ANewMemeticAlgorithmfortheAsymmetricTravelin
LUCIANABURIOL FaculdadedE e ngenhariaElétrica de eComputação UniversidadeEstadualdeCampinas –UNICAMP e-mail:
[email protected] PAULOM.FRANÇA FaculdadedE e ngenhariaElétrica de eComputação UniversidadeEstadualdeCampinas –UNICAMP e-mail:
[email protected] PABLOMOSCATO FaculdadedE e ngenhariaElétrica de eComputação UniversidadeEstadualdeCampinas –UNICAMP e-mail:
[email protected]
gSalesmanProblem
–13083-970Campinas,SP,Brazil
–13083-970Campinas,SP,Brazil
–13083-970Campinas,SP,Brazil
Corresponding author: PauloMFrança . C.P.6101 –FEEC/Densis 13083-970CampinasSP –Brazil Phone:+55197883774 Fax:+55192891395 e-mail:
[email protected]
Abstract
Thispaperintroducesanewmemeticalgorithmparti
cularlydesignedtobeeffectivewithlarge
asymmetricinstancesotfhetravelingsalesmanprob
lem(ATSP).Themethodincorporatesanew
localsearchengineandmanyother
featuresthatcontributetoitseffectiveness,such
topologicalorganizationothe f populationoagent f ii) the hierarchical organization othe f population andreproductionschemes;iii)efficientdatastruc
as:i)the
ascomplete as ternary treewiththirteennodes; in overlapping clusters leading tsopecial selecti tures.Computationalexperimentsareconducted
onallATSPinstancesavailableintheTSPLIB,and
onasetoflargerasymmetricinstanceswith
knownoptimalsolutions.Thecomparisonsshowthat
theresultsobtainedbyourmethodcompare
favorably withthoseobtainedbsyeveralotheralgo
rithmsrecently proposedfortheATSP.
KeyWords: Asymmetrictravelingsalesmanproblem,localsearc metaheuristics.
h,memeticalgorithms,
on
TheTravelingSalesmanProblem(TSP)istheproblem
offindingtheshortestrouteamong set a of
cities,havingasinputthecompletedistancematri
xamongallcities.Let
integerthatstandsforthe
i directlytocity
cost totravelfromcity
instanceisanyinstanceoT f SPsuchthat
c= ijc
instanceiasnyinstanceoTSP f thathasaleast t o formally bsetated afollows: s given as n}and
A:={( i,j ): i,j
ji
forallcities
i,j.
cij beanon-negative
j A .
An asymmetricTSP(ATSP) cij ≠c jiThe . ATSPcan
nepairofcitiessuchthat input caomplete directed graph
∈ V, i≠j }arethesetovf erticesandarcsof
solution for the ATSP iH as amiltonian circuit ( totallength,wherethelengthitshesumoftheco justtheTSPproblem restrictedtoasymmetricinsta
Itsgrowingpopularity iaslsoduetoseveralimpor
G = (
V, A ),where V:= {1, ...,
G,respectively.Afeasible
tour).The objective ito sfind tour a of the minimum stsoeach f arcinthetour.Thissaid,theATSPis nces.
TheTSPhasattracted great a dealof attention am it has been used aone s othe f most important test-
symmetricTSP(STSP)
ong researchers in recentdecades. beds for new combinatorial optimization methods. tantrealworldapplications,mainly inshopfloor
control(scheduling),distributionofgoodsandser
vices(vehiclerouting),productdesign(VLSI
layout),etc.Exactalgorithmshavebeenproposedf
orbothsymmetricandasymmetriccases.Since
theTSPhasprovedtobelongtotheclassof heuristicsandmetaheuristicsoccupy animportantp practicalsolutionsforlargeinstances.Forsurvey refertoBalasandToth(1985),Laporte(1992),Jün
NP-hardproblems(GareyandJohnson,1979), laceinthemethodsso-fardevelopedtoprovide osnsolutionmethodsfortheTSP,thereadermay ger etal.(1995).
FocusingonlyonexactmethodsfortheATSP,the by Miller and Pekny (1991) uses thewellknown
branch-and-boundalgorithmproposed AssignmentProblem (AP) relaxation othe f ATSP.
Optimalsolutionsarereportedforinstancesouf p
to500.000citiesinreasonablerunningtimes,
althoughsuchinstanceswererandomly generatedfro
m uaniformdistributionagativeninterval.In
suchacase,theinstancesseemtobeeasyforAP-b Usingasimilarapproach,Carpanetoeal. t (1995)s than minute. 1 However,there are instances in whic asthosewherecostsare“almost”symmetric(i.e. instancescomesfrom real-worldsituationsrelated
asedalgorithms(FischettiandToth,1997). olvedproblemswithupto2.000nodesinless A h P-based algorithms may face problems, such cij ≈c
ji
forall
i,j ).Anotherclassodf ifficult
tovehicle routing problems,such athose s which
ariseinpharmaceuticalproduct-deliverytaskswith
inthecityoB f ologna(Fischetieat l.,1994).
TheseATSPinstancesandtheiroptimaltourcostsa
reavailableintheTSPLIB(Reinelt,1991)and
havebeenusedinthispaper.Thepolyhedralapproa
chusedbF y ischettiandToth(1997)performs
betterforthesetwoclassesofhardinstancesthan
In fact,
AP-basedmethods.Inthecomputational
experimentsrelatedhere,theinstancesotfhesetw
oclasseswereusedtoevaluatetheproposed
method. Ontheheuristicside,w a idespectrumofmetaheur as tabusearch (Fiechter,1994),
istictechniqueshasbeenproposed,such
neuralnetworks (Potvin,1993),
1999), simulated jumping (Amin,1999) and
antcolonies (StützleandDorigo,
genetic algorithms (GAs). Of particular interest are the
GAs,duetotheeffectivenessachievedbythisclas
ostechnique f infindinggoodsolutionsinshort
computationaltimes.Thebestresultshavebeenobt
ainedbythecombinationofevolutionary
algorithmswithlocalsearchmethods.Thishybridg
eneticapproach,alsoknownas
algorithms(MAs)(Moscato,1989,1999;MoscatoandNorman,19 strength opopulation-based f methods with the inten
memetic
92),combinestherecognized
sification capability olafocal search. In aM n A,
allagentsevolvesolutionsuntiltheyturntobel
ocalminimaoacfertainneighborhood(orhighly
evolvedsolutionsoindividual f searchstrategies),
i.e.,afterstepsorecombination f andmutation,a
localsearchiapplied s totheresulting solutions. SeveralMAsforsolvingtheATSPhaverecentlybee
nproposed.Basedonthe
computational results reported for these methods,o
necansay that gaood MAimplementationmust
combineseveralessentialfeatures:i)suitablerec
ombinationandmutationoperators;ii)afastand
effectivelocalsearchalgorithm;iii)ahierarchic
allystructuredpopulation;iv)advanceddata
structuresandsmartcodificationmechanisms.Thef
irstfeatureiosbviously inherentinany genetic
algorithm,while the second icrucial s in MAs becau
se 85% 95% - of the total CPU time igenerally s
spent in local search procedures. Many experiments
conducted ipnrevious research have shown that
theadoptionofstructures
inwhichagentsareconstrainedtorecombinewithin
structuredsubpopulationshaveprovedtobemoreef
fectivethannon-structuredimplementations
(Moscato,1993;Gorges-Schleuter,1997;Françaeat The
hierarchically
l.,2001;BerrettaandMoscato,1999).
GLS(GeneticLocalSearch)methodproposedbyFreisleb
successfulMAwhichintroducedanewrecombination
enandMerz(1996)isa operator,calledDistancePreserving
Crossover (DPX).These authors use the well-known
Lin-Kernighan (LK) heuristic alasocal search
enginefortheEuclidean(symmetric)instancesand
the fast-3-Optprocedurefortheasymmetric
ones.Inamorerecentarticle,theyimprovedther
esultsbyadoptingaseriesofsophisticated
implementationmechanismsthatenhancedtheperform 1997),forinstance,by adding variant a of
anceotfhemethod(MerzandFreisleben,
4-OptmovestothesearchintheATSPcase.
Gorges-Schleuter(1997)haschosentoinvestinspa simplerlocalsearchheuristic.Inthisalgorithm, demes(localpopulations)spatiallydisposedaring as a
tiallystructuredpopulationsusinga
called Asparagos96,thepopulationiorganized s in ndreceivingthenameof
ladder-population
(Gorges-Schleuter,1989).Thelocalsearchalgorith
msemployedarethe
2-repair-mechanismand
the special 3-repair-mechanismfortheSTSPandATSP,respectively.Therecombina
tionoperator
usediM s PX2,whichdiffersslightly fromtheprevi
ously reportedMaximalPreservativeCrossover
(MPX).Limited computationaltestscarried out with
these two approaches on few a instances of the
TSPLIBshowthatforlargerinstancesotfheSTSP, performsbetter in the case oup ft7o83 cities.Fo
Asparagos96issuperior,althoughthe the r ATSP,
Asparagos96 outperformed
GLS GLS ianll
5instancesusedinthecomparison. AnewrecombinationoperatorcalledEAX(EdgeAsse proposedbyNagataandKobayashi(1997).Thisopera andtheASTPandabletogenerateawidevarietyof Heuristicinformationiasddedduringtherecombina procedurecanbedisregarded.Thealgorithm,howeve aggregatesadditionalknowledgeduring theevolutio Anotheralgorithmemployingthesame“bigfamily”c
mblyCrossover)wasrecently torisappropriateforuseinboththeSTSP individualsfromasingle
tionprocessinsuch way a thatthelocalsearch r,canbeconsideredtobeanMA,sinceit nprocess. oncept,andbasedonthe
selectionusedingeneticprogramming,wasdevelopedbyWalt parentsmay generatemany individualsbutonly the areselectedfortherestotfheprocedure.Infact iteration.Thewellknowntwo-pointrecombinationo EdgeRepair(DER)arecombinedwiththefast reducedsubsetof instancesoftheATSPdrawnfrom InthispapertheauthorsproposeanewMAforsolv thefourkeyfeaturesresponsibleforaneffective search,calledRecursiveArcInsertion(RAI),espec
pairopf arents.
softbrood
ers(1998).Inbroodselection,two best withrespecttotheirrelativefitnessvalues o, necanchoosemorethanoneoffspringper peratorandthechromosomerepairDirected 3-Optandsuccessfullyappliedtosolveavery theTSPLIB. ingtheATSPthatincorporatesallof solution.Themaincontributionisanewlocal iallydesignedfortheasymmetriccase.Anew
recombinationoperator,StrategicArcCrossover(SA
X),adaptedfromasimilarrecombination
operatorcalledStrategicEdgeCrossover(SEX)(Mos
catoandNorman,1992),wasfoundtobe
capableofdealingwithasymmetricinstances.Thep
opulationishierarchicallyorganizedasa
completeternarytreeof13agentsclusteredin4s
ubpopulations,witheachcomposedoffour
individuals.Withineachsubpopulation,aleaderan
dthreeothersubordinatedindividualsare
allowedtorecombine,yieldingoffspring.Thedata
structuresandalgorithmsusedallownear-
optimalsolutionstobefoundrapidly.Thepresent
paperisorganizedasfollows.Section1
introducesstructuresofsomeMemeticAlgorithmswh independentfeaturesof theproposedmethodsuchas Section3,theATSP-dependentproceduressuchasth
ileinSection2,themainproblempopulation structureandsize are described.In enewlocalsearchalgorithmand
recombinationoperatorsarepresented,andSection convergenceotfhepopulation.Thepaperfinishesw evaluatedbseries ay of computationalexperiments Empiricalcomparisonswithsix other leadingmetahe
4introducestheproceduresusedtoavoidfast ithSection5where , theproposedmethodis using two sets of data drawn from the literature. uristicsfor thisproblem arealsoreported.
1. MemeticAlgorithmstructures
AgeneralstructureforMAswas
proposedbyMoscato(1999),andhispseudocodecan
consideredgeneric a templateforMAscharacterizin
gtheseveraldifferentimplementationswhich
canbefoundintheliterature.Thefollowingfigur
eswillpresentsomeothese f pseudocodesfrom
theliterature,withtheirresultspointedoutlate
rinthepaper.Allofthem,incluingtheMA
proposedinthispaper,usethethe
don´tlookbits
conceptintroducedby(Bentley,1992).Figure1
presentstheMAfromMerzandFreisleben(1997).
Merz_and_Freislebenprocedure; begin initializeOrdered_Neighborhood_List(m); initializePopwith40individualsusingtheNeares t_Neighbor_Heuristic(); foreachindividuali ∈Popd:= oiLocal_Search_3-4Opt(i); repeat/* generationloop*/ for j:=1to#recombinationsdo selectparentsi Poprandomily; ai, b ∈ offspring :=Recombination_Operator_DPX(i ai, b); offspring :=Local_Search_3-4Opt(offspring); addindividualoffspring toPop; endfor; for j:=1to#mutationsdo selectanindividuali ∈Poprandomily; i:= mMutation_Random7change(i); i:= (i mLocal_Search_3-4Opt m); addindividuali to mPop; endfor; Pop:=SelectPop(Pop); untilTerminationCondition=True; end; Figure1 Merz . andFreislebenmemeticalgorithm structure. Thelocalsearchothis f algorithmperformsboth3 reversesarandomlychosensubpathoflength6(per tour).TheTerminationConditionisanumberogf ene
-Optand4-Optmovesandthemutation formingarandom7-changeonthecurrent rationsthatdependsontheinstance.The
be
length moftheneighborhoodlistisnotspecifiedforthe rateis0.5,i.e.40individualsarecreatedperit (1997)isshow inFigure2.
ATSPcase The . mutationandcrossover eration.TheMAproposedbyGorges_Schleuter
Gorges_Schleuterprocedure; begin inicializeOrdered_Neighborhood_List(20); initializePopwith20individualsusingtheNeares t_Neighbor_Heuristic_Modified(); foreachindividuali ∈Popd:= oiLocal_Search_Special3repair(i); foreachindividuali ∈PopdoEvaluate(i); repeat/* generationloop*/ for j:=1topopSizedo selectmatei from oind[j]; f a neighborhood offspring :=Recombination_Operator_MPX2(ind[j],i a); offspring :=Mutation_Non_Sequencial4Opt(offspring ); offspring :=Local_Search_Special3repair(offspring ); offspring_cost=Evaluate(offspring); If (offspring_cost