SCI论文模板Word文件下载.docx

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Abstract

Shuffledfrog-leapingalgorithm（SFLA）haslongbeenconsideredasnewevolutionaryalgorithmofgroupevolution,andhasahighcomputingperformanceandexcellentabilityforglobalsearch.KnapsackproblemisatypicalNP-completeproblem.Forthediscretesearchspace,thispaperpresentstheimprovedSFLA,andsolvestheknapsackproblembyusingthealgorithm.Experimentalresultsshowthefeasibilityandeffectivenessofthismethod.

Keywords:

shuffledfrog-leapingalgorithm;

knapsackproblem;

optimizationproblem

0Introduction

Knapsackproblem（KP）isaverytypicalNP-hardproblemincomputerscience,whichwasfirstproposedandstudiedbyDantzing​​inthe1950s.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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