運營管理第14版蔡斯IChap010

WaitingLineAnalysisandSimulationChapterTenCopyright?2021byTheMcGraw-HillCompanies,Inc.Allrightsreserved.McGraw-Hill/IrwinLearningObjectivesLO10–1:Understandwhatawaitinglineproblemis.LO10–2:Analyzewaitinglineproblems.LO10–3:Analyzecomplexwaitinglinesusingsimulation.10-2EconomicsoftheWaitingLineProblemAcentralprobleminmanyservicesettingsisthemanagementofwaitingtime.Reducingwaitingtimecostsmoney.Whenpeoplewaitingareemployees,itiseasytovaluetheirtime.Whenpeoplewaitingarecustomers,itismoredifficulttovaluetheirtime.Lostsalesisone(low)value.10-3ThePracticalViewofWaitingLines10-4MoreonWaitingLinesOneimportantvariableisthenumberofarrivalsoverthehoursthattheservicesystemisopen.Customersdemandvaryingamountsofservice,oftenexceedingnormalcapacity.Wecancontrolarrivals.ShortlinesSpecifichoursforspecificcustomersSpecialsWecanaffectservicetimebyusingfasterorslowerservers.10-5TheQueuingSystemSourcepopulationandthewaycustomersarriveatthesystemTheservicingsystemTheconditionofthecustomersexitingthesystemDotheygobacktosourcepopulationornot?10-6ComponentsoftheQueuingSystemVisuallyCustomerscomeinCustomersareservedCustomersleave10-7CustomerArrivalsFinitepopulation:limited-sizecustomerpoolthatwillusetheserviceand,attimes,formalineWhenacustomerleaveshis/herpositionasamemberofthepopulation,thesizeoftheusergroupisreducedbyone.Infinitepopulation:populationlargeenoughsothatthepopulationsizecausedbysubtractionsoradditionstothepopulationdoesnotsignificantlyaffectthesystemprobabilities10-8DistributionofArrivalsArrivalrate:thenumberofunitsarrivingperperiodConstantarrivaldistribution:periodic,withexactlythesametimebetweensuccessivearrivalsVariable(random)arrivaldistributions:arrivalprobabilitiesdescribedstatisticallyExponentialdistributionPoissondistribution10-9DistributionsExponentialdistribution:whenarrivalsataservicefacilityoccurinapurelyrandomfashionTheprobabilityfunctionis

f(t)=λe-λtPoissondistribution:whereoneisinterestedinthenumberofarrivalsduringsometimeperiodTTheprobabilityfunctionis10-10CustomerArrivalsinQueues10-11OtherArrivalCharacteristicsArrivalpatternsSizeofarrivalunitsDegreeofpatienceBalkingReneging10-12TheQueuingSystemLengthInfinitepotentiallengthLimitedlinecapacityNumberoflinesQueuediscipline:apriorityruleorsetofrulesfordeterminingtheorderofservicetocustomersinawaitingline10-13ServiceTimeDistributionConstantServiceprovidedbyautomationVariableServiceprovidedbyhumansDescribedusingexponentialdistribution10-14LineStructure10-15ExitingtheQueuingSystem10-16PropertiesofSomeSpecific

WaitingLineModels10-17NotationforEquations10-18EquationsforSolvingThreeModelProblems10-19Example10.1:CustomersinLineWesternNationalBankisconsideringopeningadrive-throughwindowforcustomerservice.Managementestimatesthatcustomerswillarriveattherateof15perhour.Thetellerwhowillstaffthewindowcanservicecustomersattherateofoneeverythreeminutes.Part1AssumingPoissonarrivalsandexponentialservice,findUtilizationofthetellerAveragenumberinlineAveragenumberinsystemAveragewaitingtimeinlineAveragewaitingtimeinsystem,includingservice10-20Example10.1:Solution10-21Example10.1:NoMoreThanThreeCars10-22Example10.1:95PercentServiceLevel10-23Example10.2:EquipmentSelectionTheRobotCompanyfranchisescombinationgasandcarwashstationsthroughouttheUnitedStates.Robotgivesafreecarwashforagasolinefill-upor,forawashalone,charges$0.50.Pastexperienceshowsthatthenumberofcustomerswhohavecarwashesfollowingfill-upsisaboutthesameasforawashalone.Theaverageprofitonagasolinefill-upisabout$0.70,andthecostofthecarwashtoRobotis$0.10.Robotstaysopen14hoursperday.Robothasthreepowerunitsanddriveassemblies,andafranchiseemustselecttheunitpreferred.Freewithfill-up50￠forwashalone50/50mixtureAverageprofitonfill-upis70￠Washcosts10￠ThreeunitsunderconsiderationUnitIwashesoneper5minutesandcosts$12perdayUnitIIwashesoneper4minutesandcosts$16perdayUnitIIIwashesoneper3minutesandcosts$22perday10-24Example10.2:BasicCalculations10-25Example10.2:Profits10-26Example10.3:DeterminingtheNumberofServersIntheservicedepartmentoftheGlenn-MarkAutoAgency,mechanicsrequiringpartsforautorepairorservicepresenttheirrequestformsatthepartsdepartmentcounter.Thepartsclerkfillsarequestwhilethemechanicwaits.Mechanicsarriveinarandom(Poisson)fashionattherateof40perhour,andaclerkcanfillrequestsattherateof20perhour(exponential).Ifthecostforapartsclerkis$6perhourandthecostforamechanicis$12perhour,determinetheoptimumnumberofclerkstostaffthecounter.(Becauseofthehigharrivalrate,aninfinite

sourcemaybeassumed.)Arrivalsof40perhourServiceat20perhourClerkcosts$30perhourMechaniccosts$60perhour10-27Example10.3:Solution10-28Example10.4:FinitePopulationSourceStudiesofabankoffourweavingmachinesattheLooseKnittextilemillhaveshownthat,onaverage,eachmachineneedsadjustingeveryhourandthatthecurrentserviceraverages7.5minutesperadjustment.AssumingPoissonarrivals,exponentialservice,andamachineidletimecostof$40perhour,determineifasecondservicer(whoalsoaverages7.5minutesperadjustment)shouldbehiredatarateof$7perhour.Eachmachineneedsadjustingeveryhour.Serviceaverages7.5minutesperadjustment.Poissonarrivalandexponentialservice.Machineidlecostis$40perhour.Shouldwehireasecondservicerepairer?Costs$7perhour10-29Example10.4:TermsN=NumberofmachinesinthepopulationS =NumberofrepairersT =TimerequiredtoserviceamachineU =AveragetimeamachinerunsbeforerequiringserviceX =ServicefactorL =AveragenumberofmachineswaitingH =AveragenumberofmachinesbeingservicedD =ProbabilitymachineneedingservicewillwaitF =Efficiencyfactor10-30Example10.4:Case1—OneRepairerN=4S=1T=7.5U=6010-31Example10.4:FindingFforCase110-32Example10.4:Case1CostL=N(1-F)=4(1-0.957)=0.172machineH=NFX=0.957(4)(0.111)=0.425machineNumberofmachinesdown

L+H=0.172+0.425=0.597Costformachinesbeingdown

L+Hx$40=0.597x$40=$23.88Totalcost=machinecost+labor

=$23.88+$7.00=$30.8810-33Example10.4:Case2CostL=N(1-F)=4(1-0.998)=0.008machineH=NFX=0.998(4)(0.111)=0.443machineNumberofmachinesdown

L+H=0.008+0.443=0.451Costformachinesbeingdown

L+Hx$40=0.451x$40=$18.04Totalcost=machinecost+labor

=$18.04+$14.00=$32.0410-34ApproximatingCustomerWaitingTimeAllyouneedisthemeanandstandarddeviationtoapproximatewaitingtime.“Quickanddirty〞mathematicalapproximationNoassumptionneededaboutaparticulararrivalrateorservicedistribution.Needdataonservicetimes.Needdataontimebetweenarrivals.Interarrivaltime10-35Equations10-36Example10.5:WaitingLineApproximation10-37BuExample10.5:Step1:CalculateArrivalRate,ServiceRate,andCoefficientsofVariation10-38Example10.5:Step2:CalculateServerUtilizationandStep3:CalculateWaitInformation10-39ComputerSimulationofWaitingLinesSomewaitinglineproblemsareverycomplex.Assumedwaitinglinesareindependent.Whenaservicesisbecomestheinputtothenext,wecannolongerusethesimpleformulas.Thisisalsotrueforanyproblemwhereconditionsdonotmeettherequirementsoftheequations.Here,computersimulationmustbeused.10-40SimulatingWaitingLinesWaitinglinesthatoccurinseriesandparallelcannotbesolvedmathematically.AssemblylinesWorkcentersThesewaitinglinesareeasilysimulatedonacomputer.Willsimulateatwo-stageassemblylineasanexample.10-41Example10.6:ATwo-StageAssemblyLineConsideranassemblylinethatmakesalargeproduct.Becausetheproductislarge,theworkstationsaredependentoneachother.RaycannotworkfasterthanBob.10-42Example10.6:ObjectiveoftheStudyWhatistheaverageperformancetimeofeachworker?Whatistheoutputrateofproductthroughthisline?HowmuchtimedoesBobwaitforRay?HowmuchtimedoesRaywaitforBob?Wouldoutputratesincreaseifstoragespacebetweentheworkerswereadded?10-43Example10.6:DataCollection10-44Example10.6:SimulationofBobandRay—Two-StageAssemblyLine10-45Example10.6:ResultsTheoutputtimeaverages60secondsperunit.UtilizationofBobis470/530=88.7percent.UtilizationofRayis430/550=78.2percent.IgnoresinitialstartupwaitAverageperformancetimeforBobis470/10=47seconds.AverageperformancetimeforRayis430/10=43seconds.10-46Example10.6:SpreadsheetSimulation10-47Example10.6:AverageTimeperUnitofOutput10-48Example10.6:AverageTimetheProductSpendsintheSystem10-49Example10.6:ResultsofSimulating1,200UnitswithaSpreadsheet10-50SimulationProgramsandLanguagesContinuousBasedonmathematicalequationsUsedforsimulatingcontinuousvaluesforallpointsintimeExample:TheamountoftimeapersonspendsinaqueueDiscreteUsedforsimulatingspecificvaluesorspecificpointsExample:Numberofpeopleinaqueue10-51TypesofSimulationProgramsGeneral-purpose:allowsprogrammerstobuildtheirownmodelsSLAMIISIMSCRIPTII.5SIMANGPSS/HGPSS/PCPC-MODELRESQSpecial-purpose:speciallybuilttosimulatespecificapplicationsExtendSIMFACTORY10-52DesirableFeaturesofSimulationSoftwareBecapableofbeingusedinteractivelyaswellasallowingcompleteruns.Beuser-friendlyandeasytounderstand.Allowmodulestobebuiltandthenconnected.Allowuserstowriteandincorporatetheirownroutines.Havebuildingblocksthatcontainbuilt-incommands.Havemacrocapability.10-53DesirableFeaturesofSimulationSoftwareContinuedHavematerial-flowcapability.Haveoutputstandardstatisticssuchascycletimes,utilization,andwaittimes.Allowavarietyofdataanalysisalternativesforbothinputandoutputdata.Haveanimationcapabilitiestodisplaygraphicallytheproductflowthr