 57 57 57) 143028 14363245 3204 5731

gil262 (261) 2391 (055) 240315 5308 15126(65 65 65 66) 4874 490645 6433 9372

pr264 (263) 49219 (017) 4981445 4875 14999(65 66 66 66) 60161 6029415 2278 9720

a280 (279) 2585 (023) 261405 10525 18703(69 70 70 70) 2740 274375 3275 9428

pr299 (298) 48375 (038) 4885706 3811 20284(74 74 75 75) 55253 5595165 4536 12361

lin318 (317) 42301 (065) 4267965 8568 25392(79 79 79 80) 52578 5281135 6229 13335

rd400 (399) 15370 (058) 1545220 20327 54674(99 100 100 100) 30821 3100660 15140 25978

fl417 (416) 11930 (058) 1200463 22738 54417(104 104 104 104) 20457 2057624 20162 34601

gr431 (430) 173270 (108) 17604720 30035 71664(107 107 108 108) 185162 18666156 29181 40647

Average 2986 7016

be better and on average solution quality by HGA is 024better than that by LSA

In terms of computational time it can be easily concludedthat HGA ismuch better than LSA though LSAwas executedon slower machine (Pentium IV PC with speed 3GHz and

448 MB RAM) Of course the nature of LSA and HGA isnot the same LSA gives exact optimal solution whereas HGAgives heuristic solution It is also seen from the table thaton average HGA hits final best solution for the first timewithin 13 of complete computational time This shows that

12 The Scientific World Journal

HGA finds best solution on average in the beginning of thegenerations for these instances

44 Proposed Solution for Some More Symmetric InstancesTable 7 presents results for some more symmetric TSPLIBinstances of sizes from 52 to 431 with various clusters andcluster sizes Since to the best of our knowledge no literaturepresents solution for these instances hence we could notprovide any comparative study on these instances Howeverwe present the results for future study of the OCTSP on theseinstances For our self-comparison we provide solution valueand percentage of error (in parentheses) by our HGA forthe instances with one cluster which are of course usualTSP instances Out of forty-seven usual TSP instances HGAfinds exact optimal solution to thirty-three instances For theremaining instances maximum percentage of error is 108That means our algorithm can provide near exact solutionif not exact Treating this study as a base for effectiveness ofthe algorithm we can conclude that the reported solutionsare near exact solution if not exact It is also seen fromthe table that for the same instances as the number ofclusters increases solution value also increases On the otherhand as the number of clusters increases computational timedecreases In general computational time for solving a singleclustered instance (ie usual TSP instance) is more thanits corresponding multiclustered instances It seems that thestructures of these multiclustered instances are less complexand hence easier than their corresponding single clusteredinstances For these symmetric instances on average HGAhits final best solution for the first time within 43 ofcomplete computational time This shows that HGA findsbest solution for these instances on average in the middleof the generations

5 Conclusions

We presented a hybrid genetic algorithm using sequentialconstructive crossover 2-opt search a local search and animmigration method to obtain heuristic solution to theOCTSPWe have used a sequential samplingmethod for gen-erating initial population The efficiency of the hybrid GA tothe problemhas been examined against the exact partitioningalgorithm (LBDCOMP) [9] for some asymmetric TSPLIBinstances and the lexisearch algorithm (LSA) [11] for somesmall sized symmetric TSPLIB instances The computationalexperiments show that ourHGA is efficient in producing highquality of solution for the benchmark instances On the basisof solution quality our HGA is found to be better than theLBDCOMP and LSA In terms of computational time alsoour algorithm is found to be the best one Finally we presentsolution to the problem for some more symmetric TSPLIBinstances Since to the best of our knowledge no literaturepresents solution for these instances we could not confirmthe quality of our solutions for the instances However forthe symmetric instances of size up to 51 we found that ourHGA obtains exact optimal solution to the instances It isto be noted that HGA does not require any modification forsolving different types of TSPLIB instances

For any instance as the number of clusters increasesthe solution value also increases Computational time forsolving a single clustered instance (ie usual TSP instance) ismore than that for solving its corresponding multiclusteredinstances For any multiclustered instance as the clustersbecome more unbalanced computational time increases

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The author is very much thankful to the honourable review-ers for their constructive comments and suggestions Thisresearch was supported by the NSTIP Strategic Technologiesprogramno 10 in the Kingdomof Saudi Arabia via Award no11-INF1788-08 The author is also very much thankful to theNSTIP for its financial and technical supports

References

[1] J A Chisman ldquoThe clustered traveling salesman problemrdquoComputers and Operations Research vol 2 no 2 pp 115ndash1191975

[2] MGendreau AHertz andG Laporte ldquoThe traveling salesmanproblem with backhaulsrdquo Computers and Operations Researchvol 23 no 5 pp 501ndash508 1996

[3] N Guttmann-Beck R Hassin S Khuller and B RaghavacharildquoApproximation algorithms with bounded performance guar-antees for the clustered traveling salesman problemrdquo Algorith-mica vol 28 no 4 pp 422ndash437 2000

[4] F C J Lokin ldquoProcedures for travelling salesman problemswith additional constraintsrdquo European Journal of OperationalResearch vol 3 no 2 pp 135ndash141 1979

[5] G Laporte J-Y Potvin and F Quilleret ldquoTabu search heuristicusing genetic diversification for the clustered traveling salesmanproblemrdquo Journal of Heuristics vol 2 no 3 pp 187ndash200 1997

[6] G Laporte and U Palekar ldquoSome applications of the clus-tered travelling salesman problemrdquo Journal of the OperationalResearch Society vol 53 no 9 pp 972ndash976 2002

[7] Z H Ahmed ldquoGenetic algorithm for the traveling salesmanproblem using sequential constructive crossover operatorrdquoInternational Journal of Biometrics amp Bioinformatics vol 3 no6 pp 96ndash105 2010

[8] Z H Ahmed ldquoA hybrid sequential constructive samplingalgorithm for the bottleneck traveling salesman problemrdquoInternational Journal of Computational Intelligence Researchvol 6 no 3 pp 475ndash484 2010

[9] T Aramgiatisiris ldquoAn exact decomposition algorithm for thetraveling salesman problemwith backhaulsrdquo Journal of Researchin Engineering and Technology vol 1 pp 151ndash164 2004

[10] TSPLIB 1995 httpcomoptifiuni-heidelbergdesoftwareTSPLIB95

[11] Z H Ahmed ldquoAn exact algorithm for the clustered travelingsalesman problemrdquo Opsearch vol 50 no 2 pp 215ndash228 2013

[12] J D E Little K G Murthy D W Sweeny and C KarelldquoAn algorithm for the travelling salesman problemrdquo OperationsResearch vol 11 pp 972ndash989 1963

The Scientific World Journal 13

[13] K Jongens and T Volgenant ldquoThe symmetric clustered trav-eling salesman problemrdquo European Journal of OperationalResearch vol 19 no 1 pp 68ndash75 1985

[14] M Gendreau G Laporte and J Y Potvin ldquoHeuristics for theclustered traveling salesman problemrdquo Tech Rep CRT-94-54Centre de Recherche sur les Transports Universite deMontrealMontreal Canada 1994

[15] J-Y Potvin and F Guertin ldquoA genetic algorithm for theclustered traveling salesman problem with an a priori order onthe clustersrdquo Tech Rep CRT-95-06 Centre de recherchesur lestransports Universite de Montreal Montreal Canada 1995

[16] J-Y Potvin and F Guertin ldquoThe clustered traveling salesmanproblem a genetic approachrdquo in Meta-Heuristics Theory ampApplications I H Osman and J Kelly Eds pp 619ndash631 KluwerAcademic Norwell Mass USA 1996

[17] S Anily J Bramel andAHertz ldquo53-Approximation algorithmfor the clustered traveling salesman tour and path problemsrdquoOperations Research Letters vol 24 no 1 pp 29ndash35 1999

[18] N Christofides ldquoWorst-case analysis of a new heuristic for thetraveling salesmanproblemrdquoTech Rep 388Graduate School ofIndustrial Administration Carnegie-Mellon University Pitts-burgh Pa USA 1976

[19] W Sheng N Xi M Song and Y Chen ldquoRobot path planningfor dimensional measurement in automotive manufacturingrdquoJournal of Manufacturing Science and Engineering Transactionsof the ASME vol 127 no 2 pp 420ndash428 2005

[20] C Ding Y Cheng andM He ldquoTwo-level genetic algorithm forclustered traveling salesman problem with application in large-scale TSPsrdquo Tsinghua Science and Technology vol 12 no 4 pp459ndash465 2007

[21] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley New York NY USA 1989

[22] Z H Ahmed ldquoA hybrid genetic algorithm for the bottlenecktraveling salesman problemrdquo ACM Transactions on EmbeddedComputing Systems vol 12 no 1 article 9 2013

[23] K Deb Optimization for Engineering Design Algorithms andExamples Prentice Hall India New Delhi India 1995

[24] Z H Ahmed ldquoMulti-parent extension of sequential construc-tive crossover for the travelling salesman problemrdquo Interna-tional Journal of Operational Research vol 11 no 3 pp 331ndash3422011

[25] Z H Ahmed ldquoAn experimental study of a hybrid geneticalgorithm for the maximum travelling salesman problemrdquoMathematical Sciences vol 7 no 1 pp 1ndash7 2013

[26] C-X Wang D-W Cui Z-R Wang and D Chen ldquoA novel antcolony system based on minimum 1-tree and hybrid mutationfor TSPrdquo in Proceedings of the 1st International Conferenceon Natural Computation (ICNC rsquo05) LNCS pp 1269ndash1278Springer Changsha China August 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 11: Research Article The Ordered Clustered Travelling Salesman Problem: A Hybrid Genetic ...downloads.hindawi.com/journals/tswj/2014/258207.pdf · 2019-07-31 · Research Article The

The Scientific World Journal 11

Table 7 Continued

Instance Clusters BestSol AvgSol FTime CTime

ch150 (149) 6528 (000) 655631 2092 3859(37 37 37 38) 13042 1308525 1020 1904

pr152 (151) 73682 (000) 7401745 1192 3425(37 38 38 38) 79941 7994100 180 2428

u159 (158) 42080 (000) 4233610 1285 3887(39 39 40 40) 42287 4230245 329 2094

si175 (174) 21407 (000) 2141210 5119 9568(43 43 44 44) 22893 2291065 761 3472

brg180 (179) 1950 (000) 201035 1429 4458(44 45 45 45) 19430 2106020 690 1529

rat195 (194) 2323 (000) 236220 2713 7136(48 48 49 49) 2544 255172 2086 3524

d198 (197) 15800 (013) 1585875 2460 7779(49 49 49 50) 17320 1733950 2976 4392

kroA200 (199) 29420 (018) 2961880 2108 7041(49 50 50 50) 62514 6294175 2345 3981

kroB200 (199) 29463 (009) 2980700 2387 7501(49 50 50 50) 62842 6325311 1828 4140

gr202 (201) 40160 (000) 4041385 3696 9725(50 50 50 51) 44176 4424820 2486 4043

ts225 (224) 126643 (000) 12700683 2851 9385(56 56 56 56) 171269 17154330 2573 4877

tsp225 (224) 3923 (018) 396735 3829 10530(56 56 56 56) 5133 517115 2011 5491

pr226 (225) 80467 (012) 8095360 1628 8769(56 56 56 57) 96508 9651010 2838 6322

gr229 (228) 134957 (026) 13618435 3733 11979(57 57 57 57) 143028 14363245 3204 5731

gil262 (261) 2391 (055) 240315 5308 15126(65 65 65 66) 4874 490645 6433 9372

pr264 (263) 49219 (017) 4981445 4875 14999(65 66 66 66) 60161 6029415 2278 9720

a280 (279) 2585 (023) 261405 10525 18703(69 70 70 70) 2740 274375 3275 9428

pr299 (298) 48375 (038) 4885706 3811 20284(74 74 75 75) 55253 5595165 4536 12361

lin318 (317) 42301 (065) 4267965 8568 25392(79 79 79 80) 52578 5281135 6229 13335

rd400 (399) 15370 (058) 1545220 20327 54674(99 100 100 100) 30821 3100660 15140 25978

fl417 (416) 11930 (058) 1200463 22738 54417(104 104 104 104) 20457 2057624 20162 34601

gr431 (430) 173270 (108) 17604720 30035 71664(107 107 108 108) 185162 18666156 29181 40647

Average 2986 7016

be better and on average solution quality by HGA is 024better than that by LSA

In terms of computational time it can be easily concludedthat HGA ismuch better than LSA though LSAwas executedon slower machine (Pentium IV PC with speed 3GHz and

448 MB RAM) Of course the nature of LSA and HGA isnot the same LSA gives exact optimal solution whereas HGAgives heuristic solution It is also seen from the table thaton average HGA hits final best solution for the first timewithin 13 of complete computational time This shows that

12 The Scientific World Journal

HGA finds best solution on average in the beginning of thegenerations for these instances

44 Proposed Solution for Some More Symmetric InstancesTable 7 presents results for some more symmetric TSPLIBinstances of sizes from 52 to 431 with various clusters andcluster sizes Since to the best of our knowledge no literaturepresents solution for these instances hence we could notprovide any comparative study on these instances Howeverwe present the results for future study of the OCTSP on theseinstances For our self-comparison we provide solution valueand percentage of error (in parentheses) by our HGA forthe instances with one cluster which are of course usualTSP instances Out of forty-seven usual TSP instances HGAfinds exact optimal solution to thirty-three instances For theremaining instances maximum percentage of error is 108That means our algorithm can provide near exact solutionif not exact Treating this study as a base for effectiveness ofthe algorithm we can conclude that the reported solutionsare near exact solution if not exact It is also seen fromthe table that for the same instances as the number ofclusters increases solution value also increases On the otherhand as the number of clusters increases computational timedecreases In general computational time for solving a singleclustered instance (ie usual TSP instance) is more thanits corresponding multiclustered instances It seems that thestructures of these multiclustered instances are less complexand hence easier than their corresponding single clusteredinstances For these symmetric instances on average HGAhits final best solution for the first time within 43 ofcomplete computational time This shows that HGA findsbest solution for these instances on average in the middleof the generations

5 Conclusions

We presented a hybrid genetic algorithm using sequentialconstructive crossover 2-opt search a local search and animmigration method to obtain heuristic solution to theOCTSPWe have used a sequential samplingmethod for gen-erating initial population The efficiency of the hybrid GA tothe problemhas been examined against the exact partitioningalgorithm (LBDCOMP) [9] for some asymmetric TSPLIBinstances and the lexisearch algorithm (LSA) [11] for somesmall sized symmetric TSPLIB instances The computationalexperiments show that ourHGA is efficient in producing highquality of solution for the benchmark instances On the basisof solution quality our HGA is found to be better than theLBDCOMP and LSA In terms of computational time alsoour algorithm is found to be the best one Finally we presentsolution to the problem for some more symmetric TSPLIBinstances Since to the best of our knowledge no literaturepresents solution for these instances we could not confirmthe quality of our solutions for the instances However forthe symmetric instances of size up to 51 we found that ourHGA obtains exact optimal solution to the instances It isto be noted that HGA does not require any modification forsolving different types of TSPLIB instances

For any instance as the number of clusters increasesthe solution value also increases Computational time forsolving a single clustered instance (ie usual TSP instance) ismore than that for solving its corresponding multiclusteredinstances For any multiclustered instance as the clustersbecome more unbalanced computational time increases

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The author is very much thankful to the honourable review-ers for their constructive comments and suggestions Thisresearch was supported by the NSTIP Strategic Technologiesprogramno 10 in the Kingdomof Saudi Arabia via Award no11-INF1788-08 The author is also very much thankful to theNSTIP for its financial and technical supports

References

[1] J A Chisman ldquoThe clustered traveling salesman problemrdquoComputers and Operations Research vol 2 no 2 pp 115ndash1191975

[2] MGendreau AHertz andG Laporte ldquoThe traveling salesmanproblem with backhaulsrdquo Computers and Operations Researchvol 23 no 5 pp 501ndash508 1996

[3] N Guttmann-Beck R Hassin S Khuller and B RaghavacharildquoApproximation algorithms with bounded performance guar-antees for the clustered traveling salesman problemrdquo Algorith-mica vol 28 no 4 pp 422ndash437 2000

[4] F C J Lokin ldquoProcedures for travelling salesman problemswith additional constraintsrdquo European Journal of OperationalResearch vol 3 no 2 pp 135ndash141 1979

[5] G Laporte J-Y Potvin and F Quilleret ldquoTabu search heuristicusing genetic diversification for the clustered traveling salesmanproblemrdquo Journal of Heuristics vol 2 no 3 pp 187ndash200 1997

[6] G Laporte and U Palekar ldquoSome applications of the clus-tered travelling salesman problemrdquo Journal of the OperationalResearch Society vol 53 no 9 pp 972ndash976 2002

[7] Z H Ahmed ldquoGenetic algorithm for the traveling salesmanproblem using sequential constructive crossover operatorrdquoInternational Journal of Biometrics amp Bioinformatics vol 3 no6 pp 96ndash105 2010

[8] Z H Ahmed ldquoA hybrid sequential constructive samplingalgorithm for the bottleneck traveling salesman problemrdquoInternational Journal of Computational Intelligence Researchvol 6 no 3 pp 475ndash484 2010

[9] T Aramgiatisiris ldquoAn exact decomposition algorithm for thetraveling salesman problemwith backhaulsrdquo Journal of Researchin Engineering and Technology vol 1 pp 151ndash164 2004

[10] TSPLIB 1995 httpcomoptifiuni-heidelbergdesoftwareTSPLIB95

[11] Z H Ahmed ldquoAn exact algorithm for the clustered travelingsalesman problemrdquo Opsearch vol 50 no 2 pp 215ndash228 2013

[12] J D E Little K G Murthy D W Sweeny and C KarelldquoAn algorithm for the travelling salesman problemrdquo OperationsResearch vol 11 pp 972ndash989 1963

The Scientific World Journal 13

[13] K Jongens and T Volgenant ldquoThe symmetric clustered trav-eling salesman problemrdquo European Journal of OperationalResearch vol 19 no 1 pp 68ndash75 1985

[14] M Gendreau G Laporte and J Y Potvin ldquoHeuristics for theclustered traveling salesman problemrdquo Tech Rep CRT-94-54Centre de Recherche sur les Transports Universite deMontrealMontreal Canada 1994

[15] J-Y Potvin and F Guertin ldquoA genetic algorithm for theclustered traveling salesman problem with an a priori order onthe clustersrdquo Tech Rep CRT-95-06 Centre de recherchesur lestransports Universite de Montreal Montreal Canada 1995

[16] J-Y Potvin and F Guertin ldquoThe clustered traveling salesmanproblem a genetic approachrdquo in Meta-Heuristics Theory ampApplications I H Osman and J Kelly Eds pp 619ndash631 KluwerAcademic Norwell Mass USA 1996

[17] S Anily J Bramel andAHertz ldquo53-Approximation algorithmfor the clustered traveling salesman tour and path problemsrdquoOperations Research Letters vol 24 no 1 pp 29ndash35 1999

[18] N Christofides ldquoWorst-case analysis of a new heuristic for thetraveling salesmanproblemrdquoTech Rep 388Graduate School ofIndustrial Administration Carnegie-Mellon University Pitts-burgh Pa USA 1976

[19] W Sheng N Xi M Song and Y Chen ldquoRobot path planningfor dimensional measurement in automotive manufacturingrdquoJournal of Manufacturing Science and Engineering Transactionsof the ASME vol 127 no 2 pp 420ndash428 2005

[20] C Ding Y Cheng andM He ldquoTwo-level genetic algorithm forclustered traveling salesman problem with application in large-scale TSPsrdquo Tsinghua Science and Technology vol 12 no 4 pp459ndash465 2007

[21] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley New York NY USA 1989

[22] Z H Ahmed ldquoA hybrid genetic algorithm for the bottlenecktraveling salesman problemrdquo ACM Transactions on EmbeddedComputing Systems vol 12 no 1 article 9 2013

[23] K Deb Optimization for Engineering Design Algorithms andExamples Prentice Hall India New Delhi India 1995

[24] Z H Ahmed ldquoMulti-parent extension of sequential construc-tive crossover for the travelling salesman problemrdquo Interna-tional Journal of Operational Research vol 11 no 3 pp 331ndash3422011

[25] Z H Ahmed ldquoAn experimental study of a hybrid geneticalgorithm for the maximum travelling salesman problemrdquoMathematical Sciences vol 7 no 1 pp 1ndash7 2013

[26] C-X Wang D-W Cui Z-R Wang and D Chen ldquoA novel antcolony system based on minimum 1-tree and hybrid mutationfor TSPrdquo in Proceedings of the 1st International Conferenceon Natural Computation (ICNC rsquo05) LNCS pp 1269ndash1278Springer Changsha China August 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 12: Research Article The Ordered Clustered Travelling Salesman Problem: A Hybrid Genetic ...downloads.hindawi.com/journals/tswj/2014/258207.pdf · 2019-07-31 · Research Article The

12 The Scientific World Journal

HGA finds best solution on average in the beginning of thegenerations for these instances

44 Proposed Solution for Some More Symmetric InstancesTable 7 presents results for some more symmetric TSPLIBinstances of sizes from 52 to 431 with various clusters andcluster sizes Since to the best of our knowledge no literaturepresents solution for these instances hence we could notprovide any comparative study on these instances Howeverwe present the results for future study of the OCTSP on theseinstances For our self-comparison we provide solution valueand percentage of error (in parentheses) by our HGA forthe instances with one cluster which are of course usualTSP instances Out of forty-seven usual TSP instances HGAfinds exact optimal solution to thirty-three instances For theremaining instances maximum percentage of error is 108That means our algorithm can provide near exact solutionif not exact Treating this study as a base for effectiveness ofthe algorithm we can conclude that the reported solutionsare near exact solution if not exact It is also seen fromthe table that for the same instances as the number ofclusters increases solution value also increases On the otherhand as the number of clusters increases computational timedecreases In general computational time for solving a singleclustered instance (ie usual TSP instance) is more thanits corresponding multiclustered instances It seems that thestructures of these multiclustered instances are less complexand hence easier than their corresponding single clusteredinstances For these symmetric instances on average HGAhits final best solution for the first time within 43 ofcomplete computational time This shows that HGA findsbest solution for these instances on average in the middleof the generations

5 Conclusions

We presented a hybrid genetic algorithm using sequentialconstructive crossover 2-opt search a local search and animmigration method to obtain heuristic solution to theOCTSPWe have used a sequential samplingmethod for gen-erating initial population The efficiency of the hybrid GA tothe problemhas been examined against the exact partitioningalgorithm (LBDCOMP) [9] for some asymmetric TSPLIBinstances and the lexisearch algorithm (LSA) [11] for somesmall sized symmetric TSPLIB instances The computationalexperiments show that ourHGA is efficient in producing highquality of solution for the benchmark instances On the basisof solution quality our HGA is found to be better than theLBDCOMP and LSA In terms of computational time alsoour algorithm is found to be the best one Finally we presentsolution to the problem for some more symmetric TSPLIBinstances Since to the best of our knowledge no literaturepresents solution for these instances we could not confirmthe quality of our solutions for the instances However forthe symmetric instances of size up to 51 we found that ourHGA obtains exact optimal solution to the instances It isto be noted that HGA does not require any modification forsolving different types of TSPLIB instances

For any instance as the number of clusters increasesthe solution value also increases Computational time forsolving a single clustered instance (ie usual TSP instance) ismore than that for solving its corresponding multiclusteredinstances For any multiclustered instance as the clustersbecome more unbalanced computational time increases

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The author is very much thankful to the honourable review-ers for their constructive comments and suggestions Thisresearch was supported by the NSTIP Strategic Technologiesprogramno 10 in the Kingdomof Saudi Arabia via Award no11-INF1788-08 The author is also very much thankful to theNSTIP for its financial and technical supports

References

[1] J A Chisman ldquoThe clustered traveling salesman problemrdquoComputers and Operations Research vol 2 no 2 pp 115ndash1191975

[2] MGendreau AHertz andG Laporte ldquoThe traveling salesmanproblem with backhaulsrdquo Computers and Operations Researchvol 23 no 5 pp 501ndash508 1996

[3] N Guttmann-Beck R Hassin S Khuller and B RaghavacharildquoApproximation algorithms with bounded performance guar-antees for the clustered traveling salesman problemrdquo Algorith-mica vol 28 no 4 pp 422ndash437 2000

[4] F C J Lokin ldquoProcedures for travelling salesman problemswith additional constraintsrdquo European Journal of OperationalResearch vol 3 no 2 pp 135ndash141 1979

[5] G Laporte J-Y Potvin and F Quilleret ldquoTabu search heuristicusing genetic diversification for the clustered traveling salesmanproblemrdquo Journal of Heuristics vol 2 no 3 pp 187ndash200 1997

[6] G Laporte and U Palekar ldquoSome applications of the clus-tered travelling salesman problemrdquo Journal of the OperationalResearch Society vol 53 no 9 pp 972ndash976 2002

[7] Z H Ahmed ldquoGenetic algorithm for the traveling salesmanproblem using sequential constructive crossover operatorrdquoInternational Journal of Biometrics amp Bioinformatics vol 3 no6 pp 96ndash105 2010

[8] Z H Ahmed ldquoA hybrid sequential constructive samplingalgorithm for the bottleneck traveling salesman problemrdquoInternational Journal of Computational Intelligence Researchvol 6 no 3 pp 475ndash484 2010

[9] T Aramgiatisiris ldquoAn exact decomposition algorithm for thetraveling salesman problemwith backhaulsrdquo Journal of Researchin Engineering and Technology vol 1 pp 151ndash164 2004

[10] TSPLIB 1995 httpcomoptifiuni-heidelbergdesoftwareTSPLIB95

[11] Z H Ahmed ldquoAn exact algorithm for the clustered travelingsalesman problemrdquo Opsearch vol 50 no 2 pp 215ndash228 2013

[12] J D E Little K G Murthy D W Sweeny and C KarelldquoAn algorithm for the travelling salesman problemrdquo OperationsResearch vol 11 pp 972ndash989 1963

The Scientific World Journal 13

[13] K Jongens and T Volgenant ldquoThe symmetric clustered trav-eling salesman problemrdquo European Journal of OperationalResearch vol 19 no 1 pp 68ndash75 1985

[14] M Gendreau G Laporte and J Y Potvin ldquoHeuristics for theclustered traveling salesman problemrdquo Tech Rep CRT-94-54Centre de Recherche sur les Transports Universite deMontrealMontreal Canada 1994

[15] J-Y Potvin and F Guertin ldquoA genetic algorithm for theclustered traveling salesman problem with an a priori order onthe clustersrdquo Tech Rep CRT-95-06 Centre de recherchesur lestransports Universite de Montreal Montreal Canada 1995

[16] J-Y Potvin and F Guertin ldquoThe clustered traveling salesmanproblem a genetic approachrdquo in Meta-Heuristics Theory ampApplications I H Osman and J Kelly Eds pp 619ndash631 KluwerAcademic Norwell Mass USA 1996

[17] S Anily J Bramel andAHertz ldquo53-Approximation algorithmfor the clustered traveling salesman tour and path problemsrdquoOperations Research Letters vol 24 no 1 pp 29ndash35 1999

[18] N Christofides ldquoWorst-case analysis of a new heuristic for thetraveling salesmanproblemrdquoTech Rep 388Graduate School ofIndustrial Administration Carnegie-Mellon University Pitts-burgh Pa USA 1976

[19] W Sheng N Xi M Song and Y Chen ldquoRobot path planningfor dimensional measurement in automotive manufacturingrdquoJournal of Manufacturing Science and Engineering Transactionsof the ASME vol 127 no 2 pp 420ndash428 2005

[20] C Ding Y Cheng andM He ldquoTwo-level genetic algorithm forclustered traveling salesman problem with application in large-scale TSPsrdquo Tsinghua Science and Technology vol 12 no 4 pp459ndash465 2007

[21] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley New York NY USA 1989

[22] Z H Ahmed ldquoA hybrid genetic algorithm for the bottlenecktraveling salesman problemrdquo ACM Transactions on EmbeddedComputing Systems vol 12 no 1 article 9 2013

[23] K Deb Optimization for Engineering Design Algorithms andExamples Prentice Hall India New Delhi India 1995

[24] Z H Ahmed ldquoMulti-parent extension of sequential construc-tive crossover for the travelling salesman problemrdquo Interna-tional Journal of Operational Research vol 11 no 3 pp 331ndash3422011

[25] Z H Ahmed ldquoAn experimental study of a hybrid geneticalgorithm for the maximum travelling salesman problemrdquoMathematical Sciences vol 7 no 1 pp 1ndash7 2013

[26] C-X Wang D-W Cui Z-R Wang and D Chen ldquoA novel antcolony system based on minimum 1-tree and hybrid mutationfor TSPrdquo in Proceedings of the 1st International Conferenceon Natural Computation (ICNC rsquo05) LNCS pp 1269ndash1278Springer Changsha China August 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 13: Research Article The Ordered Clustered Travelling Salesman Problem: A Hybrid Genetic ...downloads.hindawi.com/journals/tswj/2014/258207.pdf · 2019-07-31 · Research Article The

The Scientific World Journal 13

[13] K Jongens and T Volgenant ldquoThe symmetric clustered trav-eling salesman problemrdquo European Journal of OperationalResearch vol 19 no 1 pp 68ndash75 1985

[14] M Gendreau G Laporte and J Y Potvin ldquoHeuristics for theclustered traveling salesman problemrdquo Tech Rep CRT-94-54Centre de Recherche sur les Transports Universite deMontrealMontreal Canada 1994

[15] J-Y Potvin and F Guertin ldquoA genetic algorithm for theclustered traveling salesman problem with an a priori order onthe clustersrdquo Tech Rep CRT-95-06 Centre de recherchesur lestransports Universite de Montreal Montreal Canada 1995

[16] J-Y Potvin and F Guertin ldquoThe clustered traveling salesmanproblem a genetic approachrdquo in Meta-Heuristics Theory ampApplications I H Osman and J Kelly Eds pp 619ndash631 KluwerAcademic Norwell Mass USA 1996

[17] S Anily J Bramel andAHertz ldquo53-Approximation algorithmfor the clustered traveling salesman tour and path problemsrdquoOperations Research Letters vol 24 no 1 pp 29ndash35 1999

[18] N Christofides ldquoWorst-case analysis of a new heuristic for thetraveling salesmanproblemrdquoTech Rep 388Graduate School ofIndustrial Administration Carnegie-Mellon University Pitts-burgh Pa USA 1976

[19] W Sheng N Xi M Song and Y Chen ldquoRobot path planningfor dimensional measurement in automotive manufacturingrdquoJournal of Manufacturing Science and Engineering Transactionsof the ASME vol 127 no 2 pp 420ndash428 2005

[20] C Ding Y Cheng andM He ldquoTwo-level genetic algorithm forclustered traveling salesman problem with application in large-scale TSPsrdquo Tsinghua Science and Technology vol 12 no 4 pp459ndash465 2007

[21] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley New York NY USA 1989

[22] Z H Ahmed ldquoA hybrid genetic algorithm for the bottlenecktraveling salesman problemrdquo ACM Transactions on EmbeddedComputing Systems vol 12 no 1 article 9 2013

[23] K Deb Optimization for Engineering Design Algorithms andExamples Prentice Hall India New Delhi India 1995

[24] Z H Ahmed ldquoMulti-parent extension of sequential construc-tive crossover for the travelling salesman problemrdquo Interna-tional Journal of Operational Research vol 11 no 3 pp 331ndash3422011

[25] Z H Ahmed ldquoAn experimental study of a hybrid geneticalgorithm for the maximum travelling salesman problemrdquoMathematical Sciences vol 7 no 1 pp 1ndash7 2013

[26] C-X Wang D-W Cui Z-R Wang and D Chen ldquoA novel antcolony system based on minimum 1-tree and hybrid mutationfor TSPrdquo in Proceedings of the 1st International Conferenceon Natural Computation (ICNC rsquo05) LNCS pp 1269ndash1278Springer Changsha China August 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 14: Research Article The Ordered Clustered Travelling Salesman Problem: A Hybrid Genetic ...downloads.hindawi.com/journals/tswj/2014/258207.pdf · 2019-07-31 · Research Article The

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of