历届优秀汇总-2013年题奖1264队题

第二届“认证杯”数学中国页参赛队伍的参赛队号：（请各个参赛队提前填写好竞赛（由竞赛送至评委团前竞赛评 （由竞赛评委团评阅前进 DynamicHR:Hermitcrabs’behaviorandthetheoryofvacancychainsgiveusabeneficialmethodwhichcanbeappliedinthemanagementofHR.Thismethodcanimprovetheeffectivenessandcontrolexpenditureswithinareasonablerangeatthesametime.Inourmodels,wedividetheproblemintothestatichumanresourcesprogramming,theshortestqueueofjobapplicantsandthedynamicmodelofhumanresourcesmanagement.HRallocationoffixednumberofemployeesandjobsisthebasisofdynamicmanagement.ly,inmodelI,wedefinethesystemofmeasuringemployees’workingeffectiveness.Basedonthis,wesetuptheprogrammingmodeltoachievetheoptimaltotaleffectiveness.Secondly,weareinspiredbythecrabs’behaviorofwaitingforacertainperiodoftime.Therefore,weestablishmodelIItodeterminetheshortestqueueofjobapplicants.Thefunctionofcost-effectivenessisthebasisofthismodel.ModelIIIisadynamicmodel.InitialjobvacanciesatthebeginningofperiodtarethedrivingforceofHRadjustment.WedefinethemechanismofthisdrivingforcetoguaranteetheoptimalHRallocation.WehavereferredtotheBPneuralnetworkalgorithmandgeneticalgorithm.AccordingtotherelationshipofdifferentperiodsandthecontinuousprocessofoptimizationofHR,wesimulatetheprocessandpredictthetrendinthefuturewithcomputers.Inthelast,wesuggestsomepossiblewaystoimproveourmodel,includingthedifficultyincollectinginformationaboutapplicantsandthechangeofworkingeffectivenesswithtime,etc.Thesearealsoourdirectioninthefuture.KeyEffectivenessoptimization,mechanismofdynamiccontrol,random Problem Variablesand ModelI:StaticHumanResources Definitionof IndexesofIndividualWorking IndexesofIndividualWorking IndexesofIndividualWorking ThePrimary TheObjective TheConstraint ThePrimary TheIntermediate TheAdvanced ModelII:TheShortestQueueofJob TheDistributionofHuman TheCost TheOptimalLengthofApplicant TheModelofSelectionsofmorethanone ModelIII:TheDynamic TheTimesofJob TheShortestTransferPathof ThePredictionofTransfer TransferofaSingle Transferofmorethanone UncertainTransferof PredictionsoftheChangeofHR Thegoalof ThePredictionofTransfer ysisofSolutionsand ysisofthe Solutionsofthe AnswertoQuestion AnswertoQuestion EvaluationoftheModelandPossible AdvantagesofOur DisadvantagesofOur Possible ApplicationofRandom TheChangeofeffectiveness Advertising ProblemWhenalonecrabencounteredoneofthebeautifulnewss,itbehavedspecially.Afteraseriesofbehavior,theresourcesallocationwithinthegroupreachesSociologistsandeconomistsusetheterm“vacancychain”todescribeasequentialexchangeofresourcesthatbenefitseveryindividualinthesequence.vacancychainsareanexcellentwaytodistributeresources.Soitmakessensethathermitcrabshaveevolvedsophisticatedsocialbehaviorstomakethemostofvacancychains.Therefore,thegoalhereEvolvesocialstrategiestoexchangejobsinqueuesothateveryonebenefitsfromit.Canweapplythemtoallindustries?Ifnot,whatconditionsneedtomeet?NowanewHumanResourcewebsitewantstoadoptthisidea,Howmanyjobapplicantsneedtosatisfybothemployersandemployees.RunittopredictthechangeofHRexpendsforsomeenterprisecustomerswithinthenextfiveyears.PrepareanadvertisingsheetforthewebsitehighlightingthestrategiesandPreliminaryHermitcrabs’behaviorofexchangeofsheltershassignificantmeaningstothemanagementandoptimizationofhumanresources.Thisbehaviorisbasedonthesatisfactoryofindividualdemandandtriestodistributeresourcesinthewholesocietyinthemostbeneficialway,thoughadynamicprocessofadjustment.Givenintheproblems,wearerequiredtosetupadynamicmodeltodescribethemovementofhumanresourcesinabusinessbasedontheysisofhermitcrabs’behaviorandvacancychain.Therearemanyreasonscausingjobvacanciesinabusiness.Generally,thesevacancieswilllybetakenupbyexistingemployees panytransferofjobs.Thenthevacanciesleftwillbefilledbynewstaffsthoughrecruitmentfromtheoutsideofthebusiness.Andthenthisprocesswillberepeatedinthenextjobadjustment.ThisprocessisshowninFigure1.Tosumup,thedynamicmovementofhumanresourcescanbedividedintothreesteps:Step1:Jobvacanciesarise.Weneedtoreallocateexistingemployeeswithinthebusinesstothedifferentjobstoreachoptimization.Step2:Afterthereallocation,thejobvacanciesleftwillbetakenupbynewemployeesrecruitedfromthesociety.Step3:whentherearenewjobvacanciesandineachbusinessperiod,repeatstage1and2abovetoadjusthumanresourcesallocation.Specifically,step1istheissueofallocatinghumanresourcesundertheconditionthatthenumberofjobsandemployeesareconstant.Wewilltreatthe izationofthetotaleffectivenessfunctionasourgoal.Inthisessay,wedescribethisprocessasthestaticmodelofhumanresourcesallocation.Step2ismainlyusedinanopenprocessofrecruitment.Thisincludesthedeterminationofthenumberofjobapplicantsinordertoachievethebalancebetweenthebestchoiceofemployeesandthesmallestnumberofapplicantsrequired.Theproblemofthebalanceisduetotherandomizationofprobabilityofeachapplicant’sStep3isadynamicmodelofhumanresourcesmanagementandallocation.WehavereferredtotheBPneuralnetworkalgorithmandgeneticalgorithm.Wesetupadynamicmodelwithmultipleinputandsingleoutput.TheflowchartisshownasFigure1:FlowChartoftheDynamicModelInthisprocess,weneedtoconsiderthefollowingfactors:Theinitialjobvacanciesarethedrivingforceofthisdynamicsystem.Theprobabilitiesofjobvacanciesvaryindifferentbusiness.(Herewedefinejobvacancyasthatthepreviousemployeeleftthejobandhenceavacancyarises.)Thechoiceofnewlyrecruitedemployeesisbasedonthedemandofthebusiness.Therefore,theindividualeffectivenessofthenewlyrecruitedemployeesdependsonthenatureofthejobvacanciesofthebusiness.Upontheappearanceofthejobvacancies,weshouldlyconsidertofillthesevacancieswithexistingemployees.Thismeansthattherewillbejobexchangesandreallocationofstaffswithinthebusiness.Thisprocessissimilarwiththebehaviorofhermitcrabs.Practicalsituationalsoapprovestheprocess.Mostjobvacanciesariseasaresultofretirementandothercommonfactors.Thesevacanciesareusuallyofhighlevelsandcannotbetakenupbynewemployees.Onlywhenthepreviousemployeesgetpromotedandleavethepreviousjoboflowlevelsvacant,thevacanciescanbefilledbynewlyemployedstaffs.Theninthenextperiod,newemployeesinthisperiodcanhavethechancetoexchangejobswithexistingemployees.Expendituresincurredintheprocessofrecruitmentisonlyrelatedtothenumberofapplicants.Thereisnocontrivedfactorsandotherlimitationsintheinternaladjustmentofhumanresources.Duringtheprocessofrecruitment,theemployerscanknowalltheinformationabouttheapplicants.TherearesufficientapplicantsintheVariablesandETheeffectivenessmatrixoftheexpectationoftotaltimesoftheexpectationoftotaltheefficiencyofThelengthofModelI:StaticHumanResourcesForagroupofemployeesoffixednumberandasetoffixedjobs,weneedtotakeaccountoftheirability,theirpreferencetodifferentjobs,thevaluetheemployeescanbringintothejobsandtheextenttowhichtheycaninfluencetheperformanceandefficiencyofthegroupasawhole.DefinitionofBeforetheexchangeandreallocationofjobs,themostimportanttaskistolearnaboutemployees’abilityandpreferenceandtrytoavoidinformationasymmetry.IndexesofIndividualWorkingAstheleadersofabusiness,itisnecessarytomakesurethateachemployeeispositionedinthejobwhichheisbestat.Intermsoftheconditionsabove,wemakethefollowingassumptions:Wecanknoweachemployee’sworkingabilityoneachjobthroughinterviewsandregularcheck,etc.Andalltheinformationwegetareaccurate.Thefactoroflearningcurveeffectwhichcanimproveemployees’abilityshouldbeignored.WedonotconsiderthetransferofbusinessandtasksamongdifferentWedividethelevelofeachemployee’sworkingabilityinto5degreesasfollows:Table1:LevelofWorkingAbilityABCD5432Inordertotransferthedegreeofworkingabilityintoavailablenumericalinformationforysispurposes,wemustknowthatthelevelofworkingabilityinpracticeisnonlinear.Accordingtofuzzyevaluationmethods,weapplyCausydistributefunctiontothedegreeofworkingabilitytodothenonlinearchange.Theequationsareas𝑎={[1+𝐴(𝑥− 1≤𝑥≤𝑚𝐿𝑛𝑥+ 3≤𝑥≤WhereA,B,m,nareparameters,xisthevalueofworkingability,aisthevalueafterNowwedefinethefollowingWhenthedegreeisAandthevaluationofworkingabilityis5,thedegreeofmembershipis1,whieansf(5)=1.WhenthedegreeisCandthevaluationofworkingabilityis3,thedegreeofmembershipis0.8,whieansf(3)=0.8.Whenthevaluationofworkingabilityis1,thedegreeofmembershipis0.01, eansf(1)=0.01.Thenwecansolve𝐴={𝐵=𝑚=𝑛=Applythevalueoftheseparametersintotheequationaboveandwecangetthatthezationrangeofemployees’workingabilityis{1,0.9126,0.8,0.5245}.⋯⋯ ⋯⋯ 𝐴= where𝑎𝑖𝑗istheworkingabilityofemployee𝑖injobIndexesofIndividualWorkingEveryemployee’spreferencetowardsworkandjobsandtheiremotionwilltheirperformanceinthejobsandtheirFollowingthemethodusedin2.1.1,wesetupthematrixofindexesofemployees’workingpreferenceasfollows𝑃=[𝑃=[⋮⋯⋱⋮⋯]where𝑝𝑖𝑗istheworkingpreferenceofemployee𝑖injobIndexesofIndividualWorking⋯⋯⋯[ ⋮⋯𝐸where𝑒𝑖𝑗istheworkingeffectivenessofemployee𝑖injob𝑒𝑖𝑗=𝛼×𝑎𝑖𝑗+(1−𝛼)× 0<𝛼<whereαistheweightofworkingThePrimaryInthestaticprogrammingofhumanresources,weseektheizationofgroupeffectivenessandoptimization.Baseontheysisabove,wetakeintoconsiderationofemployees’workingabilityandpreferencetowardsjobsintheobjectivefunction,whichisrelatedtotheactualconditions.Hereweassumethatjobsinthebusinessandemployeesareone-to-onecorrespondent.TheObjectiveWedefinetheobjectivefunctionas𝑋=

⋮ where𝑥𝑖𝑗isa0-1variableanditsvalueisas

= 𝑒𝑚𝑝𝑙𝑜𝑦𝑒𝑒𝑖𝑑𝑜𝑒𝑠𝑛𝑜𝑡𝑡𝑎𝑘𝑒𝑢𝑝𝑗𝑜𝑏 𝑒𝑚𝑝𝑙𝑜𝑦𝑒𝑒𝑖𝑡𝑎𝑘𝑒𝑠𝑢𝑝𝑗𝑜𝑏Knownfromthequestion,ourequationofoptimizationofeffectiveness𝑀𝐴𝑋∑𝑒𝑖𝑗𝑥𝑖𝑗=where𝑒𝑖𝑗istheindexesofemployee𝑖’seffectivenessinjob𝑗,𝐸isthematrixofeffectiveness,𝐹=[1 1](thenumberof1is𝑛).TheConstraintTheconstraintsareasEveryemployeemusthaveajobinthe∑𝑥𝑖𝑗=Everyjobmustbetakenupbyan∑𝑥𝑖𝑗=Thedecisionvariableisa0-1variable,whi𝑥𝑖𝑗∈ThePrimaryConsideringthefactorsmentionedabove,ourmodelisas 𝑚𝑎𝑥∑∑𝑥𝑖𝑗𝑖=1∑𝑥𝑖𝑗=∑𝑥𝑖𝑗={𝑥𝑖𝑗∈TheIntermediateInpractice,theactualsituationswemeetwitharenotalwayssimilarwiththecasesinpart2.2.Inmostcases,thesupplyofemployeesisgreaterthandemand.Assumethatthenumberofjobsis𝑛,thenumberofjobapplicantsis𝑚and𝑚>𝑛.Thismeansthattherewillbe(𝑚−𝑛)applicantswhocannotgetajob.𝐸=[

⋮ 𝑋=[

]Inthismodel,weonlyneedtoconsidertheconstraintthateveryjobistakenupby 𝑚𝑎𝑥∑∑𝑥𝑖𝑗𝑒𝑖𝑗=𝑖=1∑𝑥𝑖𝑗=∑𝑥𝑖𝑗≤{𝑥𝑖𝑗∈where𝐹isanm-orderrowTheAdvancedInactualsituations,weoftenhavetoconsidersomecontingencyfactors.contingencyfactorswilllimitthemanagementandallocationofhumanresources.Forinstance,someemployeesmaynothavetheabilitytotainjobsorsomecandoonlyapartofthejobs.Weneedtoaddthefollowingconstraintsintothemodel:𝑥𝑖𝑗=where𝑑isa0-1{ 𝑒𝑚𝑝𝑙𝑜𝑦𝑒𝑒𝑖𝑑𝑜𝑒𝑠𝑛𝑜𝑡𝑡𝑎𝑘𝑒𝑢𝑝𝑗𝑜𝑏𝑑{ 𝑒𝑚𝑝𝑙𝑜𝑦𝑒𝑒𝑖𝑡𝑎𝑘𝑒𝑠𝑢𝑝𝑗𝑜𝑏Similarly,theremaybefactorsofinter-relationshipbetweenemployees.Forexample,someemployeesmayfinishtheirworkonlyiftheyworktogetherwithcertainotheremployeeortheymayonlydotheirworkiftheyareseparatedfromcertainotheremployees.Consideringthis,weaddthefollowingconstraints:f(X)=wheref(X)istheequationoflimitationsofinter-relationshipofmatrixModelII:TheShortestQueueofJobThehermitcrabs’behaviorgivesusmuchenlightenment.Whentheyfindabeautifuls,theywilllywaitforacertainperiodoftimeandmakesurethatnootherhermitcrabssuitthesmorethanthemselves.Thentheywillclaimthesastheirowns.Inbusinesshumanresourcesmanagement,thisprocessisevenmorecomplex.Afterwemakesurethatthestructureofhumanresourceswithinthebusinessachievesoptimization,wewillconsiderrecruitingnewemployeesfromthesociety.Wecannotlearnabouttheinformationaboutalltheapplicantsinthesocietybeforerecruitmentsincethiswillbringhugecoststothebusiness.Similartohermitcrabs’behaviorthattheywillwaitforonlyacertainperiodoftime,wemustdetermineaproperlengthofthequeueofjobapplicants.Thislengthisthenumberofjobapplicantsaboutwhomweneedlearninformationindetail.Andweshouldassumethatwhenwesetthislength,weknownothingaboutallthejobapplicants.Basedontheysisabove,weassumethattheconditionofrecruitmentisThetotalnumberofjobapplicantsinthesocietyismuchlargerthanthenumberinthequeue.TheabilitystructureofapplicantsinthesocietyobeysnormalThetotaltimerequiredinrecruitingemployeesisfixed.Andtheexpendituresofrecruitmentisrelatedtothelengthoftheapplicantqueue.TheDistributionofHumanWeassumethattheabilityofjobapplicantsobeynormal𝑏~𝑁(0,where𝑏istheabilityofeachFigure2:DistributionofAsisshowninFigure2,𝑥0isthelowerlimitofjobabilityrequiredfortheapplicants.Here,𝑥0,𝑥1,𝑥2,𝑥3formfourranges,dividingtheabilityoftheapplicantsintofourdegrees.Inthediagramofthenormaldistribution,sectionA,B,CandDrepresentthedegreesofworkingabilityinModelI,whichis{2,3,4,5},shownasfollowsinTableTable2:RangeofEffectivenessDegreeofworkingDCBAEffectiveness-costTotalAccordingtotheconditionsabove,wedefinethatthelengthoftheapplicantqueue𝐿.Undertheconditionthatweonlyknowabouttheinformationofthe applicants,ourgoalistoselectthebestoneapplicantasouremployeefromtheapplicantsinthequeue.Wesetupamatrixofrandomprobabilityasfollows𝑃𝑟𝑜𝑏= where𝑃𝑟𝑜𝑏𝑖istheprobabilitythatthebestapplicantinthequeuewithalengthoffallsinthedegreeof𝑃𝑟𝑜𝑏1=1−(1−𝑃𝑟𝑜𝑏2=1−(1−𝑃𝑟𝑜𝑏3=1−(1−{𝑃𝑟𝑜𝑏4=1−(1−Thenwedefinethat𝑌isthetotal𝐸(𝑌)=

3𝑒2[𝑒3where𝐸(𝑌)istheexpectationoftotaleffectiveness,𝑒𝑖istheindexofworkingeffectivenessofemployeesinthedegreeof𝑖.4𝐸(𝑌)=∑(1−(1−TheCostInthismodel,weconsiderthattheextensionoftheapplicantqueuewillincreaseexpenditures.Theseincludetheexpendituresonmaterialresourcesandhumanresourcesrequiredinlearningabouttheinformationaboutapplicants.Theexpendituresareindirectproportiontothelengthoftheapplicantqueue.E(𝐶)=𝛿𝐿+whereCisthecost,𝛿isa𝛿canbeestimatedbythehistoricalcosthappenedinthepastintheprocessofrecruitmentwithinthebusiness.Thenwedefinethat𝑊isvaluemeasuredbymoneygeneratedbyemployeesundertheunitindexofeffectivenessinunittime. 𝐸(𝑉) 𝐸(𝐶) (𝛿𝐿+ whereVisthereimbursedeffectivenessgeneratedfromtheprocessofTheOptimalLengthofApplicantWhenthemarginaleffectivenessisequaltomarginalcost,wecangettheoptimallengthofapplicantqueue.

= ThiscanbeshowninFigure3andFigure ee1ee 0 L

LFigure3:Influenceof Figure4:InfluenceofTheModelofSelectionsofmorethanoneTheysisaboveismainlyusedtosolvetheproblemwhentheenterpriseneedstorecruitonlyonenewemployee.Whenthenumberofnewemployeestheyneedtohireis𝑚,wecansetupthefollowingmatrixofprobability:

𝑃𝑟𝑜𝑏=

]((𝑃𝑟𝑜𝑏𝑖1=(1−1−𝛼𝑖1)𝐶𝐿𝑃𝑟𝑜𝑏𝑖2=(1−(1−𝛼𝑖2)𝐶𝐿𝑃𝑟𝑜𝑏𝑖3=(1−(1−𝛼𝑖3)𝐶𝐿{𝑃𝑟𝑜𝑏𝑖4=(1−(1−𝛼𝑖4)𝐶𝐿Theexpectationoftotaleffectivenesswillbeas 𝐸(𝑌)=∑∑𝑖=1HerethecostcurveisthesamewiththatinthesituationwhereonlyoneemployeeisModelIII:TheDynamicinthenextperiodInmodelI,weyzedtheallocationofhumanresourcesundertheconditionthatthenumberofemployeesandjobsarethesameandfixed.InmodelII,inthenextperiodStep1:Jobvacancies

finalvacancieswhichshouldbefilledbynewemployees.Figure5:StepsofModelInthisdynamicprocess,initialjobvacanciesarethedrivingforceofadjustmentofjobswithinthebusiness.ThentheoptimizationwillbecarriedoutthroughthemethodinmodelI.Afteraseriesofjobexchanges,thejobvacanciesleftwillbefilledbynewlyrecruitedemployeesfromthesociety.Afterthis,inthenextperiodwhenexistingemployeesleavetheirjobsandnewvacanciesarise,theadjustmentwillbeOurgoalistomakesurethatinthedynamicprocessofjobexchangeandrecruitment,highereffectivenessandlowerrateofstaffturnovercanbeachievedatthesametime.WecanshowtheprocessofjobadjustmentusingthefollowingFigure6:ChainofjobTheTimesofJobLet’sassumethatinanenterprisethetotalnumberofjobsisconstant.Beforethejobreallocation,theoptimaljoballocationmatrixisasfollows:

𝑋(𝑡)=[

⋮ Thematrixofeffectivenessisas 𝐸(𝑡)=[

⋮ Nowthereisthephenomenonthatsomeexistingemployeeswillleavetheenterprise.Assumethatthenumberofemployeesleavingtheirjobsandnewlyrecruitedemployeesis𝑞.Replacetheindexesofeffectivenessoftheemployeeswholefttheenterprisewiththatofthenewemployees.Thentherebe𝐸(𝑡+1)andwecandefinenewoptimaljoballocationmatrix𝑋(𝑡+1).ApplythesefactorsintomodelIandwecanget: 𝑚𝑎𝑥∑∑𝑥(𝑡+1)𝑖𝑗𝑒(𝑡+𝑖=1∑𝑥(𝑡+1)𝑖𝑗=∑𝑥(𝑡+1)𝑖𝑗={𝑥(𝑡+1)𝑖𝑗∈Theninperiod𝑡+1,theoptimaljoballocationandthetimesoftransferofjobswill 𝑘=∑𝑖=1

|𝑥(𝑡+1)𝑖𝑗−𝑥(𝑡)𝑖𝑗|Intheequationabove,duetotheparticularityofmatrix𝐸(𝑡+1)and𝐸(𝑡),wecansolvethetimesoftransferofjobsbymathematicaltransformations.TheShortestTransferPathofInthepracticalprocessofhumanresourcesmanagement,duetothecostoftransfersofjobswithinthegroup,wewillgenerallytrytoreducethetimesandrangeoftransfersandkeeptheeffectivenessatahighlevel.Therefore,theobjectivefunctioninperiod𝑡+1canbeasfollows:

min{𝑚𝑎𝑥∑∑𝑥(𝑡+1)𝑖𝑗𝑒(𝑡+𝑖=1Theconstrains∑𝑥(𝑡+1)𝑖𝑗=∑𝑥(𝑡+1)𝑖𝑗={𝑥(𝑡+1)𝑖𝑗∈WiththehelpofsoftwareLingo,wecansolve𝑋(𝑡+1),whichwillalsobeusedastheconditionsfor𝑋(𝑡+2).Thiswillguaranteethatineachadjustmentofhumanresources,ourgoalistoachievetheshortesttransferpathandthehighesttotalThePredictionofTransferInpracticalsituations,theappearanceofinitialjobvacanciesisrandom.Theindexesofnewlyrecruitedemployees’effectivenessarerandom,too.Therefore,theenterprisemustpredictthetrendofhumanresourcesinthesocietyforthenextperiod.TransferofaSingleInordertosimplifythemodel,wewilllylookatthepredictioninwhichonlyoneemployeeischangedforonetime.WemakethefollowingThetotalnumberofexistingemployeesis𝑛,ofwhichthenumberofemployeesrequiredtobehiredfromthesocietyis𝑚.𝐴(𝑡)istheeffectivenessmatrixofexistingemployees.𝐴′(𝑡)istheeffectivenessmatrixofapplicants.𝑋(𝑡)istheallocationofexistingemployees.Theprobabilityofleavingthejobsisthesameforalltheexistingemployees.Theprobabilityofrecruitmentisthesameforalltheapplicants.Thentheprobabilityofleavingthejobsforexistingemployees1𝑃𝑟𝑜𝑏1=Theprobabilityofrecruitmentofapplicants1𝑃𝑟𝑜𝑏2= ⋯⋯𝑌= 𝐾=[

] where𝑦𝑖𝑗isthetotaleffectivenessgeneratedwhenapplicant𝑖takestheplaceofexistingemployee𝑗.Wecanusemodel4.2tosolveeach𝑦𝑖𝑗inmatrix𝑌.𝑘𝑖𝑗isleasttimesofjobtransferswhenapplicant𝑖takestheplaceofexistingemployee𝑗.Therefore,wecan

𝐸(𝑌)

11

𝐸(𝐾)= TransferofmorethanoneIfmorethanoneemployeesneedtoberecruited,thenumberofprobabletransferswillbeS=CgCg,wheregisthenumberofemployeesneedtoberecruitedfrommThenwecangetthefollowing𝐸(𝑌)=∑𝐸(𝐾)=∑UncertainTransferofUndertheconditionoftransfersofmorethanoneemployees,if𝑚<𝑛,thenumberofprobabletransferswillbeC1C1+C2C2+⋯+m m mLet𝑆=𝐶1+𝐶2+⋯+𝐶𝑚.Then𝑌and𝐾willbeS-ordervectors. expressionof𝐸(𝑌)and𝐸(𝐾)willbethesameastheequationsinPredictionsoftheChangeofHRThegoalofTopredictthesituationofhumanresourcesinperiod𝑡,wearemainlygoingtoknowaboutthejobtransferpathandtotaleffectivenessinperiod𝑡.Throughthemethodin4.3,wecangettheexpectationoftimesoftransferandtotaleffectivenessinperiod𝑡+1basedonthejoballocationmatrixandeffectivenessmatrixinthebasicWedefinethefollowing 𝑒𝑖𝑗𝑥(𝑡+𝑟 where𝑟istheratiobetweentotaleffectivenessandtimesoftransfer.Weconsiderthattheresultandefficiencyofhumanresourcesallocationwillbebetterifthevalueof𝑟islarger.ThePredictionofTransferTheexpectationofefficiencyofhumanresourcesallocationinperiod𝑡+1𝐸(𝑟)=Intermsofthepredictionoftransferpath,werefertothegreypredictionmodel.Weusethecombinationwhichistheclosesttotheefficiencyofhumanallocationand𝐸(𝑟)as𝑚𝑖𝑛{|𝐸(𝑟)−𝑟(1)|,|𝐸(𝑟)−𝑟(2)|,⋯,|𝐸(𝑟)−Weshouldfindtheonewhichistheclosesttotheexpectationastheinitialhumanresourcesallocationinperiod𝑡+1.Thenbasedontheysisinperiod𝑡+1,wecancontinuetopredictthemanagementofhumanresourcesinperiod𝑡+2andsoon.Ourpredictionoftransferpathisledby𝐸(𝑟).5ysisofSolutionsandInordertoverifytheavailabilityofthemodelprecisely,weusecomputerstosimulatethedynamicprocessofhumanresourcesmanagement.ly,weassumethattherearedifferencesbetweenworkingeffectivenessofdifferentemployees.Nowwedefinetwotypesofhumanresourcesallocation:𝛼1:𝛼2::𝛼3:𝛼4=0.2:0.3:0.4:𝛼1:𝛼2::𝛼3:𝛼4=0.1:0.2:0.4:Itisobviousthathumanresourcesallocationtypea)isbetterthantypeForsomeenterprises,theeffectivenessofexistingemployeesisworsethanthatofthenewlyrecruitedemployeesfromthesociety.Forsomeotherenterprises,theaveragelevelofworkingabilityofnewlyrecruitedemployeesishigherthanthatofexistingWiththehelpofcomputers,wesimulatethetotaleffectivenesshumanresourcesandtimesofjobtransfersinthenext5yearsundertheconditionoftwotypesofhumanresourcesallocationrespectively.Intheprocessofsimulation,weestablishtherandomeffectivenessmatrixofexistingemployeesandnewlyrecruitedemployeesaccordingtotheweightmentionedrespectively.ThenwecanusethemethodinmodelIIItoyzeandpredictthechangeofhumanresourcesinthefuture.WesetupfourkindsofrandompatternstosimulatethepracticalhumanresourcesPattern1:Theperiodofadjustmentis1year.Thenumberofemployeeschangedis1.Theaverageeffectivenesslevelofexistingemployeesislowerthanthatofnewlyrecruitedemployees.Pattern2:Theperiodofadjustmentis1year.Thenumberofemployeeschangedis2.Theaverageeffectivenesslevelofexistingemployeesislowerthanthatofnewlyrecruitedemployees.Pattern3:Theperiodofadjustmentis1year.Thenumberofemployeeschangedis1.Theaverageeffectivenesslevelofexistingemployeesishigherthanthatofnewlyrecruitedemployees.Pattern4:Theperiodofadjustmentis1year.Thenumberofemployeeschangedis2.Theaverageeffectivenesslevelofexistingemployeesishigherthanthatofnewlyrecruitedemployees.Table4:expectationsofDifferentPatternPatternPatternPatternYear3434YearYearYearYearLong-1212ysisofthePatternonePatterntwoPatternonePatterntwo PatternonePatterntwoPatternonePatterntwo

0

Figure FigurePatternthreePatternfourPatternthreePatternfourPatternthreePatternfourPatternthreePatternfourPatternthreePatternfourPatternthreePatternfour

3

2

1 Figure FigureFigure7:changesofExpectationswithFromFigure7.1and7.2,wecanknowthatwhenthenewlyrecruitedemployees’abilityisbetterthantheexistingemployees,thetotaleffectivenessoftheenterpriseaswholewillshowaupwardtrendastimegoesby.Butthespeedoftheincreasewillslowdownandtendtobeclosetotheexpectationoftheoptimaleffectivenessoftheapplicantsinthesociety.Intheme,withtheprocessofthecombinationofexistingemployeesandnewemployees,theexpectationoftransfertimesineachperiodtendstobecloseto1.Toputitanotherway,internaladjustmentisnotrequiredsincethereisnodistinctionbetweentheeffectivenessofexistingemployeesandnewemployee.Thereforethereisnoneedtotransferjobswithinthebusiness.Situationsaredifferentinpattern3and4.Inthesetwopatterns,outsideapplicants’effectivenessislowerthanexistingemployees.Infactjobvacanciesinthesesituationsarethelossofhumanresources.Therefore,totaleffectivenessofoptimalallocationwillshowadownwardtrenduntilthetotaleffectivenessofexistingemployeesisthesamewiththatofthoseinthesociety.Inallthepatterns,timesofjobtransferswilldecreasewiththepassoftimeduetothereducedtodifferencesbetweenabilityofexistingemployeesandotherapplicants.Ineachperiod,thenumberoftransfersofemployeesdirectlydeterminesthespeedofhumanresourcesallocation.Soitcanbeseenfromgraph3andgraph4thatthepatternsinwh