177 

AdvancesinProductionEngineering&Management ISSN1854‐6250

Volume14|Number2|June2019|pp177–188 Journalhome:apem‐journal.org

https://doi.org/10.14743/apem2019.2.320 Originalscientificpaper

  

An improved flower pollination algorithm for optimization of intelligent logistics distribution center  

Hu, W.a,* aSchool of Economics and Management, Shanghai University of Electric Power, Shanghai, P.R. China   A B S T R A C T   A R T I C L E   I N F OIt iseasyto fall into localoptimalsolution insolvingtheoptimal locationofintelligentlogisticsdistributioncenterbytraditionalmethodandtheresultofoptimization is not ideal. For this, the study puts forward an optimizationmethodof intelligent logisticsdistributioncenterbasedon improved flowerpollination algorithm. This method uses the logic self‐mapping function tocarryoutchaoticdisturbancetothepollengrains,sothatthepollengrainsetlackingthemutationmechanismhasstrongself‐adaptability,andtheconver‐genceoftheoptimalsolutioninthelaterstageofthealgorithmiseffectivelyprevented.Theboundarybuffer factor isused tobuffer the cross‐boundarypollengrainsadaptivelysoastopreventthealgorithmfromthelocaloptimi‐zation,andtheconvergencespeedandtheoptimizationaccuracyofthealgo‐rithmcanbeimprovedobviouslyinprocessingtheoptimallocationofintelli‐gent logistics distribution center. The convergence of the algorithm is ana‐lyzedtheoreticallybyusingtherealnumbercodingmethod,andthebiologi‐calmodelandtheoreticalbasisofthealgorithmaregiven.Theexperimentalresults show that the proposed method has better performance than thetraditionalone,andthealgorithmoutperformsageneticalgorithmandparti‐cleswarmalgorithm.Itprovidesafeasiblesolutionfortheintelligentlogisticsdistributioncenterlocationstrategy.Itaffordsagoodreferenceforimprovingand optimizing the internal logistics of the manufacturing system and theoperationalefficiencyoftheentireintelligentlogisticssystem.

©2019CPE,UniversityofMaribor.Allrightsreserved.

  Keywords:Intelligentlogistics;Distributioncenter;Optimization;Intelligentoptimizationalgorithm;Flowerpollinationalgorithm;Intelligentlocation

*Correspondingauthor:hwshiep@shiep.edu.cn(Hu,W.)

Articlehistory:Received18October2018Revised21April2019Accepted15May2019 

  

1. Introduction 

WiththeincreaseofvehicleownershipinChina,theproblemoftraffic jambecomesmoreandmoreserious.It’sabottlenecktochoosetheoptimalrouteofdistributioncenterinarealtimemanneraccordingtotheroadconditioninthefieldofintelligentlogistics[1‐2].Thetraditionalsolution is tooptimize thedistributioncenter routeunder certain constraints, and realize theoptimizationofthedistributioncenterrouteundertheconditionofonlyconsideringthesingleindexsuchastheleastcostortheshortestpath[3‐4].Withtheincreaseofroadconditionanddistributiontaskcomplexity,itisdifficulttosatisfythedemandofintelligentlogisticsdistribu‐tionindexwithonlysingleconstraintcondition.Theoptimizationofdistributioncenter'sopti‐malroutegraduallyevolves intoamulti‐constraintoptimizationproblemthatsatisfiesseveralexpectationsofdistributionpersonnelintheroadnetwork[5‐6].Logisticsdistributioncenterisabridgebetweensupplyanddemand, its locationoptimizationstrategy is thecorecontentoflogistics network system analysis, and often determines the distribution system andmode ofintelligentlogisticssystem.Therefore,it’stheoreticallyandpracticallysignificanttoselectarea‐

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sonablelogisticsdistributioncenter,achievethebestbalancebetweensupplyanddemand,andthusimprovetheoperationefficiencyoftheentireintelligentlogisticssystem[7‐8].

Inrecentyears,researchershaveadoptedalotofmethodstostudylocationstrategiesofin‐telligentlogisticsdistributioncenter,andachievedalotofinfluentialresults.Researchscholarsathomeandabroadhavedeeplystudiedthelocationoptimizationofintelligentlogisticscenter,andclassifieditintocontinuouslocationmodelanddiscretelocationmodel[9‐10].Thecontinu‐ousmethodmainlysolvesthechoiceanddecision‐makingoflogisticsnodeinplanearea.Manyscholarshave studied theuncertaintyofparameters in themodel. Literature [11] studied thestochasticprogrammingmodelofsupplychainnetworkunderuncertainconditions.Document[12] deltwith the location of commercial facilities in uncertain environments. Literature [13]established an interval nodedecision‐makingmodel of logisticsnetworkbasedonbarycentermethod,andconstructedanintervaldecision‐makingmodelofmulti‐commodityandmulti‐nodecontinuouslogisticsfacilitylocation.Literature[14]proposesalogisticsdistributioncenterloca‐tionmodelbasedonbi‐levelprogramming.Thedistributionpointsandusercostareconsideredrespectivelyintheupperandlowerlevelsofthemodel,andasimpleheuristicalgorithmisusedtooptimizethemodel.Literature[15]usedAFSandTOPSISrelatedtheoriestoexploretheloca‐tionmodeloflogisticsdistributioncenter.Literature[16]studiedthelocationissueofdistribu‐tion center based on rough set and interactive multi‐objective fuzzy decision theory. Thesemethodsabovefocusonqualitativeorquantitativeresearchontheoptimizationof locationofintelligentlogisticsdistributioncenters,andhaveachievedcertainresults,whichprovidearef‐erence for further improvingandoptimizingtheoperationalefficiencyof theentire intelligentlogisticssystem.

Fromexistingresearches,it’sfoundthattherearetwoproblemsintherouteoptimizationofintelligentlogisticsdistributioncenter:(1)Fewresearchesfocusonthespecificapplicationre‐quirementsofintelligentoptimizationalgorithmintherouteoptimizationofintelligentlogisticsdistributioncentersbuttheinfluenceofthealgorithmontheoptimizationprecisionandconver‐gence speed is ignored; (2) The setting of the route optimization constraints is too simple tomeettheoptimizationprecisionrequirementsunderthecurrentcomplexroadconditions.Aim‐ingattheproblemsinpreviousresearches,thisstudyproposesanoptimizationmethodofintel‐ligentlogisticsdistributioncentersbasedonimprovedflowerpollinationalgorithm.

Flowerpollinationalgorithm(FPA)isanewswarmintelligenceoptimizationalgorithm.Itisastochastic global optimization algorithmwhichmimics themechanism of self‐pollination andcross‐pollinationoffloweringplantsinthebiologicalworld[17].Thealgorithmhasthecharac‐teristicsofeasyimplementation,stronguniversalityandfastconvergencespeed,andhaswideapplication.Thealgorithmissimpletoimplementandhaslessparameters,thegivenconversionprobabilitycansettheconversionthresholdofglobalsearchandlocalsearch,andthealgorithmadoptsLévy flightmechanism,whichhasexcellentglobaloptimizationeffect,so thealgorithmcan be used to solvemany complicated optimization problems. On the basis of analyzing theoptimizationnatureofstandard flowerpollinationalgorithm, thisstudypresentsan improvedflowerpollinationalgorithm,analyzestheconvergenceofthealgorithm,optimizesthelocationmodeloflogisticsdistributioncenterswiththisimprovedalgorithm,andcomparestheobtainedresultswiththeresultsofgeneticalgorithmandstandardparticleswarmalgorithm.Theexper‐imentalresultsshowthatunderthepremiseofsatisfyingsupplyanddemand,themethodpro‐posedinthisstudyobtainsthebestlocationschemeforintelligentlogisticsdistributioncenters,anditsperformanceissuperiortotheothertwoalgorithms.Theschemeinthisstudyhasagoodguidingsignificancefortheconcretepractice.

2. Intelligent logistics distribution center location model 

Theproblemofthevehicletravelpathbelongstotheoptimalschedulingproblem,andthecostisreducedbysolvingtheoptimaldrivingroute.Therelatedtheoriesandsolvingalgorithmsareofgreatsignificanceforimprovingtheefficiencyof logisticstransportation,soithasalwaysbeenthefocusofrelevantexperts.Inrecentyears,researchontheproblemofvehicletravelpathshasproducedmanyresults,suchasvehicleroutingproblemswithmultiplestations,vehiclerouting

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problemswithtimewindowconstraints,andvehicleroutingproblemsforloadingandunload‐ingcargo.Atthesametime,vehicletravelrouteshaveapplicationsinallaspectsoflife,suchasproductdistribution,cargotransportation,andmitigationoftrafficjams.

Itisdifficulttochangethelocationofanintelligentlogisticsdistributioncenteronceit’sde‐termined.Therefore, in theprocessof constructing thedistributioncenter locationmodel, thefactorssuchas fixedcost,managementcostandmaximuminventorycapacityshouldbetakenintoaccount.Inthelogisticsnetworksystem,thedemandofthedemandpointsshouldbelessthanorequaltothesizecapacityofthedistributioncenter.Undertheconditionofsatisfyingthedistanceupperlimit,itisnecessarytofindoutthedistributioncenterfromtheknowndemandpointsanddistributethegoodstoeachdemandpoint.Basedontheaboveproblems,theintelli‐gentlogisticsdistributioncenterlocationmodelcanbeexpressedas:

Min 1

s.t ,( 1,2,⋯ , )(2)

1,( 1,2,⋯ , )(3)

,( 1,2,⋯ , )(4)

(5)

,( ∈ , ∈ )(6)

where,Nrepresentsasetofordinalnumbersforalldemandpoints;Misasetofdemandpointsselectedasthedistributioncenter;Cjisthecostofbuildingadistributioncenter;hj∈ 0,1 ,whenitis1,thepointjisselectedasthedistributioncenter;gjindicatestheunitmanagementcostofmaterialcirculationinthedistributioncenters;Wijrepresentsthedemandatthedemandpointi;dij represents the distance between demand point i and its nearest distribution center j;Zij∈ 0,1 representstheserviceallocationrelationshipbetweenthedemandpointandthedis‐tributioncenter,whenit’s1,itindicatesthatthedemandofthedemandpointiissuppliedbythedistribution center j, otherwiseZij=0. l indicates the upper limit of the distance between thedemandpointand thedistributioncenter.Eq.2 indicates that thedemandofusers shouldbelessthanorequaltothesizecapacityofthedistributioncenter;Eq.3representsensuringthateachdemandpointisservedbyadistributioncenterclosesttoit;Eq.4indicatesthatthereisnocustomeratalocationwithoutadistributioncenter;Eq.5showsthatpdemandpoint(s)is(are)selectedasdistributioncenter(s);Eq.6showsthatthedistributioncentersuppliesthenearbydemandpointsonlywithinalimitedrange.

Inthematerialandproductdistributionpartofthemanufacturingsystem,thismethodcanalsobeusedtoselectthedistributioncenter,andthetarget functioncanbechangedoraddedaccordingtoitsownneeds.Forexample,whendistributingthematerialsneededformanufac‐turing,ifthearrivaltimeofmaterialshasacertainlimit,youcanaddtimeconstraints,andthenuse the improved flowerpollinationalgorithmtosolve. It canbeseen that themethodcanbeusednotonlyinthemanufacturingsystembutalsoinanysystemthatneedstobedistributed.

3. Improved flower pollination algorithm 

3.1 Standard flower pollination algorithm 

Flowerpollinationalgorithmisakindofrandomsearchalgorithmconstructedbysimulatingtheprocess of flowerpollination in thenature,which embodies thepreferencemechanismof thenature[18].Thebionicprincipleisthatflowersbreedtheiroffspringbypollination.Thepollina‐

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tionprocesscanbecarriedoutbyinsectsornaturalwindandwater.Butterfliesareattractedbythecolorandsmellofflowers,andflytoXbestaftercollectingnectaronthepollenXi,soastoreal‐izepollentransferbetweenflowers,whichiscalledcross‐pollinationorglobalpollination.Undertheactionofwind,thepollentransferbetweenadjacentflowersXiandXiisrealized.Thiskindofpollination is called self‐pollination or local pollination. Through global pollination and localpollination, flowers flexibly achieve theprocess of pollen transfer. The ideal conditionsof thealgorithmareassumedasfollows:

Intheprocessofcross‐pollination,pollinatorscarryoutpollinationthroughLévyflight. Self‐pollination is thepollinationprocessofadjacent flowersunder theactionofnatural

force. It is a kind of local pollination. The pollinationmode is determined randomly bytransitionprobability ∈ 0,1 .

Ingeneral, each floweringplantcanblooma lotof flowers,producingmillionsofpollengametes, and in order to simplify the problem,wemake the hypothesis that eachplantonlybloomsone flowerandeachfloweronlyproducesonepollengamete,whichmeansthataflowerorpollengametecorrespondstoasolutiontotheoptimizationproblem.

For the formal description of the flower pollination algorithm, the followingmathematicalmodelisestablished:

Inglobalpollination,pollinatorcarrypollengrains for large‐scaleand long‐distancesearch,soastoensuretheoptimalindividualpollinationandpropagation.Intheprocessofglobalpolli‐nation,theformulaforupdatingthepositionofpollenis:

∗ (7)

where, is thespatial locationcorresponding to the i‐thpollen in the t‐th iteration;g* is thespatialpositionof theoptimalsolution inthecurrent iteration(i.e., thet‐th iteration).Thepa‐rameterL>0isthestepsizeandfollowstheLevydistribution,asshowninEq.8.

~⁄

, 0 (8)

where,Г(λ)isastandardgammafunction,parameter1<λ≤ 3isλ=1.5inthisstudy.Theformulaofpollengrainpositionrenewalforlocalpollinationofadjacentflowersisasfol‐

lows:(9)

where, and arethespatialpositionsoftwopollengrainsrandomlyselected,and isaran‐domoperatoruniformlydistributedonthe[0,1]interval.

The transitionprobabilityP is theprobabilitydeterminingwhetherpollengrains arepolli‐natedgloballyorlocally,andisaconstant,with ∈ 0,1 .ThehigherthevalueofPis,thehighertheprobabilityofcarryingoutthelocalpollinationactivityis,andthelowerthechanceofcarry‐ingouttheglobalpollinationis.

Duetothefactthatadjacentflowersareeasiertopollinate,theprobabilityoflocalpollinationwillbehigherthanthatofglobalpollination.Throughthestudyofparameters, it isfoundthattheoptimizationproblemp=0.8isthebestchoiceofparameters.Theprocessofoptimizingthealgorithm:randomlydispersethepopulationofpollengrainsinsolutionspace,andbycompar‐ing the size of random production number randand transition probability P, and determinewhether each pollen grain will carry on global pollination or local pollination; after severalmovementsandupdatesofpositions,allthepollengrainswillconvergetotheplaceofoptimumfitness;so,thesuccessfulpollinationandinseminationofpollengrainscanberealized,andtheoptimizationcanbeachieved.Thestepsofthealgorithmareasfollows:

Step1:Initialize basic parameters of the algorithm. Set the number of pollen grains as n,transitionprobabilityasPandthemaximumvalueofthenumberofiterationsasMaxT;

Step2:Randomly initialize the positions of all pollen grains (i= 1,…,n), find out the pollengrain individuals with the best fitness in the initial situation, and record the currentpositionandfitnessoftheindividuals;

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Step3:Starttheiteration,andupdatethepositionsofallpollengrains.Firstgeneratearandomnumberrandbetween[0,1]andifRand>p,thepollengrainsarepollinatedgloballyac‐cordingtoEq.7;otherwise,thepollengrainsarelocallypollinatedaccordingtoEq.9.Af‐terthepositionupdateofallpollengrainsiscompletedinthisiteration,thecurrentop‐timalpositionandfitnessofeachpollengrainarecalculated,andthepositionandfitnessoftheoptimalindividualinthepopulationareupdated;

Step4:GotoStep5whenitreachesthemaximumnumberofiterations,otherwisegotoStep3andstartthenextiteration;

Step5:Gettheglobaloptimalsolutionandthetargetfunctionvalueatthistime.

3.2 Chaos optimization strategy 

The principle of chaotic search: first, the variable of chaotic space ismapped to the solutionspaceof theproblemby somerules, and then the solution space is traversedbyvirtueof thechaotic feature.Therearemanychaoticmappingmodels.Theresearchshowsthat thechaoticsequenceobtainedbyusinglogicalself‐mappingfunctionisbetterthanLogisticmapping.There‐fore, the logical self‐mapping function is used in this study to construct chaotic sequences torealizechaotictraversal,andthefollowingexpressionsaregiven:

1 2 (10)

where, isthej‐thdimensionalcomponentofachaoticvariable,kisthenumberofiterationsteps, ∈ 1,1 ,j = 1,2,…,d. When 0, the chaotic sequence obtained by logical self‐mappingmodelpossessesdynamiccharacteristicsofchaosandissensitivetoinitialvalues.Thealgorithmstepsareasfollows:

Step1:Setk=0,thedecisionvariable ismappedtothechaoticvariable between–1and1accordingtoEq.11.

2 ,

, ,1(11)

where j=1,2,…,d, , and , are the upper boundary and the lower boundary ofthej‐thdimensionalvariablesearch,respectively.

Step2:Accordingto ,thecarrieroperationisperformedwithEq.4,andthecalculationiter‐atestothenextgenerationofthechaoticvariable .

Step3:Thechaoticvariable isconvertedintodecisionvariable accordingtoEq.12.

, , , , (12)

wherej=1,2,…,d.Step4:The performance analysis and evaluation on the newly generated solution are carried

out according to decision variable . If the chaotic search has reached the presetlimitorthenewsolutionissuperiortotheinitialsolution,thenewsolutionisoutputastheresultofthechaoticsearch,otherwise,setk=k+1andreturntoStep2.

3.3 Boundary buffer factor 

Intheprocessofoptimizationwithflowerpollinationalgorithm,thepositionofpollengrainsislikely to break through its boundary value. In this case, the traditionalmethod is to limit thepositionofpollengrainstotheinterval[–xmax,xmax].Theadvantagesofthismethodaresimpleoperation and small computation, but the boundary compulsory processing method isunfavorabletotheconvergenceofthealgorithm,resultinginalotoferrors.Inordertosolvethisproblem, themethodof dynamically changing interval is adopted, the size of the boundary isscaled according to the situation, the out‐of‐boundary grains are buffered by the boundarybufferfactorandprocessedaccordingtotheout‐of‐boundarypositionofthepollengrains.Thespecificoperationisasfollows:

Whenxij(t)<aj,xij(t)canberepresentedas:

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182  Advances in Production Engineering & Management 14(2) 2019

1 sgn sgn | |/ (13)

Whenxij(t)>bj,xij(t)canberepresentedas:

1 sgn sgn | |/ max (14)

InEq.13andEq.14,sgn issymbolicfunctions, ∈ 0,1 ,ajandbjarelowerandupperlimitsofthegraininthej‐thdimension,respectively.FromEq.13andEq.14,theboundarybufferfac‐torsare treatedaccordingto theactualsituationofpollengrains,andtheactualsituationandmovementofpollengrainsarefullyconsidered.Thesimulationresultsshowthattheboundarybufferfactorcaneffectivelyimprovetheconvergencespeedandoptimizationaccuracyofpollenpollinationalgorithm.

3.4 Specific steps of solution 

Thespecificstepsofoptimizingthelocationproblemof intelligent logisticsdistributioncenterbyusingtheimprovedflowerpollinationalgorithmareasfollows:

Step1:Initializebasicparametersofthealgorithm.Setthenumberofpollengrainsasn,transi‐tionprobabilityasP, themaximumvalueof thenumberof iterationsasMaxT,andthelargestnumberofiterationstepsforchaoticsearchasMaxC.

Step2:Randomly initialize the positions of all pollen grains (i = 1,…,n), find out the pollengrainindividualswiththebestfitnessinthecurrentpollengrainset,andrecordthecur‐rentpositionandfitnessoftheindividuals;

Step3:Starttheiteration,andupdatethepositionsofallpollengrains.Firstgeneratearandomnumberrandbetween[0,1]andifrand>p,thepollengrainsarepollinatedgloballyac‐cordingtoEq.1;otherwise,thepollengrainsarelocallypollinatedaccordingtoEq.3.Af‐terthepositionupdateofallpollengrainsiscompletedinthisiteration,thecurrentop‐timalpositionandfitnessofeachpollengrainarecalculated,andthepositionandfitnessoftheoptimalindividualinthepopulationareupdated;

Step4:Theindividualsofpollengrainsareevaluated,thenindividuals(inpercentage)withthebestfitnessareselectedasthebestpollenset,andtheremaining1–npoorerindividuals(inpercentage)areselectedandreplacedwithrandomlygeneratedindividualsofpollengrains.

Step5:Thecurrentoptimalsolutionoftheindividualafterchaosoptimizationisobtained,andthesearchregionisdynamicallyshrunkbyEqs.7and8.

Step6:TurntoStep7whenthenumberofiterationsisequaltoMaxTorreachesthesetsearchaccuracy;otherwise,turntoStep3toperformthenextiteration;

Step7:Outputtheglobaloptimalsolutionandthecorrespondingobjectivefunctionvalueatthistime.

4. Convergence analysis of chaotic flower pollination algorithm 

Inthechaoticflowerpollinationalgorithm,thepollinationprocessofflowersisthepremiseofconvergenceofthealgorithm,thelong‐distancesearchandLévyflightmechanismofpollinatorsin the cross‐pollination guarantee the convergence stability and global optimization, the self‐pollinationenhancesthe localoptimizationabilityof thealgorithm,andthechaosstrategyen‐hancestheswingofthealgorithmnearthelocalsolution,andreducessearchrange,getsridofthelocalinterference,andacceleratesthesearchspeed.

Ifthecomponentxi(i=1,2,…,n)ofthepollensetX=(x1,x2,…,xn)isrepresentedbytheτdimen‐sionalbinarycodedstring,thecodestringcantake2τdiscretevalues,whichisequivalenttothe2τdiscretevaluesdividingthedefinedinterval[Lbi,Ubi],withtheaccuracyofε=(Ubi–Lbi)/2τ.Therefore, according to this characteristic, the convergence of the chaotic flower pollinationalgorithmcanbeanalyzedbyrealcoding.Ifthesearchaccuracyofthesolutionspaceisassumedtobeε,thesolutionspaceMcanbeunderstoodasadiscretespace,itssizeis| |

/ ,whereeachpoint ∈ isapollengrain,sosetitsfitnessF=f(x)isafunctionoffitness,

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obviously| | ,soitcanbewrittenasF={F1,F2,…F|F|},whereF1>F2>…>F|F|.Accordingtothedifferentfitnessdegree,thesearchspaceMcanbedividedintodifferentnon‐emptysubset{Mi},whichisdefinedas:

| ∈ , (15)wherei=1,2,…,|F|,then: 

| || |; , ∀ ∈ 1,2, … , | | ;

        ∩ ; ∩ , ∀ ; | |(16)

Foranytwoelementsxi∈Miandxj∈Mj,

,      ,      ,      

(17)

ItiseasytoknowthatF1canbeconsideredaglobaloptimalsolutionF*,andthatindividualswithfitnessequaltoF*shouldbeinM1.Intheiterativeprocessofthealgorithm,thenumberNofpollenpopulationsremainsunchanged,p={x1,x2,…,xN}. SetP asa set, containingallpopula‐tions,andsincethepollengrains in thepopulationareallowedtobethesame, thenumberofpossiblepopulationsis:

| | | | (18)

Then,inordertojudgethequalityofthepopulations,thefitnessfunctionofpcanbedefinedas:

max | 1,2, . . . , (19) 

Similarly,accordingtothedifferentfitness,Pcanbedividedinto|F|non‐emptysub‐sets(i=1,2,…,|F|),where |Pi| represents the sizeof the setPi.The setP1 includesallpopulationswithfitnessF1.

Setpij(i=1,2,…,|F|;j=1,2,…,|Pi|)representsthej‐thpopulationofpi.Undertheactionoftheevolutionoperator,theprobabilityPrij,klofpijtransitingtopklrepresentsthetransitionprobabil‐ityofanyoneofthepopulationsfrompijtopk,andPri,krepresentsthetransitionprobabilityofanyoneofthepopulationsfrompitopk,then:

Pr , Pr ,

| |; Pr , 1

| |; Pr , Pr , (20) 

Asquarematrix ∈ iscalled:

anonnegativematrix,ifaij 0,∀ , ∈ 1,2, . . . , ; aprimitivematrix,ifAisnonnegative,andthereisanintegerk 1,soAk>0; arandommatrix,ifAisnonnegativeand∑ 1,then∀ ∈ 1,2, . . . , ; areduciblerandommatrixcanperformarow‐columntransformationofthesameformto

obtain 0 ,wheresisthem‐orderprimitivematrix,andRandT 0.

Definition1: If an evolutionary algorithm converges to a global optimal solution, its sufficientandnecessaryconditionsare:

lim→

Pr ∗ 1(21)

where,PrrepresentsprobabilityandPtrepresentsthet‐thgenerationpopulation.

Theorem1:SetPr 0 isareduciblerandommatrix,Sisthem‐orderprimitivematrixand

RandT 0,then

Pr lim→

Pr lim→

0 00(22)

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Obviously,Pr isastablerandommatrixandPr 1,1, . . . ,1 , , . . . , , 1,

and lim→

0,when0          10      1

.  

Theorem2:Inthechaoticflowerpollinationalgorithm,for∀ , ∈ 1,2, . . . , | | ,

Pr ,0,  0,  

(23)

Prove: First, for∀ ∈ 1,2, . . . , | | ,∃ ∗ ∗, ∗, . . . , ∗ ∈ , so f(x*) = Fi,∀ ∈ ,1,2, . . . , | |, 1,2, . . . , | |,∃ , , . . . , ∈ and f(X') = Fk. Under the ac‐

tion of evolutionary operators, pij is transited to be pkl, if pij of the t‐th generation isevolvedtobepklofthe(t+1)‐thgeneration,andforconvenience,recordthemas and

.Thechaoticflowerpollinationalgorithmhastheoptimizationstrategyandthehis‐toricaloptimalsolutionF*ispreservedineachpollinationprocess.Therefore,theopti‐malpositionofpollenisF*atLt,andundertheconditionthatthepopulationremainsun‐changed,thenextgenerationLt+1isproducedthroughpollinationbehavior,andthebestpollengrainF'inLt+1iscomparedwithF*,ifF'isbetterthanF*,thepreservedhistoricaloptimalsolutionisreplacedbyF',otherwiseunchanged,then:

⇒ ⇒ ∀ , Pr , 0 ⇒ ∀ , Pr , Pr , 0| |

⇒

∀  andPr , 0.

Secondly,ineachpollinationprocess,thealgorithmisiteratedandtheoptimalsolutionisfound.Thebehaviorselectionismadeaccordingtotheprincipleofthefastestorbetterspeed.Therefore,setindividualX',withthefitnessoff(X')=Fk,k i,andX'hasrcompo‐nents( , , … , )differentfromX*,theprobabilityofx'generatedbyx*throughLevyflightoperatorisPr 1 ∗ 0,whereφ istheprobabilityden‐sity function for levy distribution. So the probability of pij evolving to be pk is greaterthanzero,so∀ andPr , Pr , 0.

Theorem3:Thechaoticflowerpollinationalgorithmisgloballyconvergent.

Prove: EachPri,i=1,2,…,|F|canbeviewedasastateonatime‐alignedfiniteMarkovchain.Ac‐cordingtoTheorem1,thetransitionprobabilitymatrixofthealgorithmisexpressedas:

Pr

Pr , 0 ⋯ 0Pr , Pr , ⋯ 0⋮ ⋮ ⋱ ⋮

Pr| |, Pr| |, ⋯ Pr| |,| |

0 

whereR>0,T 0andS=1.AccordingtoTheorem1,

Pr lim→

Pr lim→

000 

where 1and 1,1, . . . ,1 ,soPr isastablerandommatrix,and

Pr

1 0 ⋯ 01 0 ⋯ 0⋮ ⋮ ⋱ ⋮1 0 ⋯ 0

 

Therefore,regardlessofthefitnessstateofthecurrentpopulation,itwillconvergetotheop‐timalfitnessstatewithprobability1afterinfiniteevolution,sotheimprovedflowerpollinationalgorithmisgloballyconvergent.

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5. Results and discussion 

Inthissection,theresultsofoptimizationoftheproposedmethod,geneticalgorithmandstand‐ardparticle swarmoptimization algorithmare compared inMatlabenvironment toprove theeffectivenessandfeasibilityoftheproposedmethod.Duringtheexperiment,thecoordinatesof31citiesarecollected,andthepositioncoordinatesofeachdemandpointandthematerialde‐mandaregiveninTable1.Thematerialdemandinthetableisthestandardizedvalueanddoesnotrepresenttheactualvalue.Thedimensionlessprocessisappliedto31logisticspoints.Thedimensionlessprocessusesthequotientofthemeanvalueofeachfactorvariabletoobtaindi‐mensionlessdata.

Theparametersofflowerpollinationalgorithmareasfollows:p=0.8,beta=1.5,thenumberofchaositerationsk=10,andtheratioofselectingthebestpollensetisn=20.Theimprovedflower pollination algorithm is used to optimize the locationmodel of the intelligent logisticsdistributioncenter.Thepointvalue is set tobe6,which indicates that6 logisticsdistributioncentersareselectedfrom31demandpoints.

Inthesimulationexperiment,thefinaldataistheaverageresultafterrunningtheprogramfor100times,soastoreducetheerror.Theconvergencecurveoftheimprovedflowerpollina‐tionalgorithmisshowninFig.1.

ThelocationschemeoftheintelligentlogisticsdistributioncenterisshowninFig.2.Theboxrepresents the distribution center, and the dot represents the demand point. The connectionbetweentheboxandthedotindicatesthatthelogisticsdistributioncenterisresponsibleforthedistributionof thematerials at ademandpoint.The locationmodel of the intelligent logisticsdistributioncenterisoptimizedbythemethodproposedinthisstudy,andthelocationschemeis[2217629611].

Intheprocessofsolvingthelocationoptimizationoftheintelligentlogisticsdistributioncen‐ter with the improved flower pollination algorithm, chaos optimization and boundary bufferfactorareaddedtoeffectivelyavoidthealgorithmfallingintolocaloptimization,sothatthecon‐vergencespeedandoptimizationaccuracyof thealgorithmareobviously improved, reflectingthattheflowerpollinationalgorithmneedsnoaccuratedescriptionoftheprobleminsolvingthepractical problem, but can quickly obtain the optimal solutionwithin a certain limit. Table 2shows the performance comparison between the proposed scheme, genetic algorithm andstandardparticleswarmoptimization(PSO)inoptimizingthelocationmodelofintelligentlogis‐ticsdistributioncenter.

Table1Positionofdemandpointandmaterialdemand

No. Coordinates Demand No. Coordinates Demand1 (2378,3172) 50 17 (4462,3641) 902 (1689,2623) 60 18 (2685,3252) 803 (4973,4582) 60 19 (4891,3431) 504 (2732,3452) 40 20 (2541,3964) 405 (1367,2473) 70 21 (4571,3115) 906 (3874,3994) 70 22 (2714,3574) 607 (4953,2363) 30 23 (4325,3683) 708 (3852,3372) 60 24 (4232,3231) 609 (2454,3068) 60 25 (2232,3462) 5010 (2122,1374) 80 26 (4436,3285) 5011 (4132,3879) 40 27 (3340,4112) 3012 (2244,3896) 30 28 (1203,2183) 6013 (4564,1598) 30 29 (4632,3461) 8014 (3933,4212) 80 30 (2262,3365) 5015 (3284,1475) 30 31 (3422,4316) 7016 (3433,2756) 50

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186  Advances in Production Engineering & Management 14(2) 2019

Fig.1ConvergencecurveFig.2Locationscheme

Table2PerformancecomparisonofseveralalgorithmsAlgorithm Locationscheme Averagedistributioncost Numberofiterations Runningtime/s

GA 2217629611 1892 110 36.3PSO 2217629611 1363 60 15.5

ImprovedFPA 2217629611 1012 26 11

Table2showsthatwhengeneticalgorithm,standardparticleswarmoptimizationalgorithm

andthemethodinthisstudyoptimizethelocationmodeloftheintelligentlogisticsdistributioncenter, theoptimalschemecanbeobtained,Theaveragedeliverycostrequiredbythegeneticalgorithmis1892,thenumberofitsiterationsis110,anditsrunningtimeis36.3seconds.Theaveragedeliverycostrequiredbythestandardparticleswarmoptimizationalgorithmis1363,thenumberofitsiterationsis60,anditsrunningtimeis15.5seconds.Theaveragedeliverycostrequiredbytheimprovedflowerpollinationalgorithmis1012,thenumberofitsiterationsis26,anditsrunningtimeis11seconds.Therefore,theperformanceoftheimprovedflowerpollina‐tionalgorithmisgenerallybetter thantheother twoalgorithms. Inorder to furtherverify theuniversalityoftheimprovedflowerpollinationalgorithminsolvingsuchproblems,thenumberof logisticsdemandpoints ischanged,andsimulationexperimentsarecarriedoutondifferentinitialdatarespectively.Thesimulationresultsshowthatwhenthenumberoflogisticsdemandpointsislarge,theadvantagesofthealgorithmpresentedinthisstudyaremoreobvious.Atthesame time, we also find that the flower pollination algorithm has the advantages of self‐organizingability,distributedoperationandpositive feedback,andcanperceivethechangeofsurrounding road conditionparameters in real time in complex roadnetwork. It is especiallysuitableforthepathoptimizationproblemofintelligentlogisticsdistributioncenterundermul‐ti‐constraintconditions.

6. Conclusion 

Theoptimallocationmodelofintelligentlogisticsdistributioncentersisanon‐convexandnon‐smooth nonlinear model with complex constraints, which belongs to NP‐hard problem. Thisstudyusesthe intelligentoptimizationmethodtothepathoptimizationproblemof the intelli‐gent logistics distribution center, and proposes an improved flower pollination optimizationalgorithm.Accordingtotheactualcharacteristicsofthewholelogisticsnetworksystem,aloca‐tionmodeloftheintelligentlogisticsdistributioncenterisconstructed.Accordingtothebiologi‐calcharacteristicsofflowerpollination,therealizationprincipleoftheflowerpollinationalgo‐rithmisdescribedfromthemechanism,andtheconvergenceofthealgorithmisanalyzedbyrealcodingmethod. The optimization performance of the algorithm is analyzed in detail, and thebiologicalmodelandtheoreticalbasisofthealgorithmaregiven.

Inordertoavoidthelocaloptimizationandimprovetheconvergencespeedandoptimizationaccuracyofthealgorithm,thisstudyintroduceschaosoptimizationandboundarybufferfactor.

An improved flower pollination algorithm for optimization of intelligent logistics distribution center

By means of logical self-mapping function, the pollen grains are disturbed by chaos, which makes the pollen grain set without mutation mechanism have strong self-adaptive ability. Meanwhile, the size of the boundary is dynamically scaled according to the actual situation, and the boundary buffer factor is used to buffer the pollen grains crossing the boundary. The exper-imental results show that compared with other methods, the method proposed in this study is more effective in solving the location optimization of the intelligent logistics distribution center, and can quickly and accurately find the best logistics distribution center for demand points.

Acknowledgement This research is supported by Humanities and Social Sciences project of the Ministry of Education of China (No.17YJCZH062).

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