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SCI论文模板
Runningtitle:
Lietal.On….
Animprovedshuffledfrog-leapingalgorithmforknapsackproblem
Authors’name
Affiliation
Correspondenceautuor(通讯作者:
):
tel/faxXXX;e-mail:
XXX
Abstract
Shuffledfrog-leapingalgorithm(SFLA)haslongbeenconsideredasnewevolutionaryalgorithmofgroupevolution,andhasahighcomputingperformanceandexcellentabilityforglobalsearch.KnapsackproblemisatypicalNP-completeproblem.Forthediscretesearchspace,thispaperpresentstheimprovedSFLA,andsolvestheknapsackproblembyusingthealgorithm.Experimentalresultsshowthefeasibilityandeffectivenessofthismethod.
Keywords:
shuffledfrog-leapingalgorithm;knapsackproblem;optimizationproblem
0Introduction
Knapsackproblem(KP)isaverytypicalNP-hardproblemincomputerscience,whichwasfirstproposedandstudiedbyDantzinginthe1950s.Therearemanyalgorithmsforsolvingtheknapsackproblem.ClassicalalgorithmsforKParethebranchandboundmethod(BABM),dynamicprogrammingmethod(分支界定法和动态规划法),etc.However,mostofsuchalgorithmsareover-relianceonthefeaturesofproblemitself,thecomputationalvolumeofthealgorithmincreasesbyexponentially,andthealgorithmneedsmoresearchingtimewiththeexpansionoftheproblem.IntelligentoptimizationproblemforsolvingNParetheantcolonyalgorithm,greedyalgorithm,etc.Suchalgorithmsdonotdependonthecharacteristicsoftheproblemitself,andhavethestrongglobalsearchability.Relatedstudieshaveshownthatitcaneffectivelyimprovetheabilitytosearchfortheoptimalsolutionbycombiningtheintelligentoptimizationalgorithmwiththelocalheuristicsearchingalgorithm.
Shuffledfrog-leapingalgorithmisanewintelligentoptimizationalgorithm,itcombinestheadvantagesofmemealgorithmbasedongeneticevolutionandparticleswarmalgorithmbasedongroupbehavior.Ithasthefollowingcharacteristics:
simpleinconcept,fewparameters,thecalculationspeed,globaloptimizationability,easytoimplement,etc.andhasbeeneffectivelyusedinpracticalengineeringproblems,suchasresourceallocation,jobshopprocessarrangements,travelingsalesmanproblem,0/1knapsackproblem,etc.However,thebasicleapfrogalgorithmiseasytoblendintolocaloptimum,andthusthispaperimprovedtheshuffledfrog-leapingalgorithmtosolvecombinatorialoptimizationproblemssuchasknapsackproblem.Experimentalresultsshowthatthealgorithmiseffectiveinsolvingsuchproblems.
1Themathematicalmodelofknapsackproblem
KnapsackproblemisaNP-completeproblemaboutcombinatorialoptimization,whichisusuallydividedinto0/1knapsackproblem,completeknapsackproblem,multipleknapsackproblem,mixedknapsackproblem,thelatterthreekindscanbetransformedintothefirst,therefore,thepaperonlydiscussedthe0/1knapsackproblem.Themathematicalmodelof0/1knapsackproblemcanbedescribedas:
where:
nisthenumberofobjects;
wiistheweightoftheithobject(I=1,2…n);
viisthevalueoftheithobject;
xiisthechoicestatusoftheithobject;whentheithobjectisselectedintoknapsack,definingvariablexi=1,otherwisexi=0;
Cisthemaximumcapacityofknapsack.
2Thebasicshuffledfrog-leapingalgorithm
ItgeneratesPfrogsrandomly,eachfrogrepresentsasolutionoftheproblem,denotedbyUi,whichisseenastheinitialpopulation.Calculatingthefitnessofallthefrogsinthepopulation,andarrangingthefrogaccordingtothedescendingoffitness.Thendividingthefrogsoftheentirepopulationintomsub-groupof,eachsub-groupcontainsnfrogs,soP=m*n.Allocationmethod:
inaccordancewiththeprincipleofequalremainder.Thatis,byorderofthescheduled,the1,2,...,nfrogswereassignedtothe1,2,....,Nsub-groupsseparately,then+1frogwasassignedtothefirstsub-group,andsoon,untilallthefrogswereallocated.
Foreachsub-group,settingUBisthesolutionhavingthebestfitness,UWisthesolutionhavingtheworstfitness,Ugisthesolutionhavingthebestfitnessintheglobalgroups.Then,searchingaccordingtothelocaldepthwithineachsub-group,andupdatingthelocaloptimalsolution,updatingstrategyis:
where,Sistheadjustmentvectorofindividualfrog,Smaxisthelargeststepsizethatisallowedtochangebythefrogindividual.Randisarandomnumberbetween0and1.
3Theimprovedshuffledfrog-leapingalgorithmforKP
Afrogisonbehalfofasolution,whichisexpressedbythechoicestatusvectorofobject,thenfrogU=(x1,x2,…,xn),where,xiisthechoicestatusofthei-thobject;whenthei-thobjectisselectedintoknapsack,definingvariablexi=1,otherwisexi=0;
f(i),thefitnessfunctionofindividualfrogcanbedefinedas:
3.1Thelocalupdatestrategyoffrog
Thepurposeofimplementingthelocalsearchinthefrogsub-groupistosearchthelocaloptimalsolutionindifferentsearchdirections,aftersearchinganditeratingacertainnumberofiterations,makingthelocaloptimuminsub-groupgraduallytendtotheglobaloptimumindividual.
Definition1Givingafrog’sstatusvectorU,theswitchingsequenceC(i,j)isdefined:
where,Uisaidthestateofobjectibecomesfromtheselectedtothecancelstate,orinturn;Ui=Uj,objectiandobjectjexchangeplaces,thatobjectiandobjectjareselectedordeselectedatthesametime.Ui≠Uj,objectiisselectedorcanceled,orinturn.Thenthenewvectorofswitchingoperationis:
Definition2SelectinganytwovectorsUiandUjoffrogfromthegroup,D,thedistancefromUitoUjisallexchangesequencesthatUiisadjustedtoUj.
where,misthenumberofadjusting.
Basedontheabovedefinition,theupdatestrategyoftheindividualfrogisdefinedasfollows:
where,listhenumberofswitchingsequenceD(UB,UW)forupdatingUW;lmaxisthemaximumnumberofswitchingsequenceallowedtobeselected;sistheswitchingsequencerequiredforupdatingUW.
3.2Theglobalinformationexchangestrategy
Duringtheexecutionofthebasicshuffledfrog-leapingalgorithm,theoperationofupdatingthefeasiblesolutionwasisexecutedrepeatedly,itisusuallytomeetthesituationthatupdatingfail,thebasicshuffledfrog-leapingalgorithmupdatesthefeasiblesolutionrandomly,buttherandommethodoftenfallsintolocaloptimumorreducestherateofconvergenceofthealgorithm.
Obviously,thekeythatovercomingtheshortcomingsofbasicSFLAinevolutionis:
itisnecessarytokeeptheimpactoflocalandglobalbestinformationonthefrogjump,butalsopayattentiontotheexchangeofinformationbetweenindividualfrogs.Inthispaper,firsttwojumpingmethodsinbasicSFLAareimprovedasfollows:
Pn=PX+r1*(Pg-Xp1(t))+r2*(PW-Xp2(t))(5)
Pn=Pb+r3*(Pg-Xp3(t))(6)
Where,Xp1(t),Xp2(t),Xp3(t)areanythreedifferentindividualswhicharedifferentfromX.Meanwhile,removingthesortingoperationaccordingtothefitnessvalueoffrogindividualfrombasicSFLA,andappropriatelylimitingthethirdfrogjump.Thus,wegetanefficientmodifiedSFLAbasingontheimprovementsofabove.Inthemodifiedalgorithm,thefrogindividualinthesubgroupgeneratesanewindividual(thefirstjump)byusingformula(5),ifthenewindividualisbetterthanitsparententitythenreplacingtheparentindividual.otherwisere-generatinganewindividual(thefrogjumpagain)byusing(6).Ifbetterthantheparent,thenreplacingit.orwhenr4≤FS(thepre-vector,itscomponentsare0.2≤FSi≤0.4),generatinganewindividual(thethirdfrogjump)randomlyandreplacingparententity.
Thenewupdatestrategywillenhancethediversityofpopulationandthesearchthroughoftheworstindividualintheiterativeprocess,whichcanensurecommunities’evolvingcontinually,helpimprovingtheconvergencespeedandavoidfallingintolocaloptimum,andthenexpectalgorithmbothcanconvergetothenearbyofoptimalsolutionquicklyandcanapproximateaccuracy,improvedtheperformanceoftheshuffledfrog-leapingalgorithm.
4Simulationexperiment
Twoclassical0/1knapsackprobleminstanceswereusedinthepaper,example1wastakenfromtheliterature[11],example2wastakenfromtheliterature[12].Thecomparisonalgorithmusedinthepaperwasbranchandboundmethodfor0/1knapsackproblem.Underthesameexperimentalconditions,twoinstancesofsimulationexperimentswereconducted20times,theaveragestatisticalresultswereshowninTable1andTable2.
5Conclusion
Theshuffledfrog-leapingalgorithmisakindofsearchalgorithmwithrandomintelligenceandglobalsearchcapability,thispaperimprovedshuffledfrog-leapingalgorithmandsolvedthe0/1knapsackproblembyusingthealgorithm.Experimentsshowthattheimprovedalgorithmhasbetterfeasibilityandeffectivenessinsolving0/1knapsackproblem.
Acknowledgements
ThisworkwassupportedbyXXX(基金号).OurspecialthanksareduetoProf.XXX(name),XXX(affiliation),forhishelpfuldiscussionwithpreparingthemanuscript.
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