Research Article Crossover versus Mutation: A Comparative...

Research ArticleCrossover versus Mutation A ComparativeAnalysis of the Evolutionary Strategy of Genetic AlgorithmsApplied to Combinatorial Optimization Problems

E Osaba R Carballedo F Diaz E Onieva I de la Iglesia and A Perallos

Deusto Institute of Technology (DeustoTech) University of Deusto Avenue Universidades 24 48007 Bilbao Spain

Correspondence should be addressed to E Osaba eosabadeustoes

Received 24 March 2014 Revised 10 July 2014 Accepted 14 July 2014 Published 4 August 2014

Academic Editor Manuel Lozano

Copyright copy 2014 E Osaba et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Since their first formulation genetic algorithms (GAs) have been one of the most widely used techniques to solve combinatorialoptimization problems The basic structure of the GAs is known by the scientific community and thanks to their easy applicationand good performance GAs are the focus of a lot of research works annually Although throughout history there have been manystudies analyzing various concepts of GAs in the literature there are few studies that analyze objectively the influence of using blindcrossover operators for combinatorial optimization problems For this reason in this paper a deep study on the influence of usingthem is conducted The study is based on a comparison of nine techniques applied to four well-known combinatorial optimizationproblems Six of the techniques are GAs with different configurations and the remaining three are evolutionary algorithms thatfocus exclusively on the mutation process Finally to perform a reliable comparison of these results a statistical study of them ismade performing the normal distribution z-test

1 Introduction

Genetic algorithms (GAs) are one of the most successfulmetaheuristics for solving combinatorial optimization prob-lemsThanks to their easy application and good performanceGAs have been used to solve many complex problems framedin various fields as for example transport [1 2] softwareengineering [3 4] or industry [5 6] GAs were proposedin 1975 by Holland [7] in an attempt to imitate the geneticprocess of living organisms and the law of the evolutionof species Anyway their practical use to solve complexoptimization problems was shown later by Goldberg [8] andDe Jong [9]

Throughout history many researches have focused on thestudy of genetic algorithmsThese studies can be grouped into3 different categories

(i) Practical Applications of GAs These studies focusedon the application of GAs for solving specific problemsAmong these three categories this is the most common inthe literature Two subcategories can be identified in this

first group of works variations of a classic GA [10ndash12] orhybridization of a GA with some other technique [13ndash15]

(ii) Development of New Operators These researches presentnew specific operators such as crossover [16 17] or mutationfunctions [18 19] Normally these operators are heuristic andthey are applied to a particular problem in which they get agreat performance

(iii) Analysis of the Algorithm Behavior These works focuson the theoretical and practical analysis of GAs This kindof research analyzes for example behavioural characteristicsof the algorithm as the convergence [20] or the efficiencyof certain phases of the algorithm such as crossover [21 22]or mutation phases [23 24] or the influence of adaptingsome parameters as the crossover and mutation probabil-ity [25ndash27] These works attempt to overcome the draw-backs of traditional genetic algorithms and are the sourceof new problem-solving techniques such as the adaptivegenetic algorithms [28 29] or the parallel genetic algorithms[30 31]

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 154676 22 pageshttpdxdoiorg1011552014154676

2 The Scientific World Journal

In this paper a deep study on the influence of usingblind crossover operators in GAs for solving combinatorialoptimization problems is conducted This study is developedby means of a comparison between GAs with this kind ofoperators and EAs based only on mutation operators Thusthis work could be framed into the third category Previouslyother studies in the literature have had a similar purposefor example [32] where the authors tried to validate thehypothesis that the crossover phase of genetic algorithms isnot efficient when it is applied to routing problems In thatwork the authors develop several versions of the basic GAwith some blind crossover operators (eg order crossover(OX) [33] or modified order crossover (MOX) [34]) andthey apply these techniques to the traveling salesmanproblem[35] Performances of these GAs are compared with the oneof an evolutionary algorithm (EA) based solely onmutationsThe comparison is based on the quality of the solution andthe runtime Furthermore the comparison also takes intoaccount the percentage of deviation from the average valuesof each parameter

On the other hand in [22] the efficiency of six differentversions of the classic GA applied to the degree constrainedminimal spanning tree problem [36] is compared Eachversion has its own crossover function In that work the onlydata shown for each version of the GA is the average value ofthe results obtained so the comparison is performed basedonly on this criterion Moreover the authors do not performthe comparison of the results obtained by a conventional GAand an EA For this reason with this study it is not possibleto quantify the real influence of the crossover phase in theoptimization capacity of a GA

Together with the above studies in the literature there aremany others that are not comparable with the study presentedin this paper The main reason is that they are focused onother types of problems [21] or because they analyzed onlythe crossover process of a traditional GA [37ndash39]

The motivation of this work stems from the absence inthe literature of a study that proves objectively the efficiencyof using blind crossover operators in GAs for combinatorialoptimization problems Although [32] focuses on this topicit is only applicable to routing problems and it is only testedwith one problem the TSP In addition the comparison ofthe results done in [32] is not as deep as the one made in thepresent work On the other hand as it has been mentionedthe study presented in the abovementioned [22] is not trulyconclusive to prove the real influence of the crossover processin a GA

Therefore the goal of this paper is to perform an objectivestudy on the efficiency of blind crossover operators in basicGAs with respect to blindmutation operators in basic EAs Inorder to reach this goal an exhaustive comparison betweendifferent versions of genetic and evolutionary algorithms ispresented This comparison includes the following criteriaquality of the results runtime and convergence behavior ofeach of the techniques reviewed Furthermore to perform areliable comparison of these results a statistical study ismadeFor this purpose the normal distribution 119911-test is performedFor the experimentation four different problems have beenused the traveling salesman problem (TSP) the capacitated

vehicle routing problem (CVRP) [40] the N-queens problem(NQP) and the one-dimensional bin packing problem (BPP)[41]

The rest of the paper is structured as follows In Section 2the description of the experimentation is presented InSection 3 the tests for the TSP are shown After thatthe experiments performed with the CVRP (Section 4) aredisplayed followed by those conducted with the NQP andBPP (Sections 5 and 6 resp) Finally thework is finishedwiththe conclusions of the study and further work (Section 7)

2 Description of the Experimentation

In this section a description of the experimentation is madeFirst in Section 21 the problems used for the tests areintroducedThen in Section 22 the details of the techniquesdeveloped are described including the functions of thedifferent steps of the algorithms Finally in Section 23 theexperimentation setup is presented

21 Description of the Problems For this study four differentcombinatorial problems have been used Two of them areoptimization problems of routing the TSP and the CVRPIn addition to verify that the results of this study are validfor other types of problems apart from the routing onestwo constraint satisfaction problems have also been used inthe experimentation the NQP and the BPP These problemswere chosen because they are well known and easy toimplement In addition they are easily replicable In this wayany researcher can perform these same tests either to checkthe results or to perform themwith other crossover functionsor different parameters

The first problem used is the TSP The TSP is one ofthe most famous and widely studied problems throughouthistory in operations research and computer science It hasa great scientific interest and it is used in a large number ofstudies [42ndash44] This problem can be defined on a completegraph 119866 = (119881119860) where 119881 = V

1 V2 V

119899 is the set

of vertexes which represents the nodes of the system and119860 = (V

119894 V119895) V119894 V119895

isin 119881 119894 = 119895 is the set of arcs whichrepresents the interconnection between nodes Each arc hasan associated distance cost 119889

119894119895 The objective of the TSP is

to find a route that visits every customer once (and onlyonce) that is a Hamiltonian cycle in the graph 119866 and thatminimizes the total distance traveled In a formal way theTSP can be formulated as follows [45]

Minimize119891 (119883) = sum

119894=0

sum

119894 =119895119895=0

119889119894119895119909119894119895 forall119894 119895 isin 119881 (1)

where119909119894119895isin 0 1 forall 119894 119895 isin 119860 (2)

subject to constraints sum

119894=0119894 =119895

119909119894119895= 1 forall119895 isin 119881 (3)

sum

119895=0119894 =119895

119909119894119895= 1 forall119894 isin 119881 (4)

sum

119894isin119878119895isin119878119894 =119895

119909119894119895ge 1 forall119878 sub 119881 (5)

The Scientific World Journal 3

Nodes

1 2

3

4

5 6

78

9

(a)

Path

1 2

3

4

56

7

8

9

(b)

Figure 1 Example of TSP instance and possible solution

where 119909119894119895in (2) a binary variable is 1 if the arc(119894 119895) is used in

the solution Furthermore119881 is the set of nodes of the systemand119889119894119895is the distance between the nodes 119894 and 119895The objective

function (1) is the sum of all the arcs in the solution usedthat is it is the total distance of the route Constraints (3) and(4) indicate that each node have to be visited and abandonedonly once while the formula (5) guarantees the absence ofsubtours and indicates that any subset of nodes 119878 has to beabandoned at least 1 time This restriction is vital because itavoids the presence of cycles

Finally all the solutions are encoded following the pathrepresentation [46] In this way each individual119883 is encodedby a permutation of numbers which represents the pathFigure 1(a) represents a possible 9-node instance of the TSPand Figure 1(b) represents a possible solution This solutionwould be encoded as119883 = (1 2 4 6 8 9 7 5 3) and its fitnesswould be119891(119883) = 119889

12+11988924+11988946+11988968+11988989+d97+11988975+11988953+11988931

The second selected problem is the CVRP Due to itscomplexity and above all its applicability to real life theCVRP is also used inmany researches every year [47 48] Forthe TSP this problem can be defined on a complete graphIn addition the vertex V

0represents the depot and the rest

are the customers each of them with a demand 119902119894 A fleet

of vehicles 119870 is available with a limited capacity 119876 for eachvehicle The objective of the CVRP is to find a number ofroutes with a minimum cost such that (i) each route startsand ends at the depot (ii) each client is visited exactly by oneroute and (iii) the total demand of the customers visited byone route does not exceed the total capacity of the vehicle thatperforms it [49]This problem could be formulated as follows[40]

Minimize 119891 (119883) = sum

119894=0

sum

119894 =119895119895=0

119889119894119895119909119894119895

forall119894 119895 isin 119881 (6)


119894=0119894 =119895

119909119894119895= 1 forall119895 (7)

sum

119895=0119894 =119895

119909119894119895= 1 forall119894 isin 119881 (8)

sum

119894

119909119894119895ge |119878| minus V (119878) 119878 119878 sube

119881

1

|119878| ge 2 (9)

sum

119894isin119878

119902119894119910119903

119894le 119876 forall119903 isin 119870 (10)

where 119910119903

119894isin 0 1 forall119903 isin 119870 (11)

and 119909119894119895isin 0 1 forall 119894 119895 isin 119860 119894 = 119895 (12)

The formula (6) is the objective function which is thetotal distance traveled by all the routes The variable (11) is abinary variable which is 1 if the vehicle 119903 satisfies the demandof the client 119894 and 0 otherwise The binary variable (12) is 1if the arc(119894 119895) is used in the solution Formulas (8) and (9)ensure that every customer is visited by one route only andexactly once Finally clause (9) serves to eliminate subtourswhere |119878| is the number of customers and 119903(119878) the minimumnumber of vehicles to serve all Finally the restriction (10)ensures that the sum of all the demands of a route does notexceed the maximum vehicle capacity

In the case of CVRP the path representation is also usedfor the individuals encoding [50] In this case the routes arealso represented as a permutation of nodes To distinguishthe routes of one solution they are separated by zeros InFigure 2(a) an example of a CVRP is shown On the otherhand in Figure 2(b) a solution composed by three differentroutes is depicted On this occasion this solution would beencoded as 119883 = (3 1 5 0 2 4 0 7 9 8 6) and its fitnesswould be 119891(119883) = 119889

03+11988931

+11988915

+11988950

+11988902

+11988924

+11988940

+11988907

+

11988979

+ 11988998

+ 11988986

+ 11988960

The third problem is the NQP This problem is a general-ization of the problem of putting eight nonattacking queenson a chessboard [51] which was introduced by Bezzel in 1848[52] The NQP consists of placing 119873 queens on a 119873 times 119873

chess board in order that they cannot attack each other thatis on every row column and diagonal only one queen canbe placed This problem is a classical combinatorial designproblem (constraint satisfaction problem) which can alsobe formulated as a combinatorial optimization problem [53]


Depot

1

2

3

4

5

6

7

89

(a)

Vehicle routes1

2

3

4

5

6

7

89

(b)

Figure 2 Example of CVRP instance and possible solution

Although NQP is often used as benchmarking problem ithas also some real applications [54] In this study NQP hasbeen formulated as a combinatorial optimization problemwhere a solution 119883 is coded as a 119873-tuple (119902

1 1199022 119902

119899)

which is a permutation of the 119873-tuple (1 2 119873) Each 119902119894

represents the row occupied by the queen positioned in the119894th column Using this representation vertical and horizontalcollisions are avoided Thus the fitness function is defined asthe number of diagonal collisions along the board 119894th and 119895thqueens collide diagonally if

1003816100381610038161003816119894 minus 119902119894

1003816100381610038161003816=

10038161003816100381610038161003816119895 minus 119902119895

10038161003816100381610038161003816

forall119894 119895 1 2 119873 119894 = 119895 (13)

The objective is to minimize the number of conflictsbeing zero the ideal fitness An example of an individual fora 6-queens chess board could be seen in Figure 3 Accordingto the encoding explained the individual represented in thisfigure would be encoded as 119883 = (2 1 4 6 5 3) In additionits fitness would be 3 since there are three diagonal collisions(2-1 1ndash4 and 6-5) This same formulation has been widelyused in the literature [55 56]

Finally the last used problem is the BPP In distributionand production the fact of packing items into boxes or binsis a daily task Depending on the shape and size of the itemsas well as the form and capacity of bins a wide amount ofdifferent packing problems can be formulated The BPP isone of the simplest problems in this field [41 57] and it isfrequently used in the literature as benchmarking problem[58ndash60] The BPP consists in a set of items 119868 = 119894

1 1198942 119894

119899

each with an associated size 119904119894and an infinite number of bins

119861 of an equal capacity 119902 The objective of the BPP is to packall the items into a minimum number of bins Therefore theobjective function is the number of bins which has to be

Figure 3 Example of a 6 times 6 instance for the NQP

minimized In this way given 119899 items and 119899 bins the BPP canbe formulated as follows

Minimize 119891 (119883) =

119899

sum

119894=0

119910119894 (14)

subject to constraints119899

sum

119894=0

119909119894119895= 1 forall119895 isin 1 119899 (15)

119899

sum

119895=0

119904119894119909119894119895le 119902 forall119894 isin 1 119899 (16)

where119910119894isin 0 1 forall119894 isin 1 119899 (17)

and119909119894119895isin 0 1 forall 119894 119895 isin 1 119899 (18)


(1) Initialization of initial population(2) repeat(3) Parents selection process(4) Crossover phase(5) Mutation phase(6) Survivor selection process(7) until termination criterion reached(8) Return the fitness of the best individual found

Algorithm 1 Pseudocode of all the GAs

where 119909119894119895in (18) is a binary variable which is 1 if item 119895 is put

in bin 119894 and 119910119894is a variable which is 1 if bin 119894 is used

In this study the solutions of this problem are encoded asa permutation of items To count the number of bins neededfor one solution the size of the items is accumulated in avariable 119904119906119898119878119894119911119890 When 119904119906119898119878119894119911119890 exceeds 119902 the numberof bins is increased in 1 and 119904119906119898S119894119911119890 is restarted Forexample in a simple instance of 10 items every item 119894

119909has

a 119904119894= 119909 and 119902 =15 One possible solution could be 119883 =

(1 3 5)(7)(9 2 4)(6 8)(10) and its fitness would be 5

22 General Description of the Developed Techniques Forthe experiments nine different techniques have been imple-mented and compared The first six techniques (GA

1 GA2

GA3 GA4 GA5 and GA

6) are conventional GAs with

different configurations The remaining three techniques areEAs (EA

1 EA2 and EA

3) The structure used for both

GAs is represented in Algorithm 1 and it is considered theconventional one On the other hand the flowchart of theEAs is the same eliminating the parent selection process andcrossover phase

The parametrization of the GAs has been made basedon the concepts outlined in many previous studies [61ndash63]According to these researches the crossover is consideredthe main operator of genetic algorithms while the mutationis a secondary operation In this way GA

1and GA

2have a

crossover probability (119901119888) of 90 and a mutation probability

(119901119898) of 10 In addition GA

3and GA

4have a 119901

119888= 75

and 119901119898

= 25 Finally GA5and GA

6have 119901

119888= 50 and

119901119898

= 50 On the other hand all the EAs have a 119901119888

=

0 and a 119901119898of 100 For GA

1 GA2 and EA

1 an initial

population composed by 50 randomly created individuals isused Additionally for GA

3 GA4 and EA

2 the population

has 75 individuals Finally for GA5 GA6 and EA

3 a popu-

lation composed by 100 random created individuals is usedIn relation to the parents selection criteria the well-knownbinary tournament criteria has been used Regarding thesurvivor function it is 50 elitist-random (whichmeans thathalf of the population is composed by the best individualsand the remaining ones are selected at random) About theending criteria the execution of each technique finisheswhenthere are 119899 + sum

119899

119896=1119894 generations without improvements in

the best solution found where 119899 is the size of the probleminstance

To perform a rigorous comparison between differ-ent techniques it is appropriate to use neutral operators

throughout the implementation of them In other wordsheuristic operators that use characteristics of the problemand optimize by themselves have to be avoided Otherwiseby using heuristic operators the optimization capacity of thetechnique is influenced by the performance of these opera-tors and it could not be possible to determine objectivelywhich is the real efficiency of the metaheuristic In this paperthis good practice has been followed in order to make a faircomparison

With respect to TSP the well-known 2-opt [64] and theinsertion function (IF) [65] have been used as mutationfunction The first one is a classic operator which randomlyselects two arcs of the solutionThen these edges are removedfrom the route and two new arcs are created avoidingsubtours On the other hand the second operator selectsand extracts one random node of a solution and inserts itin another random position Regarding crossover functionsthe OX [33] order based crossover (OBX) [66] MOX [34]and the half crossover (HX) [67] have been usedThese samemutation and crossover functions have been used for theNQP and BPP

The OX builds the children by choosing a subroute ofone of the parents and maintaining the order of the nodesof the remaining parents First two cut points are randomlyselected identical for both parents and the segments betweenthe cut points are preserved in the children Then startingfrom the second breakpoint the remaining nodes are insertedin the same order they appear in the other parent (startingalso from the second cut point) considering that the nodesthat have already been inserted have to be omitted Whenthe end of the string is reached it continues through thebeginning of this An example of this type of crossover couldbe as follows

1198751= (12345678) 997888rarr 119875

1= (12 | 345 | 678)

997888rarr 1198741= (lowastlowast | 345 | lowast lowast lowast) 997888rarr 119874

1= (87 | 345 | 126)

1198752= (24687531) 997888rarr 119875

2= (24 | 687 | 531)

997888rarr 1198742= (lowast lowast 687 lowast lowastlowast) 997888rarr 119874

2= (45687123)

(19)

In the OBX some random positions are selected in aparent tour The order of the nodes in the selected positionsis imposed on the other parent For example consideringthe same parents (119875

1and 119875

2) and supposing that the second

third and sixth positions are selected the nodes placed in


these positions have to be inserted in the same order in thecorresponding offspring In this case in 119875

2these nodes are 4

6 and 5 and they have to be inserted in the first child in thissame order The rest of the route remains in the same orderand position as in 119875

1

1198751= (12345678) 997888rarr 119874

1= (123 lowast lowast lowast 78)

997888rarr 1198741= (12346578)

(20)

The other child would be the next one considering thatthe nodes in the second third and sixth positions of 119875 are 23 and 6

1198752= (24687531) 997888rarr 119874

2= (lowast4 lowast 875 lowast 1)

997888rarr 1198742= (24387561)

(21)

In the case of MOX a random cut point is selected Thiscutpoint divides each parent into two sections The nodesplaced on the left part of the cut point impose their positionon the other parent Then the remaining nodes are insertedin the children in the same order that they appear in theother parent An example of theworkingway of this crossoverfunction could be as follows

1198751= (1234 | 5678) 997888rarr 119874

1= (lowast2 lowast 4 lowast 6 lowast 8)

997888rarr 1198741= (72543618)

1198752= (2468 | 7531) 997888rarr 119874

2= (24 lowast lowast lowast lowast31)

997888rarr 1198742= (24567831)

(22)

The HX is a particular case of the traditional crossoverin which the cut point is made always in the middle of thepath In this way first a cut is made in the central position ofthe parents Then the order of nodes placed in the left partremains in the same order in the offspring The remainingnodes are added in the same position that they can be foundin the other parent An example of the HX could be shown asfollows

1198751= (1234 | 5678) 997888rarr 119874

1= (1234 lowast lowast lowast lowast)

997888rarr 1198741= (12346875)

1198752= (2468 | 7531) 997888rarr 119874


997888rarr 1198742= (24681357)

(23)

On the other hand for CVRP the implemented crossoverfunctions are the short route crossover (SRX) the randomroute crossover (RRX) and the large route crossover (LRX)These operators are a particular case of the traditionalcrossover in which the cut point ismade always in themiddleof the chromosome The operation of the first of them is thefollowing first of all half of the routes (the shortest ones)of one of the parents is inserted in the child After that thenodes already selected are removed from the other parentand the remaining nodes are inserted in the child in the sameorder (taking into account the vehicle capacity) Assuming

a 17-node instance (including the depot) an example couldbe the following

1198751= (1 2 3 4 0 9 10 11 12 0 13 14 15 16 0 5 6 7 8)

1198752= (1 12 6 3 0 2 4 7 11 0 5 14 16 9 0 8 13 10 15)

(24)

The resulting offprings could be as follows

1198741= (1 2 3 4 0 9 10 11 12 0 6 7 5 14 0 16 8 13 15)

1198742= (1 12 6 3 0 2 4 7 11 0 9 10 13 14 0 15 16 5 8)

(25)

RRX works similar to the SRX In this case the routesselected in the first step of the process are selected randomlyinstead of choosing the best ones Finally in the case ofLRX the selected routes are the longest ones Regarding themutation functions for CVRP the vertex insertion function(VIF) and the swapping function (SF) have been used Thefirst one selects one random node from one randomly chosenroute of the solution This node is extracted and insertedin another randomly selected route respecting the capacityconstraints On the other hand in the swapping function twonodes are selected at random from two random routes toswap their positions respecting also the capacity constraints

In order to make the experimentation more under-standable Table 1 summarizes the characteristics of the ninealgorithms used for all the problems

23 Experimentation Setup In this section the commonaspects in all the experimentations are introduced To beginwith all GA

1 GA2 and EA

1were run on an Intel Core i5 2410

laptop with 230GHz and a RAM of 4GB The rest of thetechniques were executed on an Intel Core i7 3930 com-puter with 320GHz and a RAM of 16GB Java was usedas programming language For every problem 10 differentinstances have been used and for each of them 50 runs havebeen executed For each experimentation the average resultsaverage runtime (in seconds) and convergence behaviourof every technique are shown In addition the standarddeviation of each of them is also shown Furthermore forevery problem three different experimentations have beenperformed In each experimentation the performance ofone EA is compared with the one of two different GAsThe three experimentations differ in the configuration of thetechniques

Additionally in order to make a fair and rigorous com-parison the normal distribution 119911-test has been performedfor all experiments Thanks to this statistical test it can beshown whether the differences in the results obtained byeach technique are significant or not The 119911 statistic has thefollowing form

119911 =

119883EA minus 119883GA

radic(120590EA119899EA) + (120590GA119899GA) (26)

where 119883EA is the average of an EA 120590EA is the standarddeviation of an EA119883GAis the average of the other technique


Table 1 Summary of the characteristics of all the techniques developed

Alg Pop 119901119888

119901119898

Crossover function(TSP BPP NQP)

Mutation function(TSP BPP NQP)

Crossover function(CVRP)

Mutation function(CVRP)

GA1

50 90 10 OX 2-opt SRX VIFGA2

50 90 10 OBX 2-opt RRX VIFEA1

50 0 100 No cross 2-opt No cross VIFGA3

75 75 25 HX IF LRX SFGA4

75 75 25 MOX IF SRX SFEA2

75 0 100 No cross IF No cross SFGA5

100 50 50 OBX 2-opt RRX VIFGA6

100 50 50 OX 2-opt LRX VIFEA3

100 0 100 No cross 2-opt No cross VIF

120590GAis the standard deviation of the other technique 119899EA is thesample size for an EA and 119899GA is the sample size for the othertechnique

The 119911 value can be positive (+) neutral (lowast) or nega-tive (minus) The positive value of 119911 indicates that the EA issignificantly better In the opposite case the EA obtainssubstantially worse solutions If 119911 is neutral the differenceis not significant The confidence interval has been stated at95 (119911

005= 196) Besides showing the symbolic value of

119911 its numerical value is also displayed Thus the differencein results may be seen more easily Finally as it has beenmentioned that the 119911-test has been performed for the resultsquality runtime and convergence behaviour

3 Experimentation with the TSP

In this section the experimentationwith the TSP is shownAllthe instances have been picked from the well-known TSPLIBbenchmark [68] In Table 2 the results and average runtimescan be found On the other hand in Table 3 the convergencebehaviour of each technique is displayed For this purposethe average number of generations needed to reach the finalsolution is used In Table 4 the results of the 119911-test are shown

Several conclusions can be drawn by analyzing the resultsshown First of all looking at Table 2 it can be seen thatfor the three experimentations all the EAs perform betterthan the other two techniques in all the instances Accordingto Table 4 in the first experimentation these differencesare significant only in two cases compared to GA

1 On the

other hand these improvements are significant in all butone instance respect to GA

2 In the second experimentation

the EA2gets significantly better results in all the instances

compared with the GA3and in nine instances (out of ten)

compared with GA4 Finally for the last experimentation

the EA3significantly outperforms GA

5in the 100 of the

instances and in the 60 (6 out of 10) regarding GAV6For this reason taking into account that EAs never getsworse results than the other two alternatives in the threeexperiments the following conclusion can be stated

Conclusion 1 According to the experimentation performedthe use of blind crossover operators in genetic algorithms

does not offer significant improvements in the results for theTSP

This conclusion could be explained in the followingway The main purpose of the crossover phase is to obtainnew individuals making combinations of the existing onesAlthough these operations were designed for the exploitationof the solution space several studies in the literature discussthis fact [39 63] On the other hand as it has been shownin several works before [69 70] blind crossovers betweendifferent individuals can be useful to make large jumpsalong the solution space For this reason blind crossoveroperators applied to the TSP contribute to increase theexploration capability of the algorithm instead of helping theexploitation

This way it could be said that for the TSP using blindcrossovers helps a broad exploration of the solution spacebut does not help to make an exhaustive search of promisingregions This is so because it is improbable that the resultingoffspring from blind crossovers can improve their parentsIn addition this fact is accentuated when the execution isnear to the convergence To get a deeper search the existenceof a function that makes little jumps in the solution spacebecomes necessary The mutation function can handle thisgoal and it can also contribute to perform a broad search ofpromising regions [71 72] Thus an EA can conduct a deepand wide search obtaining similar (or better) results to theGAs

Regarding the runtimes the EAs also outperform theircorresponding algorithms in all the instances and experi-mentations In addition in this case these improvements aresignificant in all of the cases Besides this the differences inthe runtimes become wider as the size of the instance growsThis is particularly important in real-time applications wherethe runtime is a key factor For these reasons the followingconclusion can be deduced

Conclusion 2 In relation to the experimentation performedthe use of blind crossover operators increases significantly theexecution time of an evolutionary algorithm applied to theTSP

This difference in runtime between the GAs and the EAscan be easily explained in the same manner as explained


Table 2 Results and runtimes of the nine techniques applied to the TSP For each instance the results average runtime and their standarddeviations are shown

TSP GA1

GA2

EA1

Instance Results Time (s) Results Time (s) Results Time (s)Instance Optimum Avg Std Avg Std Avg Std Avg Std Avg Std Avg StdSt70 675 7111 255 89 23 7144 183 46 13 7050 139 15 05Eilon75 535 5747 118 122 31 5809 140 75 22 5701 109 22 06Eil76 538 5756 114 130 30 5861 131 67 19 5740 112 20 07KroA100 21282 221295 5575 225 60 223765 5465 143 43 221170 4542 39 08KroB100 22140 231332 5613 245 56 233325 4209 131 40 230981 4016 40 08KroC100 20749 218225 7067 211 34 219244 4792 154 44 216425 5447 41 12KroD100 21294 223479 5732 244 75 225502 4635 159 52 222398 3834 42 10Eil101 629 6801 113 426 99 6858 132 228 65 6800 92 44 09Pr107 44303 462822 15289 364 137 464705 14012 231 79 455878 9364 58 18Pr124 59030 604076 7222 470 110 606783 11701 264 65 603846 9278 75 12

Instance GA3

GA4 EA2

St70 675 7446 219 36 11 7252 205 34 08 7130 109 05 01Eilon75 535 6042 266 46 11 6039 165 49 12 5796 141 07 01Eil76 538 6195 199 48 09 5975 261 54 11 5837 85 07 01KroA100 21282 224164 5184 133 30 223756 5335 100 26 222020 5394 17 03KroB100 22140 234254 4217 118 26 235426 6121 101 19 230242 4589 17 03KroC100 20749 223040 6349 118 25 223021 7337 102 33 215391 4683 18 03KroD100 21294 225923 4344 130 26 227978 6296 89 21 223708 5253 16 02Eil101 629 7189 176 164 37 7127 153 177 39 6874 111 16 03Pr107 44303 468109 11007 171 31 466612 12427 132 34 453194 6945 24 05Pr124 59030 614215 15009 271 61 611481 12862 180 30 603806 6698 36 05

Instance GA5 GA6 EA3


in the previous works [28] comparing the working wayof the crossover and mutation operators the former arecomplex operations in which two individuals combine theircharacteristics On the other hand a mutation is a smallmodification of a chromosome and requires considerably lesstime than the previous ones Thereby the fact that an EAsubstitutes the crossover phase in exchange for performingmore mutations is perfectly reflected in runtime giving agreat advantage to an EA in this aspect

Finally if the data presented in Table 3 is analyzed firstit can be seen that both GA

1and GA

2present a better

convergence behaviour compared to EA1 More specifically

GA1is better than EA

1in the 80 of the cases and GA

2

in all but one In addition comparing with the EA1 these

differences are significantly better for the GA1in 60 of

the instances while in 30 they are not significant In

the remaining cases the differences are substantially betterfor the EA

1 Regarding GA

2 these data are respectively

60 40 and 0 Regarding the second experimentationGA4shows a significantly better convergence behaviour than

EA2in the 100 of the instances On the other hand the

GA3outperforms EA

2in the 60 on the cases with these

differences being significant in four instances (out of 10) Bythe way EA

2significantly outperforms GA

3in two instances

Finally regarding the last experimentation the GA5and GA

6

present a substantially better convergence in the 90and 80of the instances respectively In the remaining instances theEA3shows a nonsignificant better performance Taken into

account all these data the following conclusion can be drawn

Conclusion 3 Considering these tests conducted for theTSP the algorithms that use blind crossover operators


Table 3 Convergence behaviour of the nine techniques applied to the TSP

TSP GA1

GA2

EA1

Instance Avg Std Avg Std Avg StdSt70 60931 15305 55907 21920 61626 15305Eilon75 79203 27156 72794 27615 84395 58529Eil76 82488 26631 66359 24812 74610 19005KroA100 95685 35492 99805 38310 123457 24044KroB100 104199 31588 100906 36554 137751 35944KroC100 92249 38539 96867 33646 136140 36234KroD100 94952 37366 99011 39190 130860 38558Eil101 186462 51444 152095 54940 150034 39362Pr107 131153 68589 124890 57375 186839 67956Pr124 136623 48512 110337 43032 189177 42398Instance GA

3GA4 EA2

St70 44002 14468 24316 7706 38957 6778Eilon75 48681 13420 31235 9405 45750 47125Eil76 49545 10988 33526 9227 47125 12693KroA100 83822 22066 77148 32580 86828 24412KroB100 73418 18633 65166 25938 90870 18799KroC100 83040 10349 31002 12992 98248 18813KroD100 81830 18862 70059 27968 87984 14855Eil101 102411 25637 72609 18766 87449 20623Pr107 89863 20213 45406 14555 127414 33437Pr124 118806 23047 94625 36872 152585 28771Instance GA5 GA6 EA3

St70 41885 15032 47488 15006 61342 11360Eilon75 57926 22908 70205 19527 86314 27277Eil76 76181 24252 76376 32744 75210 20568KroA100 33169 22450 70325 48831 118178 25745KroB100 54918 41516 59504 33187 116198 23215KroC100 39289 25973 43602 21364 128176 33638KroD100 64947 38382 64849 35666 112161 24348Eil101 107185 28773 145003 51006 144501 29389Pr107 67530 49313 127752 118068 164362 51612Pr124 63877 33422 82511 49147 180226 36100

demonstrate a better convergence behaviour needing lessgenerations to find their final solution

This improvement in the convergence behaviour can beexplained as follows As mentioned above blind crossoveroperators can be a great help to make a broad exploration ofthe solution space Comparing with the mutation functionsa blind crossover can make more sudden jumps in thesolution space On the other hand mutations are simpleoperations which move along the solution space little bylittle conducting small jumps For this reason and dependingon the problem complexity with the crossover functions abroader and faster exploration can be made and the finalsolution can be found in less generations

Furthermore as has been mentioned above mutationsare an excellent option to explore the solution space Inaddition as can be seen in the results shown in Table 2mutations can also take care of the exploitation capacity of

the technique So using them similar (or better) solutionscan be found

In conclusion all the GAs converge faster than theircorresponding EA Thus comparing with the EAs all theversions of the GA need less generations to reach thefinal solution Anyway this fact does not mean a betterperformance As can be seen in the results presented the EAsobtain similar or significantly better results for all the TSPinstances (needing a substantially smaller runtime)

4 Experimentation with the CVRP

In this section the experimentation with the CVRP is dis-played In this case instances have been picked from theCVRP set of Christofides and Eilon (httpneolccumaesvrp (Last update January 2013)) In Table 5 the results andaverage runtime can be found Moreover the convergence


Table 4 119911-test for TSP ldquo+rdquo indicates that EA is better ldquominusrdquo depicts that it is worse ldquolowastrdquo indicates that the difference between the two algorithmsis not significant (at 95 confidence level)

TSP EA1 versus GA1 EA1 versus GA2

Instance Results Convergence Time Results Convergence TimeSt70 lowast (146) lowast (minus019) + (2161) + (476) lowast (minus151) + (1499)Eilon75 + (200) lowast (minus056) + (2202) + (430) lowast (minus126) + (1653)Eil76 lowast (069) lowast (170) + (2485) + (493) lowast (minus186) + (1653)KroA100 lowast (012) minus (minus458) + (2172) + (258) minus (minus369) + (1648)KroB100 lowast (036) minus (minus495) + (2563) + (284) minus (minus508) + (1557)KroC100 lowast (142) minus (minus586) + (2895) + (274) minus (minus561) + (1757)KroD100 lowast (110) minus (minus472) + (1889) + (364) minus (minus409) + (1573)Eil101 lowast (005) + (397) + (2694) + (256) lowast (021) + (1956)Pr107 + (273) minus (minus407) + (1556) + (370) minus (minus492) + (1490)Pr124 lowast (013) minus (minus576) + (2515) lowast (139) minus (minus992) + (2012)Instance EA2 versus GA3 EA2 versus GA4

St70 + (913) + (223) + (2543) + (371) minus (minus1008) + (1984)Eilon75 + (577) lowast (042) + (2466) + (791) minus (minus213) + (2496)Eil76 + (1161) lowast (101) + (3008) + (355) minus (minus612) + (3201)KroA100 + (202) lowast (minus064) + (2242) lowast (161) lowast (minus168) + (2720)KroB100 + (455) minus (minus466) + (3087) + (455) minus (minus560) + (2728)KroC100 + (685) minus (minus500) + (1792) + (619) minus (minus2079) + (2808)KroD100 + (229) lowast (minus181) + (2446) + (368) minus (minus400) + (3091)Eil101 + (1070) + (310) + (2910) + (946) minus (minus389) + (2819)Pr107 + (810) minus (minus676) + (2222) + (666) minus (minus1590) + (3310)Pr124 + (447) minus (minus647) + (3347) + (374) minus (minus876) + (2714)Instance EA3 versus GA5 EA3 versus GA5

St70 + (342) minus (minus730) + (1677) + (323) minus (minus520) + (2496)Eilon75 + (696) minus (minus563) + (1185) + (411) minus (minus339) + (2538)Eil76 + (398) lowast (021) + (1257) lowast (158) lowast (021) + (1743)KroA100 + (621) minus (minus1759) + (831) + (457) minus (minus612) + (127)KroB100 + (174) minus (minus910) + (845) lowast (185) minus (minus989) + (1636)KroC100 + (536) minus (minus1478) + (789) + (291) minus (minus1500) + (1928)KroD100 + (474) minus (minus734) + (1121) lowast (155) minus (minus774) + (1638)Eil101 + (803) minus (minus641) + (2298) + (644) lowast (006) + (2785)Pr107 + (616) minus (minus959) + (906) + (232) minus (minus200) + (1537)Pr124 + (344) minus (minus1672) + (1446) lowast (044) minus (minus1133) + (1677)

behaviour is shown in Table 6 Finally Table 7 displays thestatistical 119911-test performed for the CVRP

The conclusions that can be drawn looking at these tablesare similar to thosementioned in the previous section In thiscase regarding the quality of the results and according tothe data shown in Table 5 EA

1outperforms GA

1in 80 of

the instances and GA2in all of them In addition looking

at Table 7 these improvements are significant in the 60 ofthe cases compared to GA

1 On the other hand 30 the

differences are not significant and in the remaining onesEA1gets substantially worse results Regarding GA

2 these

percentages are respectively 90 10 and 0Furthermore EA

2performs better than GA

3in the

90 of the instances and GA4in the 80 In the case of

GA3 the EA

2obtains significantly better results in nine

instances In the remaining instance GA3outperforms EA

2

but not substantially Moreover EA2improves significantly

GA4in the 50 of the instances In addition in the 40

these improvements are not substantially Additionally in theremaining instances EA

2gets significantly worse results

Finally regarding the third experimentation EA3outper-

forms GA5and GA

6in 80 of the cases In addition these

improvements are significant in the 60 of the instancesregarding both versions of the GAs On the other hand EA

3

gets worse results in the 20 of the instances in relation tobothGAs but these differences are not substantial in any case

With all this the following finding can be statedConclusion 4 According to the tests conducted for theCVRPthe use of blind crossover operators does not offer significantimprovements in the results


Table 5 Results and runtime of the nine techniques applied to the CVRP For each instance the results average runtime and their standarddeviations are shown

CVRP GA1 GA2 EA1

Instance Results Time (s) Results Time (s) Results Time (s)Instance Optimum Avg Std Avg Std Avg Std Avg Std Avg Std Avg StdEn22k4 375 3890 98 18 05 4109 232 25 11 4048 195 11 03En23k3 569 6227 289 21 09 6299 416 23 10 6027 308 16 06En30k3 534 5596 292 39 12 5827 433 50 20 5458 416 20 07En33k4 835 9074 319 60 18 9328 306 70 23 9119 249 22 07En51k5 521 6410 383 138 54 6943 534 182 79 6284 374 45 14En76k7 682 8500 457 444 164 8995 633 551 165 8223 429 100 33En76k8 735 9206 593 409 191 9522 446 523 175 8869 376 83 28En76k14 1021 11869 356 334 140 12196 474 381 126 11710 362 65 22En101k8 815 10614 548 1075 339 11109 716 1263 353 10167 499 157 51Pr101k14 1071 13200 465 881 296 13707 731 1149 343 12706 414 148 46


En22k4 375 3880 148 16 04 3861 103 23 04 3928 139 08 01En23k3 569 6225 311 27 11 6157 379 25 12 6018 384 09 02En30k3 534 6081 580 33 13 5576 183 40 10 5470 289 14 04En33k4 835 9170 249 34 13 9013 292 31 10 9034 237 12 04En51k5 521 7160 501 86 28 6317 343 85 32 6239 311 24 09En76k7 682 8478 485 351 135 8354 563 260 108 8096 408 48 15En76k8 735 9148 544 324 139 8952 379 244 70 8702 544 51 15En76k14 1021 11989 461 243 87 11888 451 338 105 11679 288 45 19En101k8 815 10342 578 869 243 10216 729 672 260 10070 494 80 21Pr101k14 1071 13098 510 756 167 12885 453 593 250 12532 365 87 24



This conclusion can be explained in the same waythat Conclusion 1 was explained in Section 3 Regardingthe runtime as in TSP all the EAs need less time thantheir corresponding GAs in all the instances with theseimprovements being significant in all of the cases for thefirst two experimentations In the third experimentationthe differences are substantial in the 90 of the instancesIn addition as in the previous problem these differencesbecome higher as the size of the instance grows For thisreason the following conclusion can be deducedConclusion 5 In the same way as with the TSP the useof crossover phase for the CVRP increases significantly theexecution time of an evolutionary algorithm

The reasons of this increase in the runtime are thesame as those explained in the previous section for the TSP

Anyway regarding the convergence behaviour the resultsdisplayed in Table 6 are different in relation to the previouslystudied problem Analyzing these outcomes it can beobserved how the EAs show better convergence behaviourin all the instances and experimentations Additionally theseimprovements are significant in 80 of the cases comparedto GA

2and GA

3 in 70 regarding GA

2 GA4 and GA

6 and

in 60 compared to GA5 This means that the EAs reach

the final solution in less generations than the other alter-natives The following finding can be extracted from theseobservations

Conclusion 6 Contrary to what happens for the TSP andaccording to the experimentation conducted the use ofblind crossover operators does not improve the convergencebehaviour of an evolutionary algorithm applied to the CVRP


Table 6 Convergence behaviour of the nine techniques applied to the CVRP

CVRP GA1 GA2 EA1

Instance Avg Std Avg Std Avg StdEn22k4 30209 22167 50993 42736 23587 18749En23k3 67171 50621 61623 43033 58139 37713En30k3 93925 45833 92048 49633 79260 48729En33k4 110421 47438 116284 54545 46140 31047En51k5 158483 69917 184535 91832 103874 48167En76k7 314208 130442 392203 144444 193577 76019En76k8 274601 143261 366477 143856 160326 71978En76k14 200421 104357 230841 89480 121338 61459En101k8 515259 173935 556278 154264 259250 87834Pr101k14 398349 144428 473961 146566 212766 69978Instance GA3 GA4 EA2

En22k4 32270 22866 25510 13954 23529 13845En23k3 83413 44951 55190 45859 41289 27406En30k3 78377 51423 78069 31149 76680 33858En33k4 65634 43336 69193 37607 46063 31693En51k5 104720 50025 142263 73162 97271 50627En76k7 279190 125217 258639 123699 193854 72861En76k8 261782 134427 232498 79095 190270 72348En76k14 164985 91902 164642 70820 113107 68864En101k8 482199 150137 421154 175723 275958 84801Pr101k14 388123 101299 338828 160918 238782 79606Instance GA5 GA6 EA3

En22k4 23685 14645 21750 19892 15545 13132En23k3 65434 40604 76326 54868 63008 29793En30k3 81219 48062 87079 59873 79772 58204En33k4 75862 45551 71070 42420 49429 19170En51k5 103220 71183 116192 56730 90139 46905En76k7 218572 88622 233121 92144 156880 93290En76k8 195073 69895 190864 74024 149419 48498En76k14 129454 76924 149559 67307 104770 59716En101k8 442020 166884 429676 115106 273133 90493Pr101k14 235471 92079 242058 120408 177001 72410

This change in the behavior of the EA compared to thatobserved for the previous problems can be justified as followsCrossover operators are complex functions that combine thecharacteristics of two individuals of the population Thesefunctions are easy to design and implement if the problemhas not many constraints (eg TSP and NQP) Anyway ifthe problem has a complex formulation or its restrictionsare numerous the development of a crossover function canbe very hard For this reason many operators designed forthis type of problems include problem dependent heuristics[73 74] or they do not consider some of the constraints of theproblem [75 76] In any case these operators are difficult toimplement and understand and they increase considerablythe complexity of the algorithm and its runtime

Thus blind operators are rarely used in solving thesecomplex problems In addition their performance is usuallynot good An evidence of this last statement is shown inthis study all GA techniques that prioritize the use of blind

crossover operators are outperformed by the technique thatgives more importance to the mutation phase in terms ofexploration and exploitation

5 Experimentation with the NQP

In this section the experimentation with the NQP is detailedThe characteristics of the nine techniques implemented arethe same as the algorithms used for the TSP In Table 8 theresults and average runtime can be found The name of eachinstance describes the number of queens and the size of thechessboard In this case the optimum of each instance isnot shown since it is known that it is 0 for all of themIn addition Table 9 displays the convergence behaviour ofeach algorithm On the other hand the 119911-test made for thisproblems is shown in Table 10

The conclusions that can be drawn analyzing these tablesare similar to those obtained in previous sections First


Table 7 119911-test for CVRP ldquo+rdquo indicates that EA is better ldquominusrdquo depicts that it is worse ldquolowastrdquo indicates that the difference between the twoalgorithms is not significant (at 95 confidence level)

CVRP EA1 versus GA1 EA1 versus GA2

Instance Results Convergence Time Results Convergence TimeEn22k4 minus (minus509) lowast (161) + (796) lowast (141) + (415) + (864)En23k3 + (335) lowast (101) + (276) + (371) lowast (043) + (399)En30k3 lowast (191) lowast (155) + (945) + (433) lowast (130) + (988)En33k4 lowast (minus078) + (801) + (1324) + (373) + (790) + (1369)En51k5 lowast (165) + (454) + (1163) + (714) + (550) + (1209)En76k7 + (312) + (564) + (1447) + (714) + (860) + (1890)En76k8 + (342) + (503) + (1189) + (790) + (906) + (1745)En76k14 + (220) + (461) + (1336) + (576) + (713) + (1744)En101k8 + (426) + (929) + (1886) + (763) + (1183) + (2188)En101k14 + (560) + (817) + (1726) + (842) + (1137) + (2043)Instance EA2 versus GA3 EA2 versus GA4

En22k4 lowast (minus167) + (231) + (1371) minus (minus273) lowast (071) + (2572)En23k3 + (296) + (565) + (1138) lowast (183) lowast (183) + (929)En30k3 + (666) lowast (019) + (987) + (219) lowast (021) + (1706)En33k4 + (280) + (257) + (1143) lowast (minus033) + (332) + (1247)En51k5 + (1104) lowast (074) + (1490) lowast (119) + (357) + (1297)En76k7 + (426) + (416) + (1577) + (262) + (319) + (1374)En76k8 + (409) + (331) + (1380) + (266) + (278) + (1906)En76k14 + (403) + (319) + (1572) + (276) + (368) + (1941)En101k8 + (252) + (845) + (2287) lowast (117) + (526) + (1604)En101k14 + (638) + (819) + (2803) + (429) + (394) + (1424)Instance EA3 versus GA5 EA3 versus GA6

En22k4 + (208) + (292) + (392) + (439) lowast (184) + (316)En23k3 lowast (minus118) lowast (034) lowast (187) lowast (minus062) lowast (150) lowast (147)En30k3 lowast (010) lowast (013) + (626) + (302) lowast (061) + (702)En33k4 + (233) + (378) + (1086) lowast (072) + (328) + (1135)En51k5 + (223) lowast (108) + (538) + (329) + (250) + (869)En76k7 + (247) + (339) + (1230) lowast (096) + (411) + (1463)En76k8 + (198) + (379) + (1554) + (215) + (331) + (1466)En76k14 lowast (132) lowast (179) + (921) lowast (minus119) + (351) + (1306)En101k8 lowast (minus047) + (629) + (1676) + (217) + (756) + (1754)En101k14 + (225) + (352) + (1535) + (344) + (327) + (1285)

of all as can be seen in Table 8 the EAs obtain betterresults than their corresponding GAs in all but one of theinstances In the remaining case (8-queens instance) theyget the same outcomes In addition these improvements aresignificant in 90 of the instances compared to GA

1 GA2

GA3 GA4 and GA

6 with the 8-queens instance being the

only where the differences are not significant Additionallythese improvements are substantial in the 80 of the casesregarding GA

5 being not significant in the remaining 20

For these reasons Conclusions 1 and 4 are also applicable forthe NQP

The same happens with runtime The EAs are neverovercomed by any of the genetic algorithms used obtain-ing significantly better runtimes in 90 and 60 of casesregarding GA

1and GA

2 in 80 of the instances compared

to GA3and GA

4 and in 60 and 80 in relation to GA

5

andGA6 respectivelyTherefore Conclusions 2 and 5 are also

applicable for this problemFinally regarding the convergence behaviour the results

obtained are more similar to those seen for the TSP Lookingat the data displayed in Table 8 the EA

1has a better

convergence behaviour in 40 of the instances and the GA1

and GA2in the other 60 According to Table 10 comparing

to GA1 the differences in the results are significantly better

for the EA1in 20 of the instances and significantly worse

in 30 of them In the remaining cases the differences arenot substantial On the other hand comparing to GA

2 these

percentages are respectively 30 20 and 50Regarding the second experimentation the EA

2gets

a better convergence compared to GA3and GA

4in the

40 of the instances In the remaining 60 the EA2has

been overcomed by at least one of the GAs Regarding


Table 8 Results and runtime of the nine techniques applied to the NQP For each instance the results average runtime and their standarddeviations are shown

NQP GA1 GA2 EA1

Instance Results Time (s) Results Time (s) Results Time (s)Instance Avg Std Avg Std Avg Std Avg Std Avg Std Avg Std8-queens 00 00 01 00 01 02 01 00 00 00 01 0020-queens 16 08 01 01 15 07 01 01 08 05 01 0050-queens 66 16 06 01 64 16 03 01 51 14 03 0175-queens 137 22 08 03 131 25 07 04 92 23 06 01100-queens 154 23 62 15 152 26 47 13 115 23 29 07125-queens 255 34 52 15 243 36 39 12 170 31 36 08150-queens 320 39 95 34 277 39 76 22 219 32 66 14200-queens 432 59 699 79 382 45 380 81 266 39 325 79250-queens 564 71 638 198 521 52 455 125 380 53 425 107300-queens 699 79 1233 413 652 65 1095 256 456 53 946 193Instance GA3 GA4 EA2

8-queens 00 00 01 01 00 00 01 01 00 00 01 0120-queens 14 10 01 01 13 08 01 01 08 06 01 0150-queens 59 18 02 01 56 13 02 01 46 15 01 0175- queens 109 21 07 01 100 25 08 01 87 16 05 01100-queens 147 33 22 06 153 28 18 05 121 20 15 03125-queens 198 29 42 11 183 27 48 11 172 25 31 05150-queens 237 37 81 27 222 32 93 20 213 30 58 10200-queens 333 44 267 72 304 43 271 61 269 48 187 40250-queens 435 56 526 120 416 52 564 131 371 45 448 91300-queens 578 57 986 336 504 65 1186 285 459 49 776 197Instance GA5 GA6 EA3

8-queens 00 00 01 01 00 00 01 00 00 00 01 0020-queens 13 06 01 01 11 05 01 01 08 06 01 0150-queens 52 16 02 01 49 12 02 01 42 14 01 0175-queens 100 20 09 01 87 19 08 01 76 23 06 01100-queens 127 27 26 03 134 26 25 06 118 21 21 04125-queens 178 21 63 09 156 31 51 10 144 27 47 10150-queens 212 43 82 27 212 27 86 19 195 33 77 16200-queens 303 35 286 39 305 38 258 53 270 42 229 50250-queens 369 37 591 116 362 30 625 109 321 41 528 100300-queens 467 70 935 219 469 46 1113 272 425 66 897 168

the GA3 the differences are not significant in the 60 of

the cases In addition the EA2has showed a substantial

better convergence behaviour in 30 of the instances In theremaining 10 the GA

3has significantly outperformed the

behaviour of EA2 On the other hand compared to GA

4

these percentages are different being 50 10 and 40respectively

In relation to the third experimentation the EA3has

shown a better convergence than GA5and GA

6in the 20 of

the cases being overcomed in the remaining 80 Comparedto GA

5 the difference in the behaviour is not significant

in the 70 of the cases Furthermore they are substantiallybetter for the GA in the remaining 30 On the otherhand the EA

3has significantly improved the convergence

of GA6in the 10 of the instances In addition in the 40

of the cases the differences are not substantial Ultimatelyin the remaining 50 GA

6has shown a significant better

convergence behaviourFor this reason the following finding can be drawn

Conclusion 7 According to the tests conducted the use ofblind crossover operators in the development of geneticalgorithms for the NQP entails an improvement in theconvergence behavior of the technique

The NQP is a problem with a simple formulation Forthis reason the convergence behaviour of the GAs is muchbetter than the one shown for the CVRP since the crossoverphase helps the exploration capacity of the technique In thisway the results obtained in this aspect are similar to thoseobtained for the TSP


Table 9 Convergence behaviour of the nine techniques applied to the NQP

NQP GA1 GA2 EA1

Instance Avg Std Avg Std Avg Std8-queens 33 32 49 46 28 2920-queens 369 262 379 283 391 21850-queens 2100 1266 1913 1123 1519 71475-queens 1834 988 1956 935 2248 890100-queens 8180 3857 7917 3337 5758 2558125-queens 5891 2026 5997 2178 5789 1739150-queens 6360 2909 7881 2991 7231 2091200-queens 11816 4173 15607 5636 18541 6063250-queens 16497 6159 17172 5643 18536 5676300-queens 22794 8973 24027 8435 28217 6833Instance GA3 GA4 EA2

8-queens 30 18 21 13 18 1220-queens 185 103 182 89 239 10950-queens 1280 543 1312 484 1160 38175-queens 2074 820 1766 657 2107 743100-queens 4163 2089 2527 1123 3467 1059125-queens 5117 1870 4511 1510 4624 1058150-queens 7125 2920 6541 1908 6141 1591200-queens 13631 4585 13510 3454 12085 3306250-queens 17142 4801 14615 4332 18273 4324300-queens 24658 9481 22221 6362 22508 6753Instance GA5 GA6 EA3

8-queens 16 12 23 15 14 1120-queens 212 91 217 68 211 10850-queens 892 424 800 314 916 38175-queens 1543 651 1590 553 1723 658100-queens 2408 894 3154 1242 3252 1029125-queens 3298 988 4203 1271 5695 1459150-queens 5902 2540 4844 1640 6728 1994200-queens 8470 2375 9036 2501 11937 3376250-queens 13442 3990 14915 3297 14701 3606300-queens 19745 9751 18296 5673 21370 4929

6 Experimentation with the BPP

In this section the experimentation with the BPP is shownThe characteristics of the nine techniques developed are thesame as the ones used for the TSP In Table 11 the results andaverage runtime can be found Each instance has been pickedfrom the SchollKlein benchmark (httpwwwwiwiuni-jenadeentscheidungbinppindexhtm) These cases arenamed 119873119909119862119910119882119911 119886 where 119909 is 2 (100 items) 3 (200 items)or 4 (500 items) 119910 is 1 (capacity of 100) 2 (capacity of 120)and 3 (capacity of 150) 119911 is 1 (items size between 1 and100) and 2 (items size between 20 and 100) 119886 is A or B asbenchmark indexing parameter Additionally Table 12 showsthe convergence behaviour of each technique Furthermorethe 119911-test made for the BPP is shown in Table 13

The conclusions that can be obtained in this case are verysimilar to those drawn for theNQP As can be seen in Table 11the EAs obtain better or same (in two cases only) results in the

100 of the instances being significantly better in the 90 ofthe cases Therefore Conclusions 1 and 4 can be also appliedfor this problem Regarding runtimes as already seen inthe previous experimentations all the EAs outperform theircorresponding GAs In this case the EAs obtain significantlybetter runtimes in the 100 of the instances In this wayConclusions 2 and 5 are also valid for the BPP

Concerning the convergence behavior the resultsobtained are similar to those obtained for the NQP The EAshave a better convergence in the 4333 cases (13 out of 30)while the GAs perform better in the remaining 5667 Inaddition comparing to GA

1 the differences are significantly

better for the EA1in 10 (1 out of 10) of the cases and

significantly worse in 20 (2 out of 10) In the remaining 7instances these differences are insignificant Furthermoreregarding GA

2 these percentages are 30 0 and 70

respectively In relation to the second experimentation theEA2shows a substantial better behaviour in 10 of the


Table 10 119911-test forNQP ldquo+rdquo indicates that EA is better ldquominusrdquo depicts that it is worse ldquolowastrdquo indicates that the difference between the two algorithmsis not significant (at 95 confidence level)

NQP EA1 versus GA1 EA1 versus GA2

Instance Results Convergence Time Results Convergence Time8-queens lowast (000) lowast (080) lowast (000) lowast (141) + (270) lowast (000)20-queens + (508) lowast (minus057) + (1500) + (507) lowast (minus075) + (632)50-queens + (476) + (284) + (1000) + (403) + (210) + (216)75-queens + (960) minus (minus219) + (414) + (788) lowast (minus158) lowast (106)100-queens + (823) + (369) + (1304) + (737) + (363) + (782)125-queens + (1298) lowast (026) + (630) + (1085) lowast (052) lowast (110)150-queens + (1386) lowast (171) + (540) + (1386) lowast (125) + (261)200-queens + (1648) minus (minus645) + (2356) + (1364) minus (minus250) + (339)250-queens + (1454) lowast (minus172) + (667) + (1454) lowast (120) lowast (129)300-queens + (1806) minus (minus339) + (444) + (1647) minus (minus272) + (327)Instance EA2 versus GA3 EA2 versus GA4

8-queens lowast (000) + (392) lowast (000) lowast (000) lowast (119) lowast (000)20-queens + (363) minus (minus254) lowast (000) + (353) minus (minus281) lowast (000)50-queens + (393) lowast (127) + (500) + (356) lowast (174) + (500)75-queens + (589) lowast (minus021) + (1000) + (309) minus (minus243) + (1500)100-queens + (476) + (209) + (737) + (657) minus (minus430) + (363)125-queens + (480) lowast (161) + (643) + (211) lowast (minus043) + (994)150-queens + (356) + (209) + (564) + (145) lowast (113) + (1106)200-queens + (694) lowast (193) + (686) + (384) + (210) + (814)250-queens + (629) lowast (minus123) + (366) + (462) minus (422) + (514)300-queens + (1119) lowast (130) + (381) + (392) lowast (minus021) + (836)Instance EA3 versus GA5 EA3 versus GA6

8-queens lowast (000) lowast (086) lowast (000) lowast (000) + (342) lowast (000)20-queens + (416) lowast (005) lowast (000) + (271) lowast (033) lowast (000)50-queens + (332) lowast (minus029) + (500) + (268) lowast (minus166) + (500)75-queens + (556) lowast (minus137) + (1500) + (260) lowast (minus109) + (1000)100-queens lowast (186) minus (minus437) + (707) + (338) minus (minus042) + (392)125-queens + (702) minus (minus961) + (840) + (206) minus (minus545) + (200)150-queens + (221) lowast (minus180) lowast (112) + (281) minus (minus515) + (256)200-queens + (426) minus (minus593) + (635) + (436) minus (minus488) + (281)250-queens + (614) lowast (minus165) + (290) + (570) lowast (030) + (463)300-queens + (306) lowast (minus105) lowast (097) + (386) minus (minus289) + (477)

instances and substantially worse behaviour in 25 In therest of the instances the differences are not substantialFinally for the third experimentation these percentages arerespectively 20 40 and 40Thereby looking at Table 13it can be said that Conclusion 7 is also applicable for the BPP

7 Conclusions and Further Work

In this paper a study on the influence of using blind crossoveroperators in genetic algorithms applied to combinatorialoptimization problem has been conducted For this pur-pose four different well-known combinatorial optimizationproblems have been used the traveling salesman problem(TSP) the capacitated vehicle routing problem (CVRP)the N-queens problems (NQP) and the one-dimensionalbin packing problem (BPP) For each problem 10 differentinstances have been selected making a total set of 40 cases

In the experimentation done the performance of six classicgenetic algorithms each with a different crossover functionhas been compared with the one of the three evolutionaryalgorithms

In general regarding the results the EAs obtain betterresults in 9416 of the cases (113 out of 120) In additioncomparing with the GA variants these improvements aresignificant in the 8125of the cases (195 out of 240) In 1791of the cases (43 out of 240) these differences are insignificantand in the remaining 084 (2 out of 240) one GA obtainssubstantially better results than its corresponding EA Forthese reasons we have the followingConclusion 8 Regarding the results and applicability to theexperimentation performed it is concluded that the use ofblind crossover operators in genetic algorithms for solvingcombinatorial optimization problems provides no significantimprovement in the results


Table 11 Results and runtimes of the nine techniques applied to the BPP For each instance the results average runtime and their standarddeviations are shown

BPP GA1 GA2 EA1

Instance Results Time (s) Results Time (s) Results Time (s)Instance Optimum Avg Std Avg Std Avg Std Avg Std Avg Std Avg StdN2C1W1 A 48 534 07 035 012 537 07 008 003 531 07 002 001N2C1W1 B 49 543 07 029 008 544 08 009 002 533 05 002 001N3C2W2 A 107 1214 15 184 033 1218 14 047 016 1202 13 007 002N3C2W2 B 105 1177 18 193 054 1182 22 039 020 1167 11 006 003N3C3W1 A 66 739 08 148 042 736 08 042 018 732 09 007 003N3C3W1 B 71 804 09 146 037 798 07 046 024 792 09 006 002N4C1W1 A 240 2779 24 779 290 2754 24 584 185 2734 17 037 012N4C1W1 B 262 3004 32 748 312 2988 14 593 215 2958 22 045 021N4C1W1 C 241 2779 26 767 269 2768 27 615 205 2736 16 049 018N4C2W1 A 210 2458 29 708 241 2448 21 602 199 2426 19 051 024


N2C1W1 A 48 532 09 037 010 534 08 006 002 528 06 001 001N2C1W1 B 49 540 05 025 012 541 07 008 002 535 06 001 001N3C2W2 A 107 1210 13 193 041 1220 15 051 019 1204 15 006 002N3C2W2 B 105 1174 15 212 077 1179 19 040 022 1168 10 005 001N3C3W1 A 66 742 10 182 057 733 05 058 021 730 06 008 002N3C3W1 B 71 801 07 139 028 795 11 049 032 789 10 006 003N4C1W1 A 240 2763 27 791 249 2743 21 612 209 2735 14 043 026N4C1W1 B 262 2998 34 827 393 2994 18 629 277 2953 20 051 028N4C1W1 C 241 2782 29 893 300 2771 22 700 222 2729 19 068 025N4C2W1 A 210 2452 31 811 291 2451 21 599 242 2429 21 089 032



In relation to the runtime the EAs need less time thantheir corresponding GAs in all of the instances In additionthese improvements are substantial in 9291of the cases (223out of 240) These data suggest the following finding

Conclusion 9 In relation to runtime and according to theexperimentation performed the use of blind crossover oper-ators in genetic algorithms substantially increases the execu-tion time of the technique without providing an improve-ment in results

Regarding the convergence behaviour the GAs show bet-ter performance than the EAsThis means that they need lessgenerationsiterations to find their final solution Anywaythis fact does not entail better results or less runtime ashas been mentioned in Conclusions 8 and 9 What it really

involves is a greater exploration capacity of the techniqueAdditionally this fact is subject to the problem that is beingtreated and being more effective if the problem has an easyformulation For the experimentation conducted the EAsshow better convergence behaviour in 4583 of the cases(55 out of 120) Moreover the statistical test conducted showsthat for simple formulation problems (TSP NQP and BPP)the EAs have a significantly better convergence in 1277 (23of 180) of the cases On the other hand in 4166 (75 outof 180) of the comparisons the GAs are substantially betterIn the remaining 4557 the differences are not remarkableFor the CVRP as has been seen in Section 4 the EAs show asignificantly better convergence in the 7166 (43 out of 60)of the cases As a result of this the following finding can bededuced


Table 12 Convergence behaviour of the nine techniques applied to the BPP

BPP GA1 GA2 EA1

Instance Avg Std Avg Std Avg StdN2C1W1 A 1344 850 1438 768 1287 888N2C1W1 B 648 246 1128 814 869 347N3C2W2 A 3321 1441 3849 1537 3012 1857N3C2W2 B 3564 1167 3451 1280 3148 1110N3C3W1 A 2987 1024 3108 1170 3321 986N3C3W1 B 3660 1768 4102 2184 3858 1584N4C1W1 A 15423 3127 15697 5839 13286 5869N4C1W1 B 16634 4978 16824 5977 15387 4868N4C1W1 C 13648 5994 14731 7572 14994 5847N4C2W1 A 13400 5730 14955 6746 16164 4735Instance GA3 GA4 EA2

N2C1W1 A 1517 808 1322 812 1127 907N2C1W1 B 874 431 956 422 1003 561N3C2W2 A 2327 1018 2994 814 2857 913N3C2W2 B 3715 1207 3014 1147 3500 1033N3C3W1 A 3128 1369 3587 1362 2995 770N3C3W1 B 3517 1462 4007 1874 4114 1013N4C1W1 A 15011 3047 14990 6089 14824 4999N4C1W1 B 14528 5315 15773 5190 14902 5031N4C1W1 C 16127 6714 15790 6763 15358 5553N4C2W1 A 13158 5004 13994 7412 15844 4639Instance GA5 GA6 EA3

N2C1W1 A 1140 734 1004 571 1427 904N2C1W1 B 814 211 718 274 957 438N3C2W2 A 3002 1124 3271 997 3502 1987N3C2W2 B 3764 1324 3554 1405 2994 1345N3C3W1 A 2807 1395 2730 1136 3507 1027N3C3W1 B 4818 2415 4519 2234 3714 1880N4C1W1 A 14270 2999 15002 5315 12867 4997N4C1W1 B 17018 5138 17590 6423 16120 5001N4C1W1 C 13108 5243 12104 5718 15710 6114N4C2W1 A 12740 4979 13796 5734 15271 5117

Conclusion 10 Finally regarding the convergence behaviourand according to the experimentation performed the studyconcludes that the use of blind crossover operators in geneticalgorithms for solving combinatorial optimization prob-lems with simple formulation entails a better convergencebehaviour of the technique needing less generations to obtainthe final solution Anyway this fact does not mean betterresults On the other hand for more complex problems theuse of blind crossover operators does not imply a betterconvergence behavior

Finally as a final conclusion of this work and based onthe findings that have been proposed along the paper thefollowing assertion can be concluded

Conclusion 11 Based on the experimentation performed anevolutionary algorithm (based only onmutation and survivorselection functions) is more efficient than a classic geneticalgorithm to solve combinatorial optimization problems

As a final comment the authors of this study want toclarify that they are aware that there is a large amount ofcombinatorial optimization problems in the literature In thesame way there are a lot of blind crossover operators Forthese reasons it could be pretentious to generalize the con-clusions of this study to all the combinatorial optimizationproblems In this work to perform the tests four well-knownand widely used problems have been used The goal of thisselection is to choose problems of different types and toobtain conclusions as objective as possible Following thesame philosophy all the crossover operators that have beenused in this study have been previously applied in manystudies in the literatureThereby the authors of this study areaware that the conclusions drawn are objective and rigorousbut just for the conducted experimentation

As future work and in order to verify the conclusionsof this study it could be interesting to extend this work tosome other combinatorial optimization problems such as


Table 13 119911-test for BPP ldquo+rdquo indicates that EA is better ldquominusrdquo depicts that it is worse ldquolowastrdquo indicates that the difference between the two algorithmsis not significant (at 95 confidence level)

BPP EA1 versus GA1 EA1 versus GA2

Instance Results Convergence Time Results Convergence TimeN2C1W1 A + (214) lowast (032) + (1937) + (428) lowast (090) + (1341)N2C1W1 B + (821) minus (minus367) + (2368) + (824) + (206) + (2213)N3C2W2 A + (427) lowast (092) + (3785) + (592) + (245) + (1754)N3C2W2 B + (335) lowast (182) + (2444) + (595) lowast (126) + (1153)N3C3W1 A + (411) lowast (minus166) + (2367) + (234) lowast (minus098) + (1356)N3C3W1 B + (666) lowast (minus058) + (2671) + (372) lowast (063) + (1174)N4C1W1 A + (1081) + (216) + (1807) + (480) + (197) + (2086)N4C1W1 B + (837) lowast (126) + (1589) + (813) lowast (131) + (1793)N4C1W1 C + (995) lowast (minus113) + (1883) + (720) lowast (minus019) + (1944)N4C2W1 A + (652) minus (minus262) + (1918) + (549) lowast (minus103) + (1943)Instance EA2 versus GA3 EA2 versus GA4

N2C1W1 A + (261) + (227) + (2532) + (424) lowast (116) + (1581)N2C1W1 B + (905) lowast (minus128) + (1409) + (843) lowast (minus033) + (2213)N3C2W2 A + (213) minus (minus274) + (3221) + (533) lowast (054) + (1665)N3C2W2 B + (235) lowast (095) + (1900) + (362) minus (minus208) + (1123)N3C3W1 A + (727) lowast (059) + (2157) + (271) + (298) + (1676)N3C3W1 B + (695) minus (minus237) + (3339) + (285) lowast (minus031) + (946)N4C1W1 A + (650) lowast (022) + (2112) + (224) lowast (015) + (1910)N4C1W1 B + (806) lowast (minus036) + (1392) + (1077) lowast (078) + (1467)N4C1W1 C + (1080) lowast (062) + (1937) + (1021) lowast (032) + (2000)N4C2W1 A + (434) minus (minus278) + (1743) + (523) minus (minus197) + (1477)Instance EA3 versus GA5 EA3 versus GA6

N2C1W1 A lowast (141) lowast (minus174) + (3045) lowast (000) minus (minus279) + (1115)N2C1W1 B + (714) minus (minus207) + (1702) + (500) minus (minus327) + (1920)N3C2W2 A lowast (136) lowast (minus154) + (1715) + (437) lowast (minus073) + (1776)N3C2W2 B + (239) + (288) + (1889) + (232) + (203) + (1421)N3C3W1 A + (714) minus (minus285) + (2028) lowast (000) minus (358) + (1491)N3C3W1 B + (1087) + (255) + (2468) lowast (175) lowast (194) + (1358)N4C1W1 A + (961) lowast (170) + (1934) + (522) + (206) + (2161)N4C1W1 B + (502) lowast (088) + (1782) + (404) lowast (127) + (1654)N4C1W1 C + (953) minus (minus228) + (1860) + (643) minus (minus301) + (2104)N4C2W1 A lowast (143) minus (minus250) + (1612) + (565) lowast (121) + (2037)

the minimum spanning tree problem [77] or the job-shopscheduling problem [78] Furthermore it may be worthwhileto investigate whether these same findings are also applicableto other types of optimization problems such as continuousoptimization

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

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[2] C W Ahn and R S Ramakrishna ldquoA genetic algorithm forshortest path routing problem and the sizing of populationsrdquoIEEE Transactions on Evolutionary Computation vol 6 no 6pp 566ndash579 2002

[3] A Norouzi and A H Zaim ldquoGenetic algorithm application inoptimization of wireless sensor networksrdquo The Scientific WorldJournal vol 2014 Article ID 286575 15 pages 2014

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[5] L Davis ldquoJob shop scheduling with genetic algorithmsrdquo inProceedings of the 1st International Conference on Genetic Algo-rithms pp 136ndash140 Lawrence Erlbaum Associates PittsburghPa USA 1985

[6] F Pezzella GMorganti and G Ciaschetti ldquoA genetic algorithmfor the flexible job-shop scheduling problemrdquo Computers andOperations Research vol 35 no 10 pp 3202ndash3212 2008


[7] J H Holland Adaptation in Natural and Artificial Systems AnIntroductory Analysis with Applications to Biology Control andArtificial Intelligence MIT Press Cambridge Mass USA 1975

[8] D Goldberg Genetic Algorithms in Search Optimization andMachine Learning Addison-Wesley Professional 1989

[9] K De Jong Analysis of the behavior of a class of genetic adaptivesystems [PhD thesis] University of Michigan Ann ArborMich USA 1975

[10] Z Stanimirovic ldquoA genetic algorithm approach for the capaci-tated single allocation p-hub median problemrdquo Computing andInformatics vol 29 no 1 pp 117ndash132 2012

[11] S Venkadesh G Hoogenboom W Potter and R McClendonldquoA genetic algorithm to refine input data selection for air tem-perature prediction using artificial neural networksrdquo AppliedSoft Computing Journal vol 13 no 5 pp 2253ndash2260 2013

[12] GWuY Bai andZ Sun ldquoResearch on formation ofmicrosatel-lite communication with genetic algorithmrdquo The ScientificWorld Journal vol 2013 Article ID 509508 7 pages 2013

[13] T Vidal T G Crainic M Gendreau N Lahrichi and W ReildquoA hybrid genetic algorithm formultidepot and periodic vehiclerouting problemsrdquo Operations Research vol 60 no 3 pp 611ndash624 2012

[14] M H Moradi and M Abedini ldquoA combination of genetic algo-rithm and particle swarm optimization for optimal DG locationand sizing in distribution systemsrdquo International Journal ofElectrical Power and Energy Systems vol 34 no 1 pp 66ndash742012

[15] Q Q Duan G K Yang and C C Pan ldquoA novel algorithmcombining finite statemethod and genetic algorithm for solvingcrude oil scheduling problemrdquoThe ScientificWorld Journal vol2014 Article ID 748141 11 pages 2014

[16] Z Q Chen and Y F Yin ldquoAn new crossover operator forreal-coded genetic algorithm with selective breeding basedon difference between individualsrdquo in Proceedings of the 8thInternational Conference on Natural Computation (ICNC rsquo12)pp 644ndash648 May 2012

[17] A Hara Y Ueno and T Takahama ldquoNew crossover operatorbased on semantic distance between subtrees in Genetic Pro-grammingrdquo in Proceedings of the IEEE International Conferenceon Systems Man and Cybernetics (SMC rsquo12) pp 721ndash726October 2012

[18] M Albayrak and N Allahverdi ldquoDevelopment a new mutationoperator to solve the traveling salesman problem by aid ofgenetic algorithmsrdquo Expert Systems with Applications vol 38no 3 pp 1313ndash1320 2011

[19] P M Mateo and I Alberto ldquoA mutation operator based ona pareto ranking for multi-objective evolutionary algorithmsrdquoJournal of Heuristics vol 18 no 1 pp 53ndash89 2012

[20] G Rudolph ldquoConvergence analysis of canonical genetic algo-rithmsrdquo IEEE Transactions on Neural Networks vol 5 no 1 pp96ndash101 1994

[21] K A De Jong and W M Spears ldquoAn analysis of the interactingroles of population size and crossover in genetic algorithmsrdquoin Parallel Problem Solving from Nature H-P Schwefel and RManner Eds vol 496 of Lecture Notes in Computer Science pp38ndash47 Springer Berlin Germany 1991

[22] A Kumar N Jani P Gupta et al ldquoAn empirical study oncrossover operator for degree constraint minimal spanningtree problem using genetic algorithmrdquo International Journal ofComp utational Intelligence Research vol 8 no 1 pp 1ndash15 2012

[23] W Banzhaf F D Francone and P Nordin ldquoThe effect ofextensive use of the mutation operator on generalization ingenetic programming using sparse data setsrdquo in Proceedingsof the 4th International Conference on Parallel Problem Solvingfrom Nature pp 300ndash309 Springer New York NY USA 1996

[24] E S Mresa and L Bottaci ldquoEfficiency of mutation operatorsand selective mutation strategies an empirical studyrdquo SoftwareTesting Verification and Reliability vol 9 no 4 pp 205ndash2321999

[25] A E Eiben R Hinterding and Z Michalewicz ldquoParametercontrol in evolutionary algorithmsrdquo IEEE Transactions onEvolutionary Computation vol 3 no 2 pp 124ndash141 1999

[26] J A Fernandez-Prieto J Canada-Bago M A Gadeo-Martosand J R Velasco ldquoOptimisation of control parameters forgenetic algorithms to test computer networks under realistictraffic loadsrdquo Applied Soft Computing Journal vol 11 no 4 pp3744ndash3752 2011

[27] J J Grefenstette ldquoOptimization of control parameters forgenetic algorithmsrdquo IEEE Transactions on Systems Man andCybernetics vol 16 no 1 pp 122ndash128 1986

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[29] E Osaba E Onieva R Carballedo F Diaz A Perallosand X Zhang ldquoA multi-crossover and adaptive island basedpopulation algorithm for solving routing problemsrdquo Journal ofZhejiangUniversity SCIENCEC vol 14 no 11 pp 815ndash821 2013

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[33] L Davis ldquoApplying adaptive algorithms to epistatic domainsrdquoin Proceedings of the International Joint Conference on ArtificialIntelligence vol 1 pp 161ndash163 1985

[34] S S Ray S Bandyopadhyay and S K Pal ldquoNew operators ofgenetic algorithms for traveling salesman problemrdquo in Proceed-ings of the 17th International Conference on Pattern Recognition(ICPR rsquo04) vol 2 pp 497ndash500 August 2004

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[37] F Herrera M Lozano and A M Sanchez ldquoA taxonomy forthe crossover operator for real-coded genetic algorithms anexperimental studyrdquo International Journal of Intelligent Systemsvol 18 no 3 pp 309ndash338 2003

[38] P C Pendharkar and J A Rodger ldquoAn empirical study of impactof crossover operators on the performance of non-binarygenetic algorithm based neural approaches for classificationrdquoComputers amp Operations Research vol 31 no 4 pp 481ndash4982004

[39] D B Fogel and J W Atmar ldquoComparing genetic operators withgaussian mutations in simulated evolutionary processes using


linear systemsrdquo Biological Cybernetics vol 63 no 2 pp 111ndash1141990

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[41] N Karmarkar and R M Karp ldquoAn efficient approximationscheme for the one-dimensional bin-packing problemrdquo inProceedings of the 23rd Annual Symposium on Foundations ofComputer Science pp 312ndash320 IEEE 1982

[42] Z Li Z Zhou X Sun and D Guo ldquoComparative study of arti-ficial bee colony algorithms with heuristic swap operators fortraveling salesman problemrdquo in Intelligent Computing Theoriesand Technology pp 224ndash233 Springer 2013

[43] J Bai G K Yang Y W Chen L S Hu and C C Pan ldquoAmodelinduced max-min ant colony optimization for asymmetrictraveling salesman problemrdquo Applied Soft Computing Journalvol 13 no 3 pp 1365ndash1375 2013

[44] J Sung and B Jeong ldquoAn adaptive evolutionary algorithm fortraveling salesman problem with precedence constraintsrdquo TheScientific World Journal vol 2014 Article ID 313767 11 pages2014

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[46] P Larranaga C M H Kuijpers R H Murga I Inza andS Dizdarevic ldquoGenetic algorithms for the travelling salesmanproblem a review of representations and operatorsrdquo ArtificialIntelligence Review vol 13 no 2 pp 129ndash170 1999

[47] R Baldacci A Mingozzi R Roberti and R W Calvo ldquoAnexact algorithm for the two-echelon capacitated vehicle routingproblemrdquo Operations Research vol 61 no 2 pp 298ndash314 2013

[48] M Jepsen S Spoorendonk and S Ropke ldquoA branch-and-cutalgorithm for the symmetric two-echelon capacitated vehiclerouting problemrdquo Transportation Science vol 47 no 1 pp 23ndash37 2013

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[51] J Bell and B Stevens ldquoA survey of known results and researchareas for 119899-queensrdquo Discrete Mathematics vol 309 no 1 pp 1ndash31 2009

[52] M Bezzel ldquoProposal of 8-queens problemrdquo Berliner Schach-zeitung vol 3 p 363 1848

[53] X Hu R C Eberhart and Y Shi ldquoSwarm intelligence forpermutation optimization a case study of n-queens problemrdquoin Proceedings of the IEEE Swarm Intelligence Symposium pp243ndash246 2003

[54] C ErbasMM Tanik andZ Aliyazicioglu ldquoLinear congruenceequations for the solutions of the119873-queens problemrdquo Informa-tion Processing Letters vol 41 no 6 pp 301ndash306 1992

[55] E Masehian and H Akbaripour ldquoLandscape analysis andefficient metaheuristics for solving the 119899-queens problemrdquoComputational Optimization and Applications An InternationalJournal vol 56 no 3 pp 735ndash764 2013

[56] I Martinjak and M Golub ldquoComparison of heuristic algo-rithms for the n-queen problemrdquo in Proceedings of the IEEE 29thInternational Conference on Information Technology Interfacespp 759ndash764 June 2007

[57] S Martello and P Toth Knapsack Problems JohnWiley amp SonsNew York NY USA 1990

[58] K Fleszar and C Charalambous ldquoAverage-weight-controlledbin-oriented heuristics for the one-dimensional bin-packingproblemrdquo European Journal of Operational Research vol 210no 2 pp 176ndash184 2011

[59] K Sim E Hart and B Paechter ldquoA hyper-heuristic classifierfor one dimensional bin packing problems improving classifi-cation accuracy by attribute evolutionrdquo in Proceeding of the 12thconference on Parallel Problem Solving fromNature pp 348ndash357Springer 2012

[60] K Sim and EHart ldquoGenerating single andmultiple cooperativeheuristics for the one dimensional bin packing problem using asingle node genetic programming island modelrdquo in Proceedingof the 15th Genetic and Evolutionary Computation Conference(GECCO 13) pp 1549ndash1556 ACM New York NY USA July2013

[61] E Cantu-Paz ldquoA survey of parallel genetic algorithmsrdquo Calcu-lateurs Paralleles Reseaux et Systems Repartis vol 10 no 2 pp141ndash171 1998

[62] M Tomassini ldquoA survey of genetic algorithmsrdquoAnnual Reviewsof Computational Physics vol 3 no 2 pp 87ndash118 1995

[63] D B Fogel ldquoIntroduction to simulated evolutionary optimiza-tionrdquo IEEE Transactions on Neural Networks vol 5 no 1 pp3ndash14 1994

[64] S Lin ldquoComputer solutions of the traveling salesman problemrdquoThe Bell System Technical Journal vol 44 pp 2245ndash2269 1965

[65] D B Fogel ldquoAn evolutionary approach to the traveling salesmanproblemrdquoBiological Cybernetics vol 60 no 2 pp 139ndash144 1988

[66] G Syswerda ldquoSchedule optimization using genetic algorithmsrdquoin Handbook of Genetic Algorithms pp 332ndash349 1991

[67] E Osaba F Diaz and E Onieva ldquoGolden ball a novel meta-heuristic to solve combinatorial optimization problems basedon soccer conceptsrdquo Applied Intelligence vol 41 pp 145ndash1662014

[68] G Reinelt ldquoTSPLIBmdasha traveling salesman problem libraryrdquoORSA Journal on Computing vol 3 no 4 pp 376ndash384 1991

[69] P PongcharoenW Chalnate and PThapatsuwan ldquoExplorationof genetic parameters and operators through travelling sales-man problemrdquo Science Asia vol 33 no 2 pp 215ndash222 2007

[70] R C Eberhart and Y Shi ldquoComparison between geneticalgorithms and particle swarm optimizationrdquo in Proceedings ofthe 7th International Conference on Evolutionary Programmingpp 611ndash616 Springer 1998

[71] A E Eiben and C A Schippers ldquoOn evolutionary explorationand exploitationrdquo Fundamenta Informaticae vol 35 no 1ndash4 pp35ndash50 1998

[72] Y Y Wong K H Lee K S Leung and C W Ho ldquoA novelapproach in parameter adaptation and diversity maintenancefor genetic algorithmsrdquo Soft Computing vol 7 no 8 pp 506ndash515 2003

[73] J Berger and M Barkaoui ldquoA new hybrid genetic algorithmfor the capacitated vehicle routing problemrdquo Journal of theOperational Research Society vol 54 no 12 pp 1254ndash12622003

[74] F B Pereira J Tavares P Machado and E Costa ldquoGvr anew genetic representation for the vehicle routing problemrdquo inArtificial Intelligence andCognitive Science pp 95ndash102 SpringerNew York NY USA 2002

[75] Y Nagata ldquoEdge assembly crossover for the capacitated vehiclerouting problemrdquo in Evolutionary Computation in Combinato-rial Optimization C Cotta and J van Hemert Eds vol 4446 of


Lecture Notes in Computer Science pp 142ndash153 Springer BerlinGermany 2007

[76] Y Nagata O Braysy and W Dullaert ldquoA penalty-based edgeassembly memetic algorithm for the vehicle routing problemwith time windowsrdquo Computers and Operations Research vol37 no 4 pp 724ndash737 2010

[77] R L Graham and P Hell ldquoOn the history of the minimumspanning tree problemrdquoAnnals of the History of Computing vol7 no 1 pp 43ndash57 1985

[78] D Applegate and W Cook ldquoA computational study of the job-shop scheduling problemrdquo ORSA Journal on Computing vol 3no 2 pp 149ndash156 1991

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Distributed Sensor Networks


Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014


ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014


Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications


Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia


Biomedical Imaging


ArtificialNeural Systems

Advances in


RoboticsJournal of



Computational Intelligence and Neuroscience

Industrial EngineeringJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in



In this paper a deep study on the influence of usingblind crossover operators in GAs for solving combinatorialoptimization problems is conducted This study is developedby means of a comparison between GAs with this kind ofoperators and EAs based only on mutation operators Thusthis work could be framed into the third category Previouslyother studies in the literature have had a similar purposefor example [32] where the authors tried to validate thehypothesis that the crossover phase of genetic algorithms isnot efficient when it is applied to routing problems In thatwork the authors develop several versions of the basic GAwith some blind crossover operators (eg order crossover(OX) [33] or modified order crossover (MOX) [34]) andthey apply these techniques to the traveling salesmanproblem[35] Performances of these GAs are compared with the oneof an evolutionary algorithm (EA) based solely onmutationsThe comparison is based on the quality of the solution andthe runtime Furthermore the comparison also takes intoaccount the percentage of deviation from the average valuesof each parameter

On the other hand in [22] the efficiency of six differentversions of the classic GA applied to the degree constrainedminimal spanning tree problem [36] is compared Eachversion has its own crossover function In that work the onlydata shown for each version of the GA is the average value ofthe results obtained so the comparison is performed basedonly on this criterion Moreover the authors do not performthe comparison of the results obtained by a conventional GAand an EA For this reason with this study it is not possibleto quantify the real influence of the crossover phase in theoptimization capacity of a GA

Together with the above studies in the literature there aremany others that are not comparable with the study presentedin this paper The main reason is that they are focused onother types of problems [21] or because they analyzed onlythe crossover process of a traditional GA [37ndash39]

The motivation of this work stems from the absence inthe literature of a study that proves objectively the efficiencyof using blind crossover operators in GAs for combinatorialoptimization problems Although [32] focuses on this topicit is only applicable to routing problems and it is only testedwith one problem the TSP In addition the comparison ofthe results done in [32] is not as deep as the one made in thepresent work On the other hand as it has been mentionedthe study presented in the abovementioned [22] is not trulyconclusive to prove the real influence of the crossover processin a GA

Therefore the goal of this paper is to perform an objectivestudy on the efficiency of blind crossover operators in basicGAs with respect to blindmutation operators in basic EAs Inorder to reach this goal an exhaustive comparison betweendifferent versions of genetic and evolutionary algorithms ispresented This comparison includes the following criteriaquality of the results runtime and convergence behavior ofeach of the techniques reviewed Furthermore to perform areliable comparison of these results a statistical study ismadeFor this purpose the normal distribution 119911-test is performedFor the experimentation four different problems have beenused the traveling salesman problem (TSP) the capacitated

vehicle routing problem (CVRP) [40] the N-queens problem(NQP) and the one-dimensional bin packing problem (BPP)[41]

The rest of the paper is structured as follows In Section 2the description of the experimentation is presented InSection 3 the tests for the TSP are shown After thatthe experiments performed with the CVRP (Section 4) aredisplayed followed by those conducted with the NQP andBPP (Sections 5 and 6 resp) Finally thework is finishedwiththe conclusions of the study and further work (Section 7)

2 Description of the Experimentation

In this section a description of the experimentation is madeFirst in Section 21 the problems used for the tests areintroducedThen in Section 22 the details of the techniquesdeveloped are described including the functions of thedifferent steps of the algorithms Finally in Section 23 theexperimentation setup is presented

21 Description of the Problems For this study four differentcombinatorial problems have been used Two of them areoptimization problems of routing the TSP and the CVRPIn addition to verify that the results of this study are validfor other types of problems apart from the routing onestwo constraint satisfaction problems have also been used inthe experimentation the NQP and the BPP These problemswere chosen because they are well known and easy toimplement In addition they are easily replicable In this wayany researcher can perform these same tests either to checkthe results or to perform themwith other crossover functionsor different parameters

The first problem used is the TSP The TSP is one ofthe most famous and widely studied problems throughouthistory in operations research and computer science It hasa great scientific interest and it is used in a large number ofstudies [42ndash44] This problem can be defined on a completegraph 119866 = (119881119860) where 119881 = V

1 V2 V

119899 is the set

of vertexes which represents the nodes of the system and119860 = (V

119894 V119895) V119894 V119895

isin 119881 119894 = 119895 is the set of arcs whichrepresents the interconnection between nodes Each arc hasan associated distance cost 119889

119894119895 The objective of the TSP is

to find a route that visits every customer once (and onlyonce) that is a Hamiltonian cycle in the graph 119866 and thatminimizes the total distance traveled In a formal way theTSP can be formulated as follows [45]

Minimize119891 (119883) = sum

119894=0

sum

119894 =119895119895=0

119889119894119895119909119894119895 forall119894 119895 isin 119881 (1)

where119909119894119895isin 0 1 forall 119894 119895 isin 119860 (2)


119894=0119894 =119895

119909119894119895= 1 forall119895 isin 119881 (3)

sum

119895=0119894 =119895

119909119894119895= 1 forall119894 isin 119881 (4)

sum

119894isin119878119895isin119878119894 =119895

119909119894119895ge 1 forall119878 sub 119881 (5)


Nodes

1 2

3

4

5 6

78

9

(a)

Path

1 2

3

4

56

7

8

9

(b)






12+11988924+11988946+11988968+11988989+d97+11988975+11988953+11988931





Minimize 119891 (119883) = sum

119894=0

sum

119894 =119895119895=0

119889119894119895119909119894119895

forall119894 119895 isin 119881 (6)


119894=0119894 =119895

119909119894119895= 1 forall119895 (7)

sum

119895=0119894 =119895

119909119894119895= 1 forall119894 isin 119881 (8)

sum

119894

119909119894119895ge |119878| minus V (119878) 119878 119878 sube

119881

1

|119878| ge 2 (9)

sum

119894isin119878

119902119894119910119903

119894le 119876 forall119903 isin 119870 (10)

where 119910119903

119894isin 0 1 forall119903 isin 119870 (11)




03+11988931

+11988915

+11988950

+11988902

+11988924

+11988940

+11988907

+

11988979

+ 11988998

+ 11988986

+ 11988960




Depot

1

2

3

4

5

6

7

89

(a)

Vehicle routes1

2

3

4

5

6

7

89

(b)



1 1199022 119902

119899)



1003816100381610038161003816119894 minus 119902119894

1003816100381610038161003816=

10038161003816100381610038161003816119895 minus 119902119895

10038161003816100381610038161003816

forall119894 119895 1 2 119873 119894 = 119895 (13)



1 1198942 119894

119899





Minimize 119891 (119883) =

119899

sum

119894=0

119910119894 (14)


sum

119894=0

119909119894119895= 1 forall119895 isin 1 119899 (15)

119899

sum

119895=0

119904119894119909119894119895le 119902 forall119894 isin 1 119899 (16)









119909has




1 GA2

GA3 GA4 GA5 and GA



1 EA2 and EA




1and GA

2have a



3and GA

4have a 119901

119888= 75

and 119901119898


6have 119901

119888= 50 and

119901119898


=

0 and a 119901119898of 100 For GA

1 GA2 and EA

1 an initial


3 GA4 and EA

2 the population


3 a popu-


119899







1198751= (12345678) 997888rarr 119875

1= (12 | 345 | 678)


1= (87 | 345 | 126)

1198752= (24687531) 997888rarr 119875

2= (24 | 687 | 531)


2= (45687123)

(19)


1and 119875





2these nodes are 4


1

1198751= (12345678) 997888rarr 119874


997888rarr 1198741= (12346578)

(20)


1198752= (24687531) 997888rarr 119874


997888rarr 1198742= (24387561)

(21)


1198751= (1234 | 5678) 997888rarr 119874


997888rarr 1198741= (72543618)

1198752= (2468 | 7531) 997888rarr 119874


997888rarr 1198742= (24567831)

(22)


1198751= (1234 | 5678) 997888rarr 119874


997888rarr 1198741= (12346875)

1198752= (2468 | 7531) 997888rarr 119874


997888rarr 1198742= (24681357)

(23)



1198751= (1 2 3 4 0 9 10 11 12 0 13 14 15 16 0 5 6 7 8)

1198752= (1 12 6 3 0 2 4 7 11 0 5 14 16 9 0 8 13 10 15)

(24)


1198741= (1 2 3 4 0 9 10 11 12 0 6 7 5 14 0 16 8 13 15)

1198742= (1 12 6 3 0 2 4 7 11 0 9 10 13 14 0 15 16 5 8)

(25)




1 GA2 and EA




119911 =

119883EA minus 119883GA

radic(120590EA119899EA) + (120590GA119899GA) (26)




Alg Pop 119901119888

119901119898





GA1

















1 On the







5in the 100 of the











TSP GA1

GA2

EA1


Instance GA3

GA4 EA2






1and GA

2present a better




2





1 Regarding GA




GA3outperforms EA




3in two instances


6






TSP GA1

GA2

EA1


3GA4 EA2


















1outperforms GA

1in 80 of





2 these



3in the


GA3 the EA



2






forms GA5and GA



3





CVRP GA1 GA2 EA1









2and GA


2 GA4 and GA

6 and






CVRP GA1 GA2 EA1

















1 GA2

GA3 GA4 and GA






1and GA


to GA3and GA


5




1has a better






2 these


2gets


4in the





NQP GA1 GA2 EA1









4




6in the 20 of













NQP GA1 GA2 EA1



























BPP GA1 GA2 EA1












BPP GA1 GA2 EA1


















References



























































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




Nodes

1 2

3

4

5 6

78

9

(a)

Path

1 2

3

4

56

7

8

9

(b)






12+11988924+11988946+11988968+11988989+d97+11988975+11988953+11988931





Minimize 119891 (119883) = sum

119894=0

sum

119894 =119895119895=0

119889119894119895119909119894119895

forall119894 119895 isin 119881 (6)


119894=0119894 =119895

119909119894119895= 1 forall119895 (7)

sum

119895=0119894 =119895

119909119894119895= 1 forall119894 isin 119881 (8)

sum

119894

119909119894119895ge |119878| minus V (119878) 119878 119878 sube

119881

1

|119878| ge 2 (9)

sum

119894isin119878

119902119894119910119903

119894le 119876 forall119903 isin 119870 (10)

where 119910119903

119894isin 0 1 forall119903 isin 119870 (11)




03+11988931

+11988915

+11988950

+11988902

+11988924

+11988940

+11988907

+

11988979

+ 11988998

+ 11988986

+ 11988960




Depot

1

2

3

4

5

6

7

89

(a)

Vehicle routes1

2

3

4

5

6

7

89

(b)



1 1199022 119902

119899)



1003816100381610038161003816119894 minus 119902119894

1003816100381610038161003816=

10038161003816100381610038161003816119895 minus 119902119895

10038161003816100381610038161003816

forall119894 119895 1 2 119873 119894 = 119895 (13)



1 1198942 119894

119899





Minimize 119891 (119883) =

119899

sum

119894=0

119910119894 (14)


sum

119894=0

119909119894119895= 1 forall119895 isin 1 119899 (15)

119899

sum

119895=0

119904119894119909119894119895le 119902 forall119894 isin 1 119899 (16)









119909has




1 GA2

GA3 GA4 GA5 and GA



1 EA2 and EA




1and GA

2have a



3and GA

4have a 119901

119888= 75

and 119901119898


6have 119901

119888= 50 and

119901119898


=

0 and a 119901119898of 100 For GA

1 GA2 and EA

1 an initial


3 GA4 and EA

2 the population


3 a popu-


119899







1198751= (12345678) 997888rarr 119875

1= (12 | 345 | 678)


1= (87 | 345 | 126)

1198752= (24687531) 997888rarr 119875

2= (24 | 687 | 531)


2= (45687123)

(19)


1and 119875





2these nodes are 4


1

1198751= (12345678) 997888rarr 119874


997888rarr 1198741= (12346578)

(20)


1198752= (24687531) 997888rarr 119874


997888rarr 1198742= (24387561)

(21)


1198751= (1234 | 5678) 997888rarr 119874


997888rarr 1198741= (72543618)

1198752= (2468 | 7531) 997888rarr 119874


997888rarr 1198742= (24567831)

(22)


1198751= (1234 | 5678) 997888rarr 119874


997888rarr 1198741= (12346875)

1198752= (2468 | 7531) 997888rarr 119874


997888rarr 1198742= (24681357)

(23)



1198751= (1 2 3 4 0 9 10 11 12 0 13 14 15 16 0 5 6 7 8)

1198752= (1 12 6 3 0 2 4 7 11 0 5 14 16 9 0 8 13 10 15)

(24)


1198741= (1 2 3 4 0 9 10 11 12 0 6 7 5 14 0 16 8 13 15)

1198742= (1 12 6 3 0 2 4 7 11 0 9 10 13 14 0 15 16 5 8)

(25)




1 GA2 and EA




119911 =

119883EA minus 119883GA

radic(120590EA119899EA) + (120590GA119899GA) (26)




Alg Pop 119901119888

119901119898





GA1

















1 On the







5in the 100 of the











TSP GA1

GA2

EA1


Instance GA3

GA4 EA2






1and GA

2present a better




2





1 Regarding GA




GA3outperforms EA




3in two instances


6






TSP GA1

GA2

EA1


3GA4 EA2


















1outperforms GA

1in 80 of





2 these



3in the


GA3 the EA



2






forms GA5and GA



3





CVRP GA1 GA2 EA1









2and GA


2 GA4 and GA

6 and






CVRP GA1 GA2 EA1

















1 GA2

GA3 GA4 and GA






1and GA


to GA3and GA


5




1has a better






2 these


2gets


4in the





NQP GA1 GA2 EA1









4




6in the 20 of













NQP GA1 GA2 EA1



























BPP GA1 GA2 EA1












BPP GA1 GA2 EA1


















References



























































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




Depot

1

2

3

4

5

6

7

89

(a)

Vehicle routes1

2

3

4

5

6

7

89

(b)



1 1199022 119902

119899)



1003816100381610038161003816119894 minus 119902119894

1003816100381610038161003816=

10038161003816100381610038161003816119895 minus 119902119895

10038161003816100381610038161003816

forall119894 119895 1 2 119873 119894 = 119895 (13)



1 1198942 119894

119899





Minimize 119891 (119883) =

119899

sum

119894=0

119910119894 (14)


sum

119894=0

119909119894119895= 1 forall119895 isin 1 119899 (15)

119899

sum

119895=0

119904119894119909119894119895le 119902 forall119894 isin 1 119899 (16)









119909has




1 GA2

GA3 GA4 GA5 and GA



1 EA2 and EA




1and GA

2have a



3and GA

4have a 119901

119888= 75

and 119901119898


6have 119901

119888= 50 and

119901119898


=

0 and a 119901119898of 100 For GA

1 GA2 and EA

1 an initial


3 GA4 and EA

2 the population


3 a popu-


119899







1198751= (12345678) 997888rarr 119875

1= (12 | 345 | 678)


1= (87 | 345 | 126)

1198752= (24687531) 997888rarr 119875

2= (24 | 687 | 531)


2= (45687123)

(19)


1and 119875





2these nodes are 4


1

1198751= (12345678) 997888rarr 119874


997888rarr 1198741= (12346578)

(20)


1198752= (24687531) 997888rarr 119874


997888rarr 1198742= (24387561)

(21)


1198751= (1234 | 5678) 997888rarr 119874


997888rarr 1198741= (72543618)

1198752= (2468 | 7531) 997888rarr 119874


997888rarr 1198742= (24567831)

(22)


1198751= (1234 | 5678) 997888rarr 119874


997888rarr 1198741= (12346875)

1198752= (2468 | 7531) 997888rarr 119874


997888rarr 1198742= (24681357)

(23)



1198751= (1 2 3 4 0 9 10 11 12 0 13 14 15 16 0 5 6 7 8)

1198752= (1 12 6 3 0 2 4 7 11 0 5 14 16 9 0 8 13 10 15)

(24)


1198741= (1 2 3 4 0 9 10 11 12 0 6 7 5 14 0 16 8 13 15)

1198742= (1 12 6 3 0 2 4 7 11 0 9 10 13 14 0 15 16 5 8)

(25)




1 GA2 and EA




119911 =

119883EA minus 119883GA

radic(120590EA119899EA) + (120590GA119899GA) (26)




Alg Pop 119901119888

119901119898





GA1

















1 On the







5in the 100 of the











TSP GA1

GA2

EA1


Instance GA3

GA4 EA2






1and GA

2present a better




2





1 Regarding GA




GA3outperforms EA




3in two instances


6






TSP GA1

GA2

EA1


3GA4 EA2


















1outperforms GA

1in 80 of





2 these



3in the


GA3 the EA



2






forms GA5and GA



3





CVRP GA1 GA2 EA1









2and GA


2 GA4 and GA

6 and






CVRP GA1 GA2 EA1

















1 GA2

GA3 GA4 and GA






1and GA


to GA3and GA


5




1has a better






2 these


2gets


4in the





NQP GA1 GA2 EA1









4




6in the 20 of













NQP GA1 GA2 EA1



























BPP GA1 GA2 EA1












BPP GA1 GA2 EA1


















References



























































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in









119909has




1 GA2

GA3 GA4 GA5 and GA



1 EA2 and EA




1and GA

2have a



3and GA

4have a 119901

119888= 75

and 119901119898


6have 119901

119888= 50 and

119901119898


=

0 and a 119901119898of 100 For GA

1 GA2 and EA

1 an initial


3 GA4 and EA

2 the population


3 a popu-


119899







1198751= (12345678) 997888rarr 119875

1= (12 | 345 | 678)


1= (87 | 345 | 126)

1198752= (24687531) 997888rarr 119875

2= (24 | 687 | 531)


2= (45687123)

(19)


1and 119875





2these nodes are 4


1

1198751= (12345678) 997888rarr 119874


997888rarr 1198741= (12346578)

(20)


1198752= (24687531) 997888rarr 119874


997888rarr 1198742= (24387561)

(21)


1198751= (1234 | 5678) 997888rarr 119874


997888rarr 1198741= (72543618)

1198752= (2468 | 7531) 997888rarr 119874


997888rarr 1198742= (24567831)

(22)


1198751= (1234 | 5678) 997888rarr 119874


997888rarr 1198741= (12346875)

1198752= (2468 | 7531) 997888rarr 119874


997888rarr 1198742= (24681357)

(23)



1198751= (1 2 3 4 0 9 10 11 12 0 13 14 15 16 0 5 6 7 8)

1198752= (1 12 6 3 0 2 4 7 11 0 5 14 16 9 0 8 13 10 15)

(24)


1198741= (1 2 3 4 0 9 10 11 12 0 6 7 5 14 0 16 8 13 15)

1198742= (1 12 6 3 0 2 4 7 11 0 9 10 13 14 0 15 16 5 8)

(25)




1 GA2 and EA




119911 =

119883EA minus 119883GA

radic(120590EA119899EA) + (120590GA119899GA) (26)




Alg Pop 119901119888

119901119898





GA1

















1 On the







5in the 100 of the











TSP GA1

GA2

EA1


Instance GA3

GA4 EA2






1and GA

2present a better




2





1 Regarding GA




GA3outperforms EA




3in two instances


6






TSP GA1

GA2

EA1


3GA4 EA2


















1outperforms GA

1in 80 of





2 these



3in the


GA3 the EA



2






forms GA5and GA



3





CVRP GA1 GA2 EA1









2and GA


2 GA4 and GA

6 and






CVRP GA1 GA2 EA1

















1 GA2

GA3 GA4 and GA






1and GA


to GA3and GA


5




1has a better






2 these


2gets


4in the





NQP GA1 GA2 EA1









4




6in the 20 of













NQP GA1 GA2 EA1



























BPP GA1 GA2 EA1












BPP GA1 GA2 EA1


















References



























































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





2these nodes are 4


1

1198751= (12345678) 997888rarr 119874


997888rarr 1198741= (12346578)

(20)


1198752= (24687531) 997888rarr 119874


997888rarr 1198742= (24387561)

(21)


1198751= (1234 | 5678) 997888rarr 119874


997888rarr 1198741= (72543618)

1198752= (2468 | 7531) 997888rarr 119874


997888rarr 1198742= (24567831)

(22)


1198751= (1234 | 5678) 997888rarr 119874


997888rarr 1198741= (12346875)

1198752= (2468 | 7531) 997888rarr 119874


997888rarr 1198742= (24681357)

(23)



1198751= (1 2 3 4 0 9 10 11 12 0 13 14 15 16 0 5 6 7 8)

1198752= (1 12 6 3 0 2 4 7 11 0 5 14 16 9 0 8 13 10 15)

(24)


1198741= (1 2 3 4 0 9 10 11 12 0 6 7 5 14 0 16 8 13 15)

1198742= (1 12 6 3 0 2 4 7 11 0 9 10 13 14 0 15 16 5 8)

(25)




1 GA2 and EA




119911 =

119883EA minus 119883GA

radic(120590EA119899EA) + (120590GA119899GA) (26)




Alg Pop 119901119888

119901119898





GA1

















1 On the







5in the 100 of the











TSP GA1

GA2

EA1


Instance GA3

GA4 EA2






1and GA

2present a better




2





1 Regarding GA




GA3outperforms EA




3in two instances


6






TSP GA1

GA2

EA1


3GA4 EA2


















1outperforms GA

1in 80 of





2 these



3in the


GA3 the EA



2






forms GA5and GA



3





CVRP GA1 GA2 EA1









2and GA


2 GA4 and GA

6 and






CVRP GA1 GA2 EA1

















1 GA2

GA3 GA4 and GA






1and GA


to GA3and GA


5




1has a better






2 these


2gets


4in the





NQP GA1 GA2 EA1









4




6in the 20 of













NQP GA1 GA2 EA1



























BPP GA1 GA2 EA1












BPP GA1 GA2 EA1


















References



























































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





Alg Pop 119901119888

119901119898





GA1

















1 On the







5in the 100 of the











TSP GA1

GA2

EA1


Instance GA3

GA4 EA2






1and GA

2present a better




2





1 Regarding GA




GA3outperforms EA




3in two instances


6






TSP GA1

GA2

EA1


3GA4 EA2


















1outperforms GA

1in 80 of





2 these



3in the


GA3 the EA



2






forms GA5and GA



3





CVRP GA1 GA2 EA1









2and GA


2 GA4 and GA

6 and






CVRP GA1 GA2 EA1

















1 GA2

GA3 GA4 and GA






1and GA


to GA3and GA


5




1has a better






2 these


2gets


4in the





NQP GA1 GA2 EA1









4




6in the 20 of













NQP GA1 GA2 EA1



























BPP GA1 GA2 EA1












BPP GA1 GA2 EA1


















References



























































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





TSP GA1

GA2

EA1


Instance GA3

GA4 EA2






1and GA

2present a better




2





1 Regarding GA




GA3outperforms EA




3in two instances


6






TSP GA1

GA2

EA1


3GA4 EA2


















1outperforms GA

1in 80 of





2 these



3in the


GA3 the EA



2






forms GA5and GA



3





CVRP GA1 GA2 EA1









2and GA


2 GA4 and GA

6 and






CVRP GA1 GA2 EA1

















1 GA2

GA3 GA4 and GA






1and GA


to GA3and GA


5




1has a better






2 these


2gets


4in the





NQP GA1 GA2 EA1









4




6in the 20 of













NQP GA1 GA2 EA1



























BPP GA1 GA2 EA1












BPP GA1 GA2 EA1


















References



























































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





TSP GA1

GA2

EA1


3GA4 EA2


















1outperforms GA

1in 80 of





2 these



3in the


GA3 the EA



2






forms GA5and GA



3





CVRP GA1 GA2 EA1









2and GA


2 GA4 and GA

6 and






CVRP GA1 GA2 EA1

















1 GA2

GA3 GA4 and GA






1and GA


to GA3and GA


5




1has a better






2 these


2gets


4in the





NQP GA1 GA2 EA1









4




6in the 20 of













NQP GA1 GA2 EA1



























BPP GA1 GA2 EA1












BPP GA1 GA2 EA1


















References



























































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in











1outperforms GA

1in 80 of





2 these



3in the


GA3 the EA



2






forms GA5and GA



3





CVRP GA1 GA2 EA1









2and GA


2 GA4 and GA

6 and






CVRP GA1 GA2 EA1

















1 GA2

GA3 GA4 and GA






1and GA


to GA3and GA


5




1has a better






2 these


2gets


4in the





NQP GA1 GA2 EA1









4




6in the 20 of













NQP GA1 GA2 EA1



























BPP GA1 GA2 EA1












BPP GA1 GA2 EA1


















References



























































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





CVRP GA1 GA2 EA1









2and GA


2 GA4 and GA

6 and






CVRP GA1 GA2 EA1

















1 GA2

GA3 GA4 and GA






1and GA


to GA3and GA


5




1has a better






2 these


2gets


4in the





NQP GA1 GA2 EA1









4




6in the 20 of













NQP GA1 GA2 EA1



























BPP GA1 GA2 EA1












BPP GA1 GA2 EA1


















References



























































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





CVRP GA1 GA2 EA1

















1 GA2

GA3 GA4 and GA






1and GA


to GA3and GA


5




1has a better






2 these


2gets


4in the





NQP GA1 GA2 EA1









4




6in the 20 of













NQP GA1 GA2 EA1



























BPP GA1 GA2 EA1












BPP GA1 GA2 EA1


















References



























































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in










1 GA2

GA3 GA4 and GA






1and GA


to GA3and GA


5




1has a better






2 these


2gets


4in the





NQP GA1 GA2 EA1









4




6in the 20 of













NQP GA1 GA2 EA1



























BPP GA1 GA2 EA1












BPP GA1 GA2 EA1


















References



























































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





NQP GA1 GA2 EA1









4




6in the 20 of













NQP GA1 GA2 EA1



























BPP GA1 GA2 EA1












BPP GA1 GA2 EA1


















References



























































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





NQP GA1 GA2 EA1



























BPP GA1 GA2 EA1












BPP GA1 GA2 EA1


















References



























































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in
















BPP GA1 GA2 EA1












BPP GA1 GA2 EA1


















References



























































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





BPP GA1 GA2 EA1


















References



























































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in












References



























































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in























































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in



Research Article Crossover versus Mutation: A Comparative...

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