Key Words: Asymmetric traveling salesman problem, local searc metaheuristics

ANewMemeticAlgorithmfortheAsymmetricTravelin LUCIANABURIOL FaculdadedE e ngenhariaElétrica de eComputação UniversidadeEstadualdeCampinas –UNICAMP e-m...
Author: Muriel Burns
0 downloads 0 Views 231KB Size
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

Suggest Documents