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Please cite this article in press as: P. Victer Paul, et al., A new population seeding technique for permutation-coded Genetic Algorithm:Service transfer approach, J. Comput. Sci. (2013), http://dx.doi.org/10.1016/j.jocs.2013.05.009

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A new population seeding technique for permutation-coded GeneticAlgorithm: Service transfer approach

P. Victer Paul a,, A. Ramalingam b , R. Baskaran c , P. Dhavachelvan a ,K. Vivekanandan d , R. Subramanian aa Department of Computer Science, Pondicherry University, Puducherry, Indiab Department of MCA, Sri Manakula Vinayagar Engineering College, Puducherry, Indiac Department of Computer Science and Engineering, Anna University, Chennai, Indiad Department of Computer Science and Engineering, Pondicherry Engineering College, Puducherry, India

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Article history:Received 31 December 2012Received in revised form 16 May 2013Accepted 26 May 2013Available online xxx

Keywords:Genetic AlgorithmPopulation seeding techniqueTraveling Salesman ProblemOrder distance vectorCombinatorial problem

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Genetic Algorithm (GA) is a popular heuristic method for dealing complex problems with very large searchspace. Among various phases of GA, the initial phase of population seeding plays an important role indeciding the span of GA to achieve the best t w.r.t. the time. In other words, the quality of individualsolutions generated in the initial population phase plays a critical role in determining the quality of naloptimal solution. The traditional GA with random population seeding technique is quite simple and of course efcient to some extent; however, the population may contain poor quality individuals whichtake long time to converge with optimal solution. On the other hand, the hybrid population seedingtechniques which have the benet of good quality individuals and fast convergence lacks in terms of randomness, individual diversity and ability to converge with global optimal solution. This motivates todesign a population seeding technique with multifaceted features of randomness, individual diversityand good quality. In this paper, an efcient Ordered Distance Vector (ODV) based population seedingtechnique has been proposed for permutation-coded GA using an elitist service transfer approach. Oneof the famous combinatorial hard problems of Traveling Salesman Problem (TSP) is being chosen as the

testbed and the experiments are performed on different sized benchmark TSP instances obtained fromstandard TSPLIB [54] . The experimental results advocate that the proposed technique outperforms theexisting popular initialization methods in terms of convergence rate, error rate and convergence time.

2013 Elsevier B.V. All rights reserved.

1. Introduction

Genetic Algorithm(GA) is a well-known method forglobal opti-mization of complex problem very large search space based onthe survival of the ttest concept of natural evolution [55] . Thesignicant features of the GA, which makes it perform compe-tently could be dened as follows: GA operates on a populationof feasible solutions rather than on a single solution; the vari-ety of genetic operators helps to explore unrevealed solutions inthe large search space effectively; possibility to construct prob-lem the specic genetic operators which can offer better solutionsearch; population diversity helps to avoid the drawback of gettingtrapped in local optima and premature convergence. These exible

Corresponding author. Tel.: +91 0413 2274430.E-mail addresses: [email protected] , [email protected]

(P. Victer Paul), [email protected] (A. Ramalingam),[email protected] (R. Baskaran), [email protected] (P. Dhavachelvan),[email protected] (K. Vivekanandan), [email protected] (R. Subramanian).

congurations encourage researchers to design novel GA with amodied operators and population seeding techniques to improvefurther its performance.

GA had been proven to be efcient at searching optimal solu-tion among a large and complex search space in an adaptable way,controlled by the equivalent biological evolutionary mechanismsof reproduction, crossover, and mutation. Various phases of GAcan be dened as population seeding (initial population), selection,reproduction, crossover, mutation and termination constraint, inwhich rst step occurs once and the rest of the steps are repeateduntil the nal condition is satised [16,52] . The rst step of anyGA is to generate a set of possible solutions randomly as an initialpopulation or population seeding [34,37,41] . The quality of indi-vidual solutions in the initial population plays a critical role indetermining the quality of the nal solution that can be obtainedusing GA [32,40] . However, in traditional GA, population seedingis performed randomly which can be simple but, the whole popu-lation contains much of individuals with worst quality, infeasiblesolutions sometimes [36] . As a consequence, GA with a randompopulation seeding technique requires longer search time to nd

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an optimal solution,number of generationsrequiredto evolve opti-mal solution increases,the search possibility foran optimalsolutiondecreases and more importantly the convergence rate or quality of theoptimal solution obtained is reduced. Thus, therequirementfora modied population seeding technique in GA is clear and in fact,several research works were proposed to support the dispute [61] .

Lawrence and Amini [61] d iscussed about different GA cong-uration issues and claims that seeding the initial population withheuristics can improve the efciency of the GA greatly. Togan andDaloglu [40] believes that performance and convergence ability of GA are critically inuenced by the population seeding method andproposed twonew self-adaptive membergrouping strategies andanew strategy for population seeding. The large search space is col-lected into different groups and the list of cross section values areassigned as initial values to set the initial populationautomatically.This method is dedicatedly proposed for application in the area of structural engineering and tested in truss structures and transmis-sion towers. Nearest Neighbor (NN) tour construction heuristic isone of the familiar alternatives for random population seeding inGA, particularly for TSP [3,16,4246] . In NN technique, individualsin the population seeding are constructed with the city nearest tothe current city and such good individuals can rene the subse-quent search in the next generations [3] . Though NN works ne,it suffers with some critical factors: several cities are not includedin the individuals created initially and have to be inserted at highcosts in the end; neglecting several cities at the population seedingstage leads to severe errors in optimal solution construction andthe diversity among the individuals created is very minimum.

Yingziet al. [32] proposed a GreedyGA (GGA),in which thepop-ulation seeding is performed using Gene Bank (GB). The GB is builtby assembling the permutation of n cities based on their distance.In GGAmethod, thepopulation of individuals is generated from theGB such that the individuals are of above-average tness and shortdening length. In GGA, with the increase in number of cities leadsto augmented problem complexity and performance degradation,and large collection of GB individuals enlarges the cost of com-putation at each generation. The improved performance of GGA is justied using TSP with maximum of hundred cities and its per-formance deteriorates with large number of cities. In [38] , Fuyanet al. proposed K-means algorithm, based on the work reported in[46] , which is considered to generate much infeasible solutions, toobtain the initial population in which N number of individuals arepartitioned and assigned to one of K clusters. As a result, usingK-means algorithm, generating infeasible solutions in the popula-tion seeding stage is avoided. Performance evaluation is performedwith a maximum of 10 cities with only time based analysis andbe decient to validate the proposed technique for large number of cities and convergence capability. Yugayet al. [36] proposes a mod-ied GA with sorted initial population method based on theory of better parents produce better offsprings. In this approach, a largeinitial pool of population is generated and ranked in accordance

to their tness values and at last, a certain number of individ-uals with bad tness are omitted. This approach also suffers withthe issues discussed with NN tour construction heuristic techniquesuch as premature convergence, reduced search space explorationand minimum population diversity. Hence, the traditional GA doesnot provide effective performance when applied to some of thecombinatorial problems like TSP [33] , so each stage in the tradi-tional GAhas been modied in order to achieve a better outputandthus resulted in hybrid GAs [2832,34,49,50] .

Recently, many researchers proposed modied versions of GA,particularly for solving TSP using randompopulation seeding tech-niques [3,16,3032,34,3941,48,6466] . Though several modiedpopulation seeding techniques for GA have been proposed, manyresearchers still continue to work with a random population seed-

ing technique because of the complexitynatureof algorithm, which

is difcult to understand and implement, problem specic mod-ications are required to apply and problem such as prematureconvergence, ineffective search space exploration and less popu-lation diversity. This implies that the researchers are interestedin random population seeding technique to accomplish a bettersearch space exploration and nding best optimal solution at thecost of high convergence time. In [64] , Xing et al.proposed a hybridapproach combining an improvedGenetic Algorithmand optimiza-tion strategies using random population seeding technique. Anefcient hybrid mutation Genetic Algorithm has been proposedusing a random population seeding technique in [34] . Chang andRamakrishna [39] proposed a GA forshortestpath routing problem,in which author reveals the reason for preferring random popu-lation seeding method than heuristic initialization. Although themean tness of the individuals generated using heuristic initial-ization are high so that it may help the GA to obtain the bettersolutionsfaster,but itends upin exploringa small part of thesearchspace and never nd the global optimal solutions because of thelack of diversity in the initial population generated [47] . Insome of the works, the authors used hybrid population seeding techniquewhich combines random and any of the modied population seed-ing technique [33,61] . In [62] , Qu and Sun proposed a synergeticapproach to GA by adding some new randomly generated individ-uals into the population after each generation in order to preventpremature convergence and to obtain nal optimum.

To summarize, the modied population seeding techniques hasthe advantages of good quality or generating potential sequenceindividuals at an early stage and the ability to nd near optimalsolutions at few generations; however they lacks in randomness,diversity of individuals generated, ability to explore more searchspace and nding the global solution. On the other hand, the ran-dompopulation seeding techniquehas the advantages of individualdiversity, can explore search space efciently and nding optimalsolution; however they have the disadvantages of individuals withworst potential sequence and requires longer search time to con-verge an optimal solution. The controversies between these twocategories of population seeding techniques motivate to proposean efcient population seeding technique with characteristics of randomness, individual diversity and potential sequence. Thus inthis paper, an efcient Ordered Distance Vector (ODV) based pop-ulation seeding technique with three different varieties has beenproposed for permutation-coded GA. The popular combinatorialoptimization problem of Traveling Salesman Problem (TSP) is beingchosen as the testbed to validate and claim the efcacy of the pro-posedpopulationseeding technique. Experiments were performedover the different sized benchmark TSP instances obtained formthe TSPLIB [54] . The organization of the paper is as follows: Sec-tion 2 offers sufcient background information over GA and TSP toimprove the understanding of this paper. Section 3 describes theproposed technique and its variants along with the correspondingalgorithms. The different phases of the experiments are summa-

rized in Section 4. This section also reports experimental resultswith corresponding analyses. And nally, Section 5 presents theconclusive remarks of the work reported in this paper.

2. Background information

As stated earlier, this section offers a brief introduction onGenetic Algorithm (GA) and Traveling Salesman Problem (TSP) toimprove the understandability of this paper.

2.1. Genetic Algorithm

Genetic Algorithms (GAs), a subclass of evolutionary algorithms,

is a stochastic optimization technique based on the principles
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Select Individual Representation

Design Fitness Function, Population Size andGeneration Limit

Generate Initial Population (Gen=0)

Begin

IfGen>Limi

If Best indiv isFittest

Individual

Select Individual(s)

Perform Reproduction

Perform Crossover

Perform Mutation

New Population (Gen=Gen+1) withPopulation Size

Find Best individual in Population (Best indiv)Stop

Fig. 1. The generic ow of Genetic Algorithm.

and mechanisms of natural selection. In 1975, John Hollandintroduced the concept of GA in his book Adaptation in Natu-ral and Articial Systems [55] . GA is a panacea for resolving alloptimization problems and typically used to solve complex opti-mization problems like aircraft design, robot trajectory planning,gas pipelining control, traveling salesman, routing, game playing, job shop scheduling, etc. [52] . GA has two important features: it isa stochastic algorithm where selection and reproduction are per-formed randomly; it always works with a population of solution,which offers the benet of assortment and robustness to the tech-nique. The need of GAis to nd the best solution in the large searchspace, which is the collection of all feasible solutions. An individual(or) chromosome is a representation of possible and legal solutionto a problem. A population is a collection of individuals that can behandled by GA at a specic point of time. Fitness of an individualis the value associated with the corresponding individual whichdepends on its representation. The tness function, which is prob-lem specic, corresponds to evaluation of quality of individual tocheck for solution optimality for the problem in hand. The termi-nation criteria for GA can be nding the best solution or reachingmaximum generation limit.

Fig. 1 illustrates a generic ow of Genetic Algorithm. GA startsby selecting suitable format for individual representation.Followedby designing tness function, population size and generation limit,population size and generation limit are set as necessary. An initialpopulation (Gen= 0) of predened size is generated by random orheuristics fashion. Then, theGA loops through the iterationprocesstill any of the termination criteria satised. Each iteration processconsists of the following steps:

Check whether number of generation exceeds the generationlimit (Gen> limit). If so, exit with best solution of previous gen-

eration, otherwise continue.

Evaluate the tness value of each individual based on a tnessfunction.

Check whether any of the individual satises the condition forbestor optimal solution(Best indiv ). Ifso, exit with thecorrespond-ing solution, otherwise continue.

Select the better individual(s) from the population to applygenetic operators, namely reproduction, crossover and muta-tion, to produce next generation. These genetic operators canbe applied either individually or combination of any two. Vari-ous techniques used for selection are random [38] , tournamentselection [16,52] and the roulette wheel selection [35,36] .

Reproduction is copying the best (elite) solutions of the previ-ous population to the next. The elite preservation [52] strategyensures the best solution of the population survives into the nextgeneration.

Crossover is the probabilistic process of the creating newindividuals by exchanging information of two selected parentindividuals to produce a better t individual.

Mutation is the probabilistic process to introduce constructivechanges in the information stored in the individual.

This new population generated has to replace the previous pop-ulation (Gen= Gen + 1) and continue with the iterations.

Thus, at the end of GA, the optimal or near optimal solution canbe obtained for the problem considered.

2.2. Traveling Salesman Problem

The Traveling Salesman Problem (TSP) is a well-known NP-hardproblem exceedingly studied in theeld of operations research andcomputer science [13] . TSP is commonly considered as a standardtestbed for various combinatorial optimization techniques. In TSP,a salesman wants to visit each of a set of cities exactly once andreturn to the starting city with minimal distance traveled. Thusthe objective of TSP is to nd a minimum total cost closed tourthat visits each city exactly once for a given number of cities andthe distance (or the cost) of traveling between any two cities. Theproblem can be formalized as follows.

Let G = (C n , E n ) be the complete undirected graph suchthat C {c 1 , c 2 , c 3 , . . . , c n }and E {(c 1 , c 2 ), (c 1 , c 3 ), . . . , (c 1 , c n ), (c 2 ,c 3 ), (c 2 , c 4 ), . . . , (c 2 , c n ), . . . , (c (n 1) , c n )}. In the graph G, C and E corresponds to the cities and path between the cities respectively.

For each pair, ( c i, c j) and i /= j, Distance between the city c i andc j can be given as d(c i , c j) TSP aims to nd the minimum distancetour (optimal solution) between cities which can be given as,

Optimal solution = minn

i= 1

d(c i , c (i+ 1) ), 1 (n + 1) (1)

The search space for the TSP is a set of permutation of n cities.Any permutation of n cities gives a possible solution and the size of the TSP search space with n cities is (( n 1)!/2). Let be the set of permutation of cities (search space),

= { 1 , 2 , . . . , n ! } (2)

and

1 ...n ! = { c i, c (i+ 1) , c (i+ 2) , . . . , c n } (3)

TSP has to search for the optimal (best) solution which existsin the permutation set and can be given as,

n

i= 1

d(c i, c (i+ 1) ) n

i= 1

d(c i , c (i+ 1) ) (4)

Such that, 1 (n + 1) and { }


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Thus, be therequired best solution forthe considered TSPwithn number of cities.

By considering the importance and extended applications of TSP in various elds, several meta-heuristicsearch techniqueshavebeen proposed to solve it, such as Tabu Search [9,10] , Genetic Algo-rithm [1117] , Ant Colony Optimization [1822] , Particle swarmOptimization [23,29] , Neural Networks [24,25] , Simulated Anneal-ing [26] , Multi-Agent System [27,35,41,63] and Hybrid-Heuristics[2830] . Major applications of TSP are Vehicle routing [4] , Drillingof printed circuit boards [5] , Overhauling gas turbine engines [6] ,X-ray crystallography [7] , computer wiring [4] , sequencing andscheduling [47,52] and the order-picking problem in warehouses[8] . In this paper, TSP is being used as a testbed for demonstratingthe effectiveness of the proposed technique. On the other hand, itcould also be considered that the proposed technique is a bettersolution for solving TSP.

3. Proposed system

3.1. Problem statement

As describedin Section 1, the controversies between therandom

and heuristic population seeding techniques motivated to proposean efcient population seeding technique with the characteristicsof randomness, individual diversity and potential sequence. Theproposed Ordered Distance Vector (ODV) based Population Seed-ing, which depends on the ODV matrix, is different from those inthe literature in terms of its simplicity, fastness and global optimalsolution convergence. The features such as randomness, potentialsequence anddiversity are considered as critical factors to improvethe effectiveness of the proposed ODV population seeding tech-nique.

3.2. Ordered Distance Vector matrix

For any TSP, ODV for any city c i can be formed by the permuta-

tion of ( n 1) cities, which are arranged in the increasing order of distance from the city c i. ODV for the city c i can be given as,

ODV (c i) = (c j, c j+ 1 , c j+ 2 , . . . , c n 1 ) (5)

such that,

d(c i, c j) d(c i , c j+ 1 ) d(c i, c j+ 2 ) . . . d(c i , c n 1 ) (6)

where, d(c i, c j) refers to the distance between the cities c i and c j.An ODV Matrix is a ( n (n 1)) matrix, which contains OD for

each city and arranged in the sequence of its distance matrix D. Ingeneral ODM can be represented as,

ODM =

ODV c 1

ODV c 2ODV c 3

. . .ODV c n

=

C 1( j) C 1( j+ 1) C 1( j+ 2) C 1( n 1)

C 2( j) C 2( j+ 1) C 2( j+ 2) C 2( n 1)C 3( j) C 3( j+ 1) C 3( j+ 2) C 3( n 1)

. . . . . . . . . . . .C n( j) C n ( j + 1) C n( j+ 2) C n(n 1)

(7)

Rank of distance of city c x with c y denoted as R(c x, c y) refers tothe position of the city c x in the row corresponding to the city c y inthe ODM.

R(d(c 1( j+ 2) , c 1 )) = 3 (8)

R(d(c 2( n 1) , c 2 )) = n 1 (9)

The rank of distance of the city is used to choose the city atspecic position in the ODM matrix for solution generation.

3.3. ODV based population seeding techniques

Ordered Distance Vector based population seeding techniquegenerates a set of permutation of n cities using the Ordered Dis-tance Vectormatrix.In each permutation, thesequence of thecitiesis chosen such that the sum of distances between the cities is nearminimum. The initial population generated using ODV populationseeding technique P ODM can be represented as,

P ODM =

1 (c i) 1 (c (i+ 1) ) 1 (c (i+ 2) ) . . . 1 (c n )

2 (c i) 2 (c (i+ 1) ) 2 (c (i+ 2) ) . . . 2 (c n )

3 (c i) 3 (c (i+ 1) ) 3 (c (i+ 2) ) . . . 3 (c n )

. . . . . . . . . . . .

o(c i) o(c (i+ 1) ) o(c (i+ 2) ) o(c n )

(10)

where k is the kth permutation of n cities, k = (1, 2, 3, . . . , o). ois the total number of individuals generated in the population.

There are two signicant characteristics that makes the ODVpopulation seeding technique distinct from the others andthey areas follows:

Potential sequence : This factorhelps tokeepthe good quality orderof cities in each individual.

Individual diversity : This factor refers to the difference in the per-mutation of cities in each individual generated. It helps to avoidthe drawback of getting trapped in local optima and prematureconvergence.

Potential sequence : For GA, the capability to nd the optimalsolution is critically affected by the tness (potential sequence) of individualsgenerated in thepopulationinitialization [9] . This factorcan be applied in ways as follows:

Equi-begin (Eb) : It is claimed that forcing individual to alwaysbegin thepermutationof city at a specic city c x willallowthe GAto take thebenet of good building block or high tnesssequenceof cities in generating new individuals [7,10] .

Vari-begin (Vb) : This method is used in many existing techniquesin which the beginning of the permutation of the city is not nec-essarily the same city. Creating the individuals beginning withdifferent cities provide the possibility of creating new individ-uals, the size of the initial population can be more, repetition of individual sequence can be avoided and mainly it helps to avoidthe algorithm to get trapped in premature convergence.

Individual diversity : Individual diversity refers to the variationin the permutation of cities in each individual in the population.The initial population of the individual must offer a wide diver-sity because preserving diversity in the population, particularlyat the initial stages of GA, represents a condition for avoiding thesearch from getting stuck in local optima and also from prematureconvergence.

Best adjacent (ba) number : A detailed study on existing work[5658] helps to derive an assumption that, in an optimal solution,anycity c i is connectedto city c j such that c j isoneof the c is nearestba number of cities. The value of ba (an integer) can be expressedin range and it highly depends on the size of the population n. Therange of ba value can be assigned as follows:

If 1 n 10, then 2 ba 3. If 11 n 100, then 2 ba 4. If 101 n 1000, then 2 ba 5.

If n 1001, then 2 ba 6.


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Fig. 2. Types of ODV based population seeding techniques.

The above said ranges are not exact and are derived as theexperimental outcomes of the literature study. It shows that if thenumber of cities n is between 101 n 1000, then in the optimalsolution, every city would be connected to any of its ba 2 ba 5cities which are nearest to it.

In ODV population seeding technique, the value of ba num-ber plays a vital role in providing individual diversity. The adjacentcities in each individual of the initial population depends on theba value. An integer value generated between the ranges of ba

decides which cities are kept as adjacent in each individual. Thereare two different ways of initializing population based on diversityin ba number generation,

Equal ba diversity (Ed) : An integer value bax is generatedbetween the ranges of ba before creating each individual, forevery city, the cityat bax position is added as adjacent. The rankof distance between each city in the same individual is same. If itis already added then the city at next position is added and thesame is repeated until the complete individual is created. In thismethod, bax value is generated once and the diversity (rank of distance) between each city is equal for each individual in thepopulation.

Variable ba diversity (Vd) : An integer value bax is generatedbetween the ranges of ba before adding each city in the individ-ual, for every city, the city at currently generated bax position isadded as adjacent city. Therank of distance betweeneach city in thesameindividual is different . Inthismethod, bax value is generatedfor n 1 times and the diversity between each city is differentfor each individual.

Based on these two characteristic factors, there are threeeffective ways for generating initial population in ODV seeding

techniqueas shown in Fig. 2 and the comparison among those vari-ants are given in Table 1 . These variants are explained as follows:

3.3.1. Type 1: Equi-begin with Variable diversity (EV)Inthis type, a new bax number is generatedbefore adding each

city into every individual and each individual starts with the samecity. The individuals created in this type have high potential per-mutation of cities and thus the convergence time can be reduced.The maximum number of individuals in the initial population thatis being created using this type can be given as:

Max(tot (P ODM )) = ba (n 1) (11)

where tot (P ODM ) refers to the total number individuals in the pop-ulation P ODM and ba refers to the Best Adjacent number assignedand n refers to the total number of cities.

The P ODM generated using this type can be represented as:

P ODM =

1 (1) 1 (c 2 ) 1 (c 3 ) . . . 1 (c n )

2 (1) 2 (c 2 ) 2 (c 3 ) . . . 2 (c n )

3 (1) 3 (c 2 ) 3 (c 3 ) . . . 3 (c n )

. . . . . . . . . . . .

o(1) o(c 2 ) o(c 3 ) o(c n )

(12)

where the rst city remains same for each individual (i.e.) 1 (1) 2 (1) 3 (1) , . . . , o(1).

This type of population should be handled with suitablecrossover operatorsbecause it maybe suffered by pre-convergenceproblem due to lack of much diversity.

3.3.2. Type 2: Vari-begin with Equal diversity (VE)Inthis type, a new bax number is generated for each individual

created and the same bax number is used to add every the cities in

individual. In the population, there is exactly ba

number of indi-viduals that starts with same initial city. The maximum numberof individuals in the initial population that is be created using thistype can be given as,

Max(tot (P ODM )) = n ba (13)

where tot (P ODM ) refers to the total number individual in the pop-ulation P ODM and ba refers to the Best Adjacent number assignedand n refers to the total number of cities.

Table 1Comparison between the variants of ODV population seeding technique.

Population seeding techniques

Type 1 (EV) Type 2 (VE) Type 3 (VV)

CharacteristicsInitial city of individuals Same Different DifferentIndividual diversity Random bax value

Huge diversity amongindividuals

Same bax value Little diversity amongindividuals

Random bax value Huge diversity amongindividuals

Rank of distance of cities Different Equal DifferentNo. of times bax generated ( n 1) o n ba n oMax. possible initial population ba (n 1) n ba ba n

Advantages High potential sequence of cities Limited size of population

Limited size of population High potential sequence of cities No cyclic repetition problem No pre-convergence problem

Disadvantages Suitable crossover operatorshave to used to avoidpre-convergence problem

Suffer cyclic repetitionproblem Stuck in local optimalsolution

Very large size of population


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Fig. 3. List of variables used in thealgorithm forODV based population seeding technique.

The P ODM generated using this type can be represented as:

P ODM =

1 (1) 1 (c 2 ) 1 (c 3 ) 1 (c n )

2 (1) 2 (c 2 ) 2 (c 3 ) 2 (c n )

......

......

ba (1) ba (c 2 ) ba (c 3 ) ba (c n )

1(2)

2(c

2)

3(c

3)

3(c

n)

......

......

ba (2) ba (c 2 ) ba (c 3 ) ba (c n )

1 (n) 2 (c 2 ) 2 (c 3 ) 2 (c n )

......

......

ba (n) ba (c 2 ) ba (c 3 ) ba (c n )

(14)

where The rst city remains same for ba number of indi-viduals (i.e.) 1 (1) 2 (1) . . . ba (1) , 1 (2) 2 (2) . . . ba (2) and so on and R( 1 (c 2 ), 2 (1)) R( 1 (c 3 ), 1 (c 2 )) . . . R ( 1 (c n ), 1 (c n 1 ))

This type of population seeding is not so efcient because theindividuals created using this technique suffer from cyclic rep-etition problem and only ba number of individuals are unique.The cyclic repetition problem could be referred as the sequence of the adjacent cities remains identical for more than one individualthough its initial and nal cities are different. The crossover andmutation rate has to be set high in order to overcome the cyclicrepetition problem.

3.3.3. Type 3: Vari-begin with Variable diversity (VV)In this type, a new bax numberis generated beforeadding each

city into every individual and each individual starts with the ran-dom city. The individuals created in this type have high potentialpermutation of cities and also have good individual diversity. This

type of population seeding is most effective and can produce best

solution with minimum convergence time. The maximum numberof individuals in the initial population that is be created using thistype can be given as,

Max(tot (P ODM )) = ba n (15)

where tot (P ODM ) refers to the total number individual in the pop-ulation P ODM and ba refers to the Best Adjacent number assignedand n refers to the total number of cities.

The P ODM

generated using this type can be represented as,

P ODM =

1 (c 1 ) 1 (c 2 ) 1 (c 3 ) . . . 1 (c n )2 (c 1 ) 2 (c 2 ) 2 (c 3 ) . . . 2 (c n )3 (c 1 ) 3 (c 2 ) 3 (c 3 ) . . . 3 (c n ). . . . . . . . . . . .o(c 1 ) o(c 2 ) o(c 3 ) o(c n )

This type of population seeding technique is the most recom-mended for effective search for best solution. The main difcultyconcerned with this technique is to determine the limit of popula-tion size o, to support small population size techniques, becausegood initial population individuals which may occur outside thelimit are left unrevealed.

As the last, the fourth possible type population seeding tech-

nique,Equi-begin withEqual diversity(EE), whichis notconsideredas an effective one since it can produce the initial population withonly n individuals. It is very hard to nd the optimal solution withless initial population containing un-potential sequence of cities,which leads to pre-mature convergence and large number of gen-erations.

3.4. Algorithm for ODV population seeding technique

The algorithm for ODV population seeding technique consistsof two stages. First an Order Distance Matrix (ODM) will be cre-ated from the Distance Matrix (DM) and followed by generatingthe initial population based on the ODM. The second stage of thetechnique can be performed in three ways; EV, VE and VV depends

on the type of population seeding technique to be chosen. List of


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Fig. 4. Algorithm for ODV based population seeding technique and its variants.

variables used in the proposed algorithm is shown in Fig. 3 . Theproposed ODV population seeding algorithm, as shown in Fig. 4 ,consists of ve procedures namely Generate INIT POP(), Gener-ate ODM(DM), EV(ODM), VE(ODM), and VV(ODM). The algorithmbegins with the procedure Generate INIT POP() and returns the

initial population. This procedure gets n and DM as inputs andinvoke other procedures conditionally. The functionalities of otherprocedures are explained as follows:

Generate ODM (DM) : This procedure creates ODM from theDM, which is passed as an argument to the procedure. In this


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Table 2Experimental results of random population initialization method.

S.no. Instance Optimaltness

Time taken(s)

Besttness Worsttness

Averagetness

Error rate Convergencerate

Convergencediversity (%)

Averageconvergence(%)

Best indiv.(%)

Worstindiv. (%)

Best indiv.(%)

Worstindiv. (%)

1 Eil51 426 0.51 803.56 1176.34 931.38 88.63 176.14 11.37 76.14 87.51 18.632 Pr76 108,159 0.61 195,678.73 247,890.85 21,7356.19 80.92 129.19 19.08 29.19 48.27 0.963 KroA100 21,282 1.25 39,789.3 58,475.04 44,782.09 86.96 174.76 13.04 74.76 87.80 10.424 Pr144 58,537 1.35 111,987.32 154,899.24 124,689.52 91.31 164.62 8.69 64.62 73.31 13.015 Gil262 2378 3.51 4469.94 6402.06 5173.77 87.97 169.22 12.03 69.22 81.25 17.576 Fl417 11,861 7.12 22,646.99 33,757.7 27,848.63 90.94 184.61 9.06 84.61 93.67 34.797 Pr1002 259,045 15.04 511,527 755,689.63 627,364.63 97.47 191.72 2.53 91.72 94.25 42.18

procedure, each row from the DM is retrieved and sorted, thechanges in each of DM is tracked with the corresponding changesin the sequence of cities in the row. The sequence of citieswith corresponding changes, neglecting the rst element, whichwill be always zero (symmetric TSP), is assigned to a row inODM. The sorting technique followed in the procedure is BubbleSort.

EV (ODM): This procedure will be invoked if the type of popula-tion seeding is Type 1, which takes ODMas argument and returnsthe initial population (Pop) generated using equi-begin with thevariable diversity (EV) method. This procedure gets initial city andba value either from the user or may be a random value chosenby the system ( ba range will be chosen based on problem size).Each individual is generated using EV method until the number of individuals in generating population reaches the insisted Pop Sizevalue.

VE (ODM): This procedure will be invoked if the type of popula-tion seeding is Type 2, which takes ODMas argument and returnsthe initial population generated using Vari-begin with Equal diver-sity (VE) method. For each individual, the initial city has beenselected randomly and the value of ba is chosen by the systembased on the problem size. In this procedure, each individual is

generated using VE method and returns the generated population(Pop) until the Pop Size limit.

VV (ODM): This procedure will be invoked if the type of popula-tion seeding is Type 3, which takes ODM as argument and returnsthe initial population (Pop) generated using Vari-begin with Varidiversity (VV) method. In this procedure, the value of the initialcity and the ba value have been chosen randomly for the additionof each individual and city in the individual respectively. Once thepopulation size reaches the Pop Size limit,the procedure stops andreturns the generated population.

At theend of thealgorithm, an initial population of size Pop Sizehas been generated using one of these three population seedingmethods and stored in the variable POP.

4. Experimentation and result analysis

4.1. Experimentation setup essentials

4.1.1. Individual representationThe signicance of the proposed technique is demonstrated

by assessing their performance w.r.t. the problem space of Traveling Salesman Problem (TSP). Holland [55] classied the

Table 3Experimental results of Nearest Neighbor population initialization method.


Time taken(s)


Averagetness




Best indiv.(%)

Worstindiv. (%)

Best indiv.(%)

Worstindiv. (%)

1 Eil51 426 1.18 495.77 688.01 572.65 16.38 61.50 83.62 38.50 45.13 65.582 Pr76 108,159 1.26 121,567.43 186,540.75 169,746.4 12.40 72.47 87.60 27.53 60.07 43.063 KroA100 21,282 1.96 24,421.8 32,689.65 29,765.76 14.75 53.60 85.25 46.40 38.85 60.144 Pr144 58,537 2.01 62,952.97 82,577.56 72,458.43 7.54 41.07 92.46 58.93 33.53 76.225 Gil262 2378 5.62 2622.75 3532.06 2734.75 10.29 48.53 89.71 51.47 38.24 85.006 Fl417 11,861 12.46 14,123.99 18,554.38 17,754.67 19.08 56.43 80.92 43.57 37.35 50.317 Pr1002 259,045 20.12 312,135.76 409,843.75 379,744.72 20.49 58.21 79.51 41.79 37.72 53.41

Table 4Experimental results of Gene Bank population Initialization method.


Time taken(s)


Averagetness




Best indiv.(%)

Worstindiv. (%)

Best indiv.(%)

Worstindiv. (%)

1 Eil51 426 1.14 505.65 711.21 593.91 18.70 66.95 81.30 33.05 48.25 60.582 Pr76 108,159 1.03 130,237.83 161,930.85 159,436.19 20.41 49.72 79.59 50.28 29.30 52.593 KroA100 21,282 1.96 25,059.4 35,973.04 29,654.83 17.75 69.03 82.25 30.97 51.28 60.664 Pr144 58,537 2.18 69,189.73 90,528.27 74,363.72 18.20 54.65 81.80 45.35 36.45 72.965 Gil262 2378 4.56 2937.24 3824.45 3138.27 23.52 60.83 76.48 39.17 37.31 68.036 Fl417 11,861 9.96 15,190.99 19,913.1 18,346.11 28.08 67.89 71.92 32.11 39.81 45.327 Pr1002 259,045 16.62 340,468.3 431,877.73 394,532.29 31.43 66.72 68.57 33.28 35.29 47.70


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P l e a s e c i t e t h i s a r t i c l e i npr e s s a s : P .V i c t e r P a ul ,e t a l .,A ne wpopul a t i ons e e d i ngt e c h ni que f or pe r mut a t i on-c od e d G e ne t i c A l gor i t h m:

Table 5Experimental results of EV population seeding method.

S. no. Instance Optimaltness

ba value Time taken(s)

Best tness Worsttness

Averagetness


Best indiv.(%)

Worstindiv. (%)

Best indiv.(%)

1 Eil51 426 2 1.04 480.23 648.01 552.38 12.73 52.11 87.273 1.13 468.72 675.38 555.20 10.03 58.54 89.974 1.13 459.91 660.55 565.52 7.96 55.06 92.045 1.16 486.02 674.68 586.51 14.09 58.38 85.91 41

2 Pr76 108,159 2 0.89 122,815.73 161,930.85 150,166.19 13.55 49.72 86.453 1.04 129,988.13 180,861.62 152,180.65 20.18 67.22 79.82 32.784 0.96 122,222.99 181,133.91 156,608.47 13.00 67.47 87.00 32.535 0.98 117,737.35 185,199.42 160,039.16 8.86 71.23 91.14

3 KroA100 21,282 2 1.83 22,579.30 31,671.04 26,797.09 6.10 48.82 93.90

3

2.36 23,835.01 33,867.96 28,967.86 12.00 59.14 88.00 40.84 1.96 24,705.75 36,461.39 30,623.82 16.09 71.33 83.91 28.65 2.57 27,226.07 36,658.67 31,916.62 27.93 72.25 72.07

4 Pr144 58,537 2 1.97 62,952.97 77,253.26 68,031.87 7.54 31.97 92.463 1.89 65,118.87 84,841.73 71,417.70 11.24 44.94 88.76 55.04 1.96 67,942.88 93,087.71 78,660.40 16.07 59.02 83.93 40.95 2.47 68,776.48 104,878.52 81,943.07 17.49 79.17 82.51

5 Gil262 2378 2 4.61 2695.27 3419.06 3138.27 13.34 43.78 86.663 4.56 2775.68 3608.30 3293.49 16.72 51.74 83.28 484 4.94 3039.58 3797.79 3395.44 27.82 59.71 72.18 40.25 4.50 3094.87 3860.18 3504.37 30.15 62.33 69.85

6 Fl417 11,861 2 9.64 13,320.99 18,086.68 16,149.93 12.31 52.49 87.693 9.83 13,742.98 18,992.64 16,856.82 15.87 60.13 84.13 39.84 9.98 14,738.10 19,650.10 17,295.57 24.26 65.67 75.74 34.35 9.85 15,034.46 20,188.81 17,858.83 26.76 70.21 73.24

7 Pr1002

259,045 2 15.95 311,056.00 362,563.17 335,898.99 20.08 39.96 79.923 20.43 318,419.76 381,566.91 338,586.44 22.92 47.30 77.08 52.704 24.53 344,430.94 392,676.91 367,596.27 32.96 51.59 67.04 48.415 19.97 360,853.18 406,318.12 380,040.78 39.30 56.85 60.70


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Table 6Experimental results of VE population seeding method.




Averagetness


Best indiv.(%)

Worstindiv. (%)

Best indiv.(%)

1 Eil51 426 2 0.85 514.61 635.02 571.92 20.80 49.07 79.203 0.68 510.67 635.02 568.51 19.88 49.07 80.124 0.79 510.67 638.16 570.44 19.88 49.80 80.12 50.25 0.64 510.67 688.21 583.60 19.88 61.55 80.12

2 Pr76 108,159 2 0.58 142,396.36 159,821.61 152,344.17 31.65 47.77 68.353 0.71 142,396.36 170,453.18 155,229.69 31.65 57.60 68.354 0.66 142,396.36 187,050.56 161,093.74 31.65 72.94 68.355 0.67 142,396.36 187,050.56 161,892.75 31.65 72.94 68.35

3 KroA100 21,282 2 1.50 25,125.22 28,541.55 26,544.47 18.06 34.11 81.94

3

1.36 25,125.22 31,983.78 27,817.70 18.06 50.29 81.94 49.74 1.54 25,125.22 36,397.00 30,195.35 18.06 71.02 81.94 28.985 1.59 25,125.22 37,333.38 31,326.19 18.06 75.42 81.94

4 Pr144 58,537 2 1.31 61,496.32 80,851.07 68,600.74 5.06 38.12 94.943 1.50 61,496.32 80,851.07 71,180.47 5.06 38.12 94.94 61.884 1.48 61,496.32 84,518.93 72,375.21 5.06 44.39 94.94 55.615 1.39 61,496.32 97,687.98 72,375.21 5.06 66.88 94.94

5 Gil262 2378 2 3.87 3002.62 3593.52 3199.15 26.27 51.12 73.733 3.85 3002.62 3685.80 3343.51 26.27 55.00 73.73 45.04 4.34 3002.62 3973.25 3407.15 26.27 67.08 73.73 32.95 4.51 3002.62 4120.66 3487.54 26.27 73.28 73.73

6 Fl417 11,861 2 8.94 15,104.49 19,043.95 16,820.81 27.35 60.56 72.653 7.96 15,104.49 19,043.95 17,326.59 27.35 60.56 72.65 39.44 8.23 15,104.49 19,043.95 17,628.11 27.35 60.56 72.65 39.445 8.11 15,104.49 19,607.30 17,492.75 27.35 65.31 72.65

7 Pr1002

259,045 2 17.40 315,597.59 351,116.55 329,395.18 21.83 35.54 78.173 17.10 315,597.59 377,213.31 345,410.40 21.83 45.62 78.17 54.384 17.11 315,597.59 410,111.38 360,368.97 21.83 58.32 78.17 41.685 18.90 315,597.59 436,384.90 374,376.88 21.83 68.46 78.17


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Fig. 7. Therange of variation in convergence rate of VE method.

of the city x. Experimental results of NN population seeding tech-nique areshown in Table 3 . From Table 3 , the following observationcan be made:

Observation 3 : For all the problem instances, NN techniqueproduces theindividual with least error rate, consequentlythe best

individual, when compared to the other existing techniques.

4.2.1.3. Gene Bank (GB) technique. In this technique, the gene bankis built by assembling the permutation of N cities based on theirdistance. The population of individuals is generated from the genebank such that the individuals are of above-average tness andshortdening length. Table 4 shows the experimentalresults of GBpopulation seeding technique and from that the following obser-vation can be made:

Observation 4 : It can be observed that the quality of best indi-viduals generated decreases with increase in the size of probleminstance. It is because of the constant gene size used for the gener-ation of initial population.

4.2.2. Experiments on proposed techniques4.2.2.1. Type 1: EV technique. The experimental results of EVmethodof ODVbasedpopulation seeding techniqueis presentedinTable 5 . In Table 5 , the best results are emphasized in bold. The fol-lowing observations can be made based on the statistical analysesshown in Table 5 .

Observation 5 : For all the problem instances and sizes, the EVpopulation seeding technique tends to give the best t individualwith the ba value of 2.

Observation 6 : The error rate of the best individuals gener-ated using the EV population seeding technique increases with anincrease in the number of cities in the instance.

Fig. 8. Therange of variation in error rate of VE method.

Fig. 9. The range of variation in convergence rate of VV method.

Observation 7 : The Convergence rate of the best individuals cre-ated using the EV population seeding technique decreases with anincrease in the number of cities in the instance. The maximum andminimum values obtained are 93.90% for instance KroA100 with

ba value of 2 and 80.92% for instance Pr1002 with ba

value 2.This is because with the same population size of 100, the initialconvergence rate of thetechnique decreaseswith an increase in thenumber of cities in the instance. However, at least 80.92% of con-vergence has been achieved at the population initialization stageof the GA for solving TSP problem.

Observation 8 : The convergence diversity of the individualsremains constant for all the instances except Pr1002 for whichit decreases comparatively. The most suitable practical diversityrange canbe 3545%. Themaximumand minimum valuesobtainedare 62.37% for instance of Pr76 with ba value of 2 and 17.55% forPr1002 with ba value of 5.

Observation 9: The average convergence of EV techniqueremains between the range of 6050%, which shows that thepopulation contains a good composition of good and bad qualityindividuals leads for exploring more search space.

Theranges of variations in convergence rate anderrorrateof thepopulation generatedby EV techniqueare illustrated in Figs.5and6respectively.

4.2.2.2. Type 2: VE technique. The experimental results of the VEmethod of ODV based population seeding technique is recorded inTable 6 and the best results are emphasized in bold. The followingobservations are made based on the results presented in Table 6 .

Fig. 10. The range of variation in error rate of VV method.


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Table 7Experimental results of VV population seeding method.




Averagetness


Best indiv.(%)

Worstindiv. (%)

Best indiv.(%)

1 Eil51 426 2 1.16 481.93 650.60 562.88 13.13 52.72 86.87 47.23 0.96 466.72 681.36 556.12 9.56 59.94 90.444 1.12 464.15 677.47 572.71 8.96 59.03 91.045 1.30 488.30 694.68 589.75 14.63 63.07 85.37

2 Pr76 108,159 2 0.96 132,815.73 164,233.18 151,647.85 22.80 51.84 77.203 1.10 116,449.76 172,650.39 153,976.41 7.67 59.63 92.334 1.24 121,277.86 184,779.43 157,896.94 12.13 70.84 87.87 29.165 0.85 133,252.43 186,572.67 162,671.50 23.20 72.50 76.80

3 KroA100 21,282 2 2.09 23,081.90 32,911.35 27,110.27 8.46 54.64 91.54

3

2.29 24,763.78 34,533.24 29,062.30 16.36 62.27 83.64 37.734 2.33 25,720.73 37,224.15 30,894.57 20.86 74.91 79.14 25.095 2.03 27,389.15 37,511.53 31,862.64 28.70 76.26 71.30

4 Pr144 58,537 2 1.81 63,213.76 78,883.82 68,178.38 7.99 34.76 92.013 2.23 65,249.00 82,901.59 72,129.39 11.47 41.62 88.53 58.384 1.83 67,995.22 92,248.32 79,161.49 16.16 57.59 83.84 42.415 1.91 68,069.09 107,500.38 78,761.06 16.28 83.65 83.72

5 Gil262 2378 2 5.04 2639.70 3443.90 3125.53 11.01 44.82 88.993 5.29 2728.50 3731.19 3294.28 14.74 56.90 85.26 43.14 4.85 3054.15 3772.09 3407.21 28.43 58.62 71.57 41.35 5.14 3145.82 3986.36 3523.60 32.29 67.63 67.71

6 Fl417 11,861 2 9.90 12,994.83 19,207.45 16,214.79 9.56 61.94 90.443 10.97 14,309.41 19,140.47 16,761.76 20.64 61.37 79.36 38.634 10.40 14,806.25 19,705.09 17,464.90 24.83 66.13 75.17 33.875 11.03 15,081.88 20,633.11 17,881.99 27.16 73.96 72.84

7 Pr1002

259,045 2 21.54 306,292.97 369,643.84 336,460.31 18.24 42.69 81.763 21.84 317,395.30 375,577.45 352,951.65 22.53 44.99 77.47 55.014 22.07 345,069.13 390,673.04 367,582.75 33.21 50.81 66.79 49.195 18.02 360,241.58 404,670.15 380,772.15 39.07 56.22 60.93


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Table 8Performance order for average computation time.

S. no. Instance Performance order ( best worst )

1 Eil51 Random VE VV EV GB NN 2 Pr76 Random VE VV EV GB NN 3 Kroa100 Random VE EV VV NN GB4 Pr144 VE Random VV EV NN GB5 Gil262 Random VE EV VV GB NN 6 Fl417 Random VE EV VV GB NN

7

Pr1002 Random EV GB VE VV NN

0

5

10

15

20

25

Eil51 Pr76 KroA100 Pr144 Gil262 F l417 Pr1002

Random Generation Nearest NeighbourGeneBank EVVE VV

TSP Instance

T i m e

( s e c

)

Fig. 11. Computation time of different population seeding techniques.

0

10

20

3040

50

60

70

80

90

100

Eil51 Pr76 KroA100 Pr144 Gil262 Fl417 Pr1002


TSP Instance

C o n v e r g e n c e

D i v e r s i

t y ( % )

Fig. 12. Convergence diversity of different population seeding techniques.

Table 9Performance order for convergence diversity.

S. no. Instance Performance order ( high low )

1 Eil51 Random VV EV GB NN VE

2 Pr76 Random EV NN VV VE GB3 Kroa100 Random VE EV VV GB NN 4 Pr144 Random VV VE EV GB NN 5 Gil262 Random VE VV EV NN GB6 Fl417 Random VV EV GB VE NN 7 Pr1002 Random VV EV GB NN VE

Observation 10 : The convergence rate of the best individualscreated using the VE population seeding technique remains equaldespite the value of ba . This shows that the individuals generatedsuffers a cyclic repetition problem though at least 72.65% of con-vergence has been achieved at the population initialization stage.

Observation 11 : For each instance, the convergence diversity of individuals increases with ba value. The maximum and minimum

values obtained are 61.83% for instance of Pr144 with ba

value

-50

-30

-10

10

30

50

70

90

Eil51 Pr76 Kr oA100 Pr144 Gil262 Fl417 Pr 1002


TSP Instance

A v e r a g e


( % )

Fig. 13. Average convergence of different population seeding techniques.

of 5 and 13.71% for Pr1002 with ba value of 2 respectively. Onaverage, the convergence diversity remains between 20% and 35%which reveals that the individuals are identical and may stick inlocal optimal solution.

The convergence rate and error rate ranges of the populationgeneratedby VEtechnique aredepicted in Figs.7and8 re spectively.4.2.2.2.1. Type 3: VV technique. Table 7 illustrates the exper-

imental results of VV method of ODV based population seedingtechniqueand thebestresultsare emphasized in bold.Theseresultsmay derive the possible observations as follows:

Observation 12 : The maximum and minimum values of the bestconvergence rate obtained are 92.33% for instance Pr76 with ba value of 2and 81.76%for instance Pr1002 with ba value2 respec-tively. Thus, at least 82% of convergence has been achieved at thepopulation initialization stage of the GA for solving TSP problem.

Observation 13 : The convergence diversity of the individuals inthe population is maintained between the ranges of 2050% whichis most suitable for exploring search space effectively. The max-imum and minimum values obtained are 67.36% for instance of Pr144 with ba value of 5 and 17.15% for Pr1002 with ba value of 5 respectively.

The convergence rate range and error rate range of the pop-ulation generated by VV technique are shown in Figs. 9 and 10respectively.

Table 10Performance order for average convergence.


1 Eil51 EV VV VE NN GB Random2 Pr76 VV EV VE GB NN Random3 Kroa100 VE EV VV GB NN Random4 Pr144 EV VV VE NN GB Random

5 Gil262 NN VV EV VE GB Random6 Fl417 VV EV VE NN GB Random7 Pr1002 VE VV EV NN GB Random

Table 11GA conguration parameters.

S. no. Parameter Value/technique

1 Population size 1002 Generation limit 2003 Crossover method Ordered crossover4 Crossover probability 0.6 [3,35]5 Mutation method Swap Mutation [35]6 Mutation probability 0.02 [3,32]7 Elitism True (3 individuals)8 Termination condition Generation Limit


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0

5

10

15

20

25

30

35

40

Eil51 Pr76 Kroa100 Pr144 Gil262 Fl417 Pr1002 Average

Random Populaton Neare st NeighbourGeneBank EVVE VV

TSP Instance

E r r o r

R a t e

( % )

Fig. 16. Error rate of different methods analyzed.

60

65

70

75

80

8590

95

100

Eil51 Pr76 Kroa100 Pr144 Gil262 Fl417 Pr1002 Average

Random Populaton Nearest NeighbourGeneBank EVVE VV

TSP Instance


R a t e ( % )

Fig. 17. Convergence rate of different methods analyzed.

Table 18

Performance order for computation time.S. no. Instance Performance order ( best worst )

1 Eil51 VE Random EV GB VV NN 2 Pr76 Random VE EV GB NN VV 3 Kroa100 VE Random EV GB NN VV 4 Pr144 VE Random EV GB VV NN 5 Gil262 Random VE EV GB VV NN 6 Fl417 Random VE EV GB VV NN 7 Pr1002 Random VE EV GB NN VV

performance order of each populationseeding technique in respectof computation time is shown in Table 8 .

4.2.3.2. Convergent diversity. The convergence diversity is a prop-

erty that elucidates the distribution of good and bad qualityindividuals among the population generated. Fig. 12 shows thecomparative performance of proposed and existing techniquesw.r.t.convergence diversityfor thebenchmarkTSP instances. From

Table 19Performance order for error rate.


1 Eil51 VV EV NN VE GB Random2 Pr76 VV EV NN GB Random VE 3 Kroa100 VV EV NN GB Random VE 4 Pr144 VV EV NN GB VE Random5 Gil262 VV EV NN Random GB VE 6 Fl417 VV EV NN GB VE Random7 Pr1002 VV NN EV GB VE Random

Table 20Performance order for convergence rate.


1 Eil51 VV EV NN VE GB Random2 Pr76 VV EV NN GB Random VE 3 Kroa100 VV EV NN GB Random VE 4 Pr144 VV EV NN GB VE Random5 Gil262 VV EV NN Random GB VE 6 Fl417 VV EV NN GB VE Random

7

Pr1002 VV NN EV GB VE Random

Fig. 12 , it is understood that the VV technique outperform othertechniques except the random method, which perform betterthan VV technique. The performance order of all population seed-ing techniques in respect of convergence diversity is depicted inTable 9 . The NN and VE methods have the least convergence diver-sity, which preventthem from evolving optimal solution and stuckwith the local optimal solution. The EV method has better con-vergence diversity can produce the individuals with high potentialsequence, butthe experimentalresults show that it lacks in explor-ing the search space completely, which resulted in poor abilityto produce optimal solution sometime. The VV method has bet-ter convergence diversity than the other two proposed methods,which proves that it could explore the problem search space effec-tively and has a high possibility of nding the optimal solution forproblem of any size.

4.2.3.3. Average convergence. Average convergence explores thequality of population generated by nding the average of tness of individuals. Fig. 13 shows the comparison between the proposedand existing techniques in terms of average convergence. This g-ureillustratesthatthe average convergence of thepopulation of EV,VE and VV methods tends to be equal for large sized problem spacebesides with slight variations for small sized problems. The possi-bility of generating identical individuals is very high in VE methodwhen compared with other two methods. The performance orderof each population seeding technique in respect of average conver-

gence is shown in Table 10 . From the table, it could be observedthat the proposed VV, EV and VE techniques offer better averageconvergence rate compared to Random, NN and GB techniques.

4.2.3.4. Best tness value. The best tness value refers to the actualoutcome of the optimal solution. i.e. it refers to the destinationof the solution. The performance comparison in terms of the t-ness value of the best solution obtained using random, NN, GB,EV, VE and VV (with different ba values) population seeding tech-niques w.r.t. the known optimal tness value is shown in Fig. 14 .Fig.14 (a)(g)refers to theperformance of theabove saidtechniquesfor the benchmark TSP instances Eil51, Pr76, KroA100, Pr144,Gil262, Fl417 and Pr1002 respectively. The EV method producedthe near optimal solution for Eil51, KroA100 and Pr144 instances

with ba

value of 4, 2,and 2 respectively. Thenear optimal solutionsfor Pr76, Gil262, Fl417 and Pr1002 instances are produced by VVmethod with ba value 3, 2, 2 and 2 respectively. From the graphs,it is understood that the VV method outperforms the other tech-niques. In addition to that it is also observed that the EV methodgenerated equally good solution several times, by generating bestsolution whose tness is near the optimal value, but not for all theinstances. As to conclude, the EV method of population seedingworks well for small sized TSP instances and VV method performbetter for large sized TSP instances compared to other techniques.

4.3. Phase II: Experimentation and result analyses

As said earlier, in this phase, the scope of the experiments is

extended to cover the whole life cycle of the Genetic Algorithm.


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Fig. 18. Convergence graph for TSP instances usingdifferent population seeding techniques.

This extended scope of the experiments facilitates to assess theoverall impactof the proposed techniques compared with random,NN and GB techniques. The crossover and mutation operator usedareOrdered Crossover (OX) [3,5153] and Swap Mutation operator[30,34,38] respectively. The ordered crossover has been found tobe one of the best in terms of quality and speed to solve TSP andits working principle is exemplied in [3] . The GA parameters and

the corresponding values are depicted in Table 11 . Elitist strategy

is followed to ensure that the ttest individuals in each generationis carried out to the next generation in order to avoid the replace-ment of best t individuals with poor individuals in the successivegenerations.

For each technique, the executions are carried out for 50 timeswith different ba values ( ba values are notapplicable for random,NN and GB methods) and the average of 50 runs of each case has

beenconsideredfor experimental analyses.Theexecutionproceeds


19/21



until the termination condition is satised and the tness values of the converged solutions are used. Tables 1217 shows the Perfor-mance of the Random, NN, GB, EV, VE and VV population seedingmethods respectively. The analyses are carried out based on theperformance factors as discussed in Eqs. (17) and (18) . The com-parative analyses of various population seeding methods in termsof computation time, error rate and convergence rate are shown inFigs. 1517 respectively.

Analysis based on computation time elucidates that Randomand VE methods take less time than EV and VV methods, whichare better than GB and NN methods. The order of performance interms of computation time for different TSP instances is given inTable 18 . Bothconvergenceand errorratesbasedassessmentsshowthat VV method leads to evolve better convergence with less errorfor all the TSP instances despite of problem size. The performanceordersfor error andconvergence rates aregiven in Tables19 and20respectively. The VE and NN methods that posses less populationdiversity records worst convergence rate for all the TSP instances,which justify that they have stuck in local optimal solution. Theconvergence graph visualizes the improvement in the tness of the solution by each generation. The convergence graphs for theTSP instances Eil51, Pr76, Kroa100,Pr144, Gil262, Fl417 and Pr1002are shown in Fig. 18 (a )(g) respectively. From these graphs, the t-ness improvement of the solutionsof VV seeding method convergesbetter than EV, which outperforms all the other techniques.

4.4. Discussion

In summary, the investigations in the Phase I shows that theVV population seeding method offers better convergence diversityand initial convergence rate than the EV which is better than NN,GB, VE and random techniques. It is also noted that, the compu-tation time of VV method is relatively more than the random, EVand VE methods. The Phase II investigation proves that the VV andEV methods of population seeding outperforms the random,GB andNN,which areclaimed tobe thebetter population seeding methods.

Thus, the required characteristics of population seeding like ran-domness, individual diversity and potential sequence are assuredin the proposed ODV based VV and EV population seeding meth-ods. Though, the VV method works best, it is necessary to pointthat it could not achieve the complete 100% of convergence for anyof the test instances after certain point of generations. The tnessvalue of best solution generated by the VV method at the popula-tion seeding stage for instance Eil51 is 91.04%, for which the tnessof the best solution after 200 generations of crossover is 98.98%.This implies that the OX operator does not completely exploit thepotential sequence of individuals generated at the initial stage.Therefore,genetic operators like edge-recombination and partitioncrossover [59,60] , which perform well with local optima solutions,canbe used to make useof potentialsolutions generated using ODVpopulation seeding technique. And also, designing an ODV basedcrossover operator that exclusively exploit the advantages of indi-viduals generated using ODV methods may enable to attain the100% of convergence, particularly for large sized TSP instances.

5. Conclusion

In this paper, an efcient Ordered Distance Vector (ODV) basedpopulation seeding techniquehas been proposed forGA to enhanceits overall performance. The proposed technique consists of threedifferent methods namely EV, VE and VV, which generate a pop-ulation of individuals with characteristics such as randomness,diversity and potential sequence. The Traveling Salesman Problem(TSP) has been chosen as the testbed and the experiments are per-

formed on different sized TSP benchmark datasets obtained from

TSPLIB [54] . The experiments in this research are carried out intwo different phases. The scope of the experiments in the Phase Iis limited to the initialization phase alone, whereas this restrictedscope helps to assessthe performance of theproposedtechniqueinits intended phase alone. In Phase II, the scope of the experimentsis extended to cover the whole life cycle of the Genetic Algorithm,whereas this extended scope facilitates to assess the overall impactof the proposed technique. In both the phases, the performanceof the proposed technique has been investigated and comparedwith the existing techniques like Random Initialization technique,Nearest Neighbor Technique andGeneBankTechnique. The experi-mentalanalysis based on the performance factors like convergenceanderrorrate shows that theVV performs betterthan EV with rela-tively highcomputation timecomparedto existing techniques.Thiswork can be extended to design an ODV based crossover operatorthat exclusively exploit the potential of individuals generated toimprove the performance further, to accomplish100% convergencein lesser computation time.

Acknowledgements

This work is a part of the Research Projects sponsored under

the Major Project Scheme, UGC, India, Reference Nos: F. No. 40-258/2011(SR), dated 29 June 2011 and F. No. 41-639/2012(SR)/Dt.16-07-2012. The authors would like to express their thanks for thenancial supports offered by the Sponsored Agency.

Appendix A. Supplementary data

Supplementary material related to this article can be found, inthe online version, at http://dx.doi.org/10.1016/j.jocs.2013.05.009 .

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