运筹学与供应链管理-第5讲课件

第五讲TransportationandNetworkModels第五讲1IntroductionSeveralspecificmodels(whichcanbeusedastemplatesforreal-lifeproblems)willbeintroduced.TRANSPORTATIONMODEL

ASSIGNMENTMODEL

NETWORKMODELS

IntroductionSeveralspecificm2IntroductionTRANSPORTATIONMODEL

ASSIGNMENTMODEL

Determinehowtosendproductsfromvarioussourcestovariousdestinationsinordertosatisfyrequirementsatthelowestpossiblecost.Allocatingfixed-sizedresourcestodeterminetheoptimalassignmentofsalespeopletodistricts,jobstomachines,taskstocomputers…NETWORKMODELS

Involvethemovementorassignmentofphysicalentities(e.g.,money).

IntroductionTRANSPORTATIONMOD3TransportationModelAnexample,theAutoPowerCompanymakesavarietyofbatteryandmotorizeduninterruptibleelectricpowersupplies(UPS’s).AutoPowerhas4finalassemblyplantsinEuropeandthedieselmotorsusedbytheUPS’sareproducedintheUS,shippedto3harborsandthensenttotheassemblyplants.Productionplansforthethirdquarter(July–Sept.)havebeenset.Therequirements(demandatthedestination)andtheavailablenumberofmotorsatharbors(supplyatorigins)areshownonthenextslide:TransportationModelAnexample4DemandSupplyAssemblyPlant

No.ofMotorsRequiredLeipzig 400(2)Nancy

900(3)Liege 200(4)Tilburg 500 2000

Harbor

No.ofMotorsAvailable(A)Amsterdam 500(B)Antwerp

700(C)LeHavre 800 2000BalancedDemandSupplyAssemblyPlant N5GraphicalpresentationofLeHavre(C)800Antwerp(B)700Amsterdam(A)500SupplyLiege(3)200Tilburg(4)500Leipzig(1)400Nancy(2)900andDemand:GraphicalpresentationofLeHa6TransportationModelAutoPowermustdecidehowmanymotorstosendfromeachharbor(supply)toeachplant(demand).Thecost($,onapermotorbasis)ofshippingisgivenbelow.

TODESTINATION

LeipzigNancyLiegeTilburgFROMORIGIN

(1)(2)(3)(4)

(A)Amsterdam 1201304159.50(B)Antwerp

6140100110(C)LeHavre102.509012242TransportationModelAutoPower7Thegoalistominimizetotaltransportationcost.Sincethecostsintheprevioustableareonaperunitbasis,wecancalculatetotalcostbasedonthefollowingmatrix(wherexijrepresentsthenumberofunitsthatwillbetransportedfromOriginitoDestinationj):TransportationModelThegoalistominimizetotal8

TODESTINATIONFROMORIGIN

1234

A 120xA1130xA241xA359.50xA4

B 61xB140xB2100xB3110xB4C102.50xC190xC2122xC342xC4TotalTransportationCost=

120xA1+130xA2+41xA3+…+122xC3+42xC4TransportationModel TODESTINATIONTota9Twogeneraltypesofconstraints.1.

Thenumberofitemsshippedfromaharbor

cannotexceedthenumberofitemsavailable.ForAmsterdam:xA1+xA2+xA3+xA4

<500ForAntwerp:

xB1+xB2+xB3+xB4

<700ForLeHavre:xC1+xC2+xC3+xC4

<800Note:Wecouldhaveusedan“=“insteadof“<“sincesupplyanddemandarebalancedforthismodel.TransportationModelTwogeneraltypesofconstrain102.

Demandateachplantmustbesatisfied.ForLeipzig:xA1+xB1+xC1

>400ForNancy:

xA2+xB2+xC2

>900ForLiege:xA3+xB3+xC3>200Note:Wecouldhaveusedan“=“insteadof“>“sincesupplyanddemandarebalancedforthismodel.ForTilburg:xA4+xB4+xC4>500TransportationModelTwogeneraltypesofconstraints.2.Demandateachplantmust11VariationsontheTransportationModelSupposewenowwanttomaximizethevalueoftheobjectivefunctioninsteadofminimizingit.Inthiscase,wewouldusethesamemodel,butnowtheobjectivefunctioncoefficientsdefinethecontributionmargins(i.e.,unitreturns)insteadofunitcosts.SolvingMaxTransportationModelsVariationsontheTransportati12Whensupplyanddemandarenotequal,thentheproblemisunbalanced.Therearetwosituations:Whensupplyisgreaterthandemand:WhenSupplyandDemandDifferInthiscase,whenalldemandissatisfied,theremainingsupplythatwasnotallocatedateachoriginwouldappearasslackinthesupplyconstraintforthatorigin.Usinginequalitiesintheconstraints(asinthepreviousexample)wouldnotcauseanyproblems.VariationsontheTransportationModelWhensupplyanddemandarenot13Inthiscase,theLPmodelhasnofeasiblesolution.However,therearetwoapproachestosolvingthisproblem:1.

Rewritethesupplyconstraintstobe

equalitiesandrewritethedemand

constraintstobe<.Unfulfilleddemandwillappearasslackoneachofthedemandconstraintswhenoneoptimizesthemodel.Whendemandisgreaterthansupply:VariationsontheTransportationModelInthiscase,theLPmodelhas142.

Revisethemodeltoappendaplaceholder

origin,calledadummyorigin,withsupply

equaltothedifferencebetweentotal

demandandtotalsupply.Thepurposeofthedummyoriginistomaketheproblembalanced(totalsupply=totaldemand)sothatonecansolveit.Thecostofsupplyinganydestinationfromthisoriginiszero.Oncesolved,anysupplyallocatedfromthisorigintoadestinationisinterpretedasunfilleddemand.VariationsontheTransportationModel2.Revisethemodeltoappend15Certainroutesinatransportationmodelmaybeunacceptableduetoregionalrestrictions,deliverytime,etc.Inthiscase,youcanassignanarbitrarilylargeunitcostnumber(identifiedasM)tothatroute.Thiswillforceonetoeliminatetheuseofthatroutesincethecostofusingitwouldbemuchlargerthanthatofanyotherfeasiblealternative.EliminatingUnacceptableRoutesChooseMsuchthatitwillbelargerthananyotherunitcostnumberinthemodel.VariationsontheTransportationModelCertainroutesinatransporta16Generally,LPmodelsdonotproduceintegersolutions.TheexceptiontothisistheTransportationmodel.Ingeneral:IntegerValuedSolutionsIfallofthesuppliesanddemandsinatransportationmodelhaveintegervalues,theoptimalvaluesofthedecisionvariableswillalsohaveintegervalues.VariationsontheTransportationModelGenerally,LPmodelsdonotpr17AssignmentModelIngeneral,theAssignmentmodelistheproblemofdeterminingtheoptimalassignmentofn“indivisible”agentsorobjectstontasks.Forexample,youmightwanttoassignSalespeopletosalesterritoriesComputerstonetworksConsultantstoclientsServicerepresentativestoservicecallsCommercialartiststoadvertisingcopyTheimportantconstraintisthateachpersonormachinebeassignedtooneandonlyonetask.AssignmentModelIngeneral,th18WewillusetheAutoPowerexampletoillustrateAssignmentproblems.AutoPowerEurope’sAuditingProblemAutoPower’sEuropeanheadquartersisinBrussels.Thisyear,eachofthefourcorporatevice-presidentswillvisitandauditoneoftheassemblyplantsinJune.Theplantsarelocatedin:Leipzig,GermanyLiege,BelgiumNancy,FranceTilburg,theNetherlandsAssignmentModelWewillusetheAutoPowerexam19Theissuestoconsiderinassigningthedifferentvice-presidentstotheplantsare:1.

Matchingthevice-presidents’areasof

expertisewiththeimportanceofspecific

problemareasinaplant.2.

Thetimethemanagementauditwillrequire

andtheotherdemandsoneachvice-

presidentduringthetwo-weekinterval.3.

Matchingthelanguageabilityofavice-

presidentwiththeplant’sdominantlanguage.Keepingtheseissuesinmind,firstestimatethe(opportunity)costtoAutoPowerofsendingeachvice-presidenttoeachplant.AssignmentModelTheissuestoconsiderinassi20Thefollowingtableliststheassignmentcostsin$000sforeveryvice-president/plantcombination.

PLANT

LeipzigNancyLiegeTilburgV.P.(1)(2)(3)(4)

Finance(F) 24102111Marketing(M)

14221015Operations(O)15172019Personnel(P)11191413AssignmentModelThefollowingtableliststhe21

PLANT

LeipzigNancyLiegeTilburgV.P.(1)(2)(3)(4)

Finance(F) 24102111Marketing(M)

14221015Operations(O)15172019Personnel(P)11191413Considerthefollowingassignment:Totalcost=24+22+20+13=79Thequestionis,isthistheleastcostassignment?AssignmentModel PLANTConsi22Completeenumerationisthecalculationofthetotalcostofeachfeasibleassignmentpatterninordertopicktheassignmentwiththelowesttotalcost.SolvingbyCompleteEnumerationThisisnotaproblemwhenthereareonlyafewrowsandcolumns(e.g.,vice-presidentsandplants).However,completeenumerationcanquicklybecomeburdensomeasthemodelgrowslarge.AssignmentModelCompleteenumerationistheca231.

Fcanbeassignedtoanyofthe4plants.2.

OnceFisassigned,Mcanbeassignedtoany

oftheremaining3plants.3.

NowOcanbeassignedtoanyofthe

remaining2plants.4.

Pmustbeassignedtotheonlyremaining

plant.Thereare4x3x2x1=24possiblesolutions.Ingeneral,iftherearenrowsandncolumns,thentherewouldben(n-1)(n-2)(n-3)…(2)(1)=n!

(nfactorial)solutions.Asnincreases,n!increasesrapidly.Therefore,thismaynotbethebestmethod.AssignmentModel1.Fcanbeassignedtoanyo24Forthismodel,let

xij=numberofV.P’softypeiassignedtoplantjwherei=F,M,O,P

j=1,2,3,4TheLPFormulationandSolutionNoticethatthismodelisbalancedsincethetotalnumberofV.P.’sisequaltothetotalnumberofplants.Remember,onlyoneV.P.(supply)isneededateachplant(demand).AssignmentModelForthismodel,let

xij=num25Asaresult,theoptimalassignmentis:

PLANT

LeipzigNancyLiegeTilburgV.P.(1)(2)(3)(4)

Finance(F) 24102111Marketing(M)

14221015Operations(O)15172019Personnel(P)11191413TotalCost($000’s)=10+10+15+13=48AssignmentModelAsaresult,theoptimalassig26TheAssignmentmodelissimilartotheTransportationmodelwiththeexceptionthatsupplycannotbedistributedtomorethanonedestination.RelationtotheTransportationModelIntheAssignmentmodel,allsuppliesanddemandsareone,andhenceintegers.Asaresult,eachdecisionvariablecellwilleithercontaina0(noassignment)ora1(assignmentmade).Ingeneral,theassignmentmodelcanbeformulatedasatransportationmodelinwhichthesupplyateachoriginandthedemandateachdestination=1.AssignmentModelTheAssignmentmodelissimila27Case1:SupplyExceedsDemandUnequalSupplyandDemand:Intheexample,supposethecompanyPresidentdecidesnottoaudittheplantinTilburg.Nowthereare4V.P.’stoassignto3plants.Hereisthecost(in$000s)matrixforthisscenario:

PLANT NUMBEROFV.P.sV.P. 1 2 3 AVAILABLE

F 24 10 21 1

M 14 22 10 1

O 15 17 20 1

P 11 19 14 1No.ofV.P.s 4Required 1 1 1 3

AssignmentModelCase1:SupplyExceedsDemand28Toformulatethismodel,simplydroptheconstraintthatrequiredaV.P.atplant4andsolveit.AssignmentModelUnequalSupplyandDemand:Toformulatethismodel,simpl29Case2:DemandExceedsSupplyUnequalSupplyandDemand:Inthisexample,assumethattheV.P.ofPersonnelisunabletoparticipateintheEuropeanaudit.Nowthecostmatrixisasfollows:

PLANT NUMBEROFV.P.sV.P. 1 2 3 4 AVAILABLE

F 24 10 21 11 1

M 14 22 10 15 1

O 15 17 20 19 1No.ofV.P.s 3Required 1 1 1 1 4

AssignmentModelCase2:DemandExceedsSupply30

1.Modifytheinequalitiesintheconstraints

(similartotheTransportationexample)

2.AddadummyV.P.asaplaceholdertothe

costmatrix(shownbelow).

PLANT NUMBEROFV.P.sV.P. 1 2 3 4 AVAILABLE

F 24 10 21 11 1

M 14 22 10 15 1

O 15 17 20 19 1Dummy 0 0 0 0 1No.ofV.P.s 4Required 1 1 1 1 4

ZerocosttoassignthedummyDummysupply;nowsupply=demandAssignmentModel 1.Modifytheinequalitiesi31Inthesolution,thedummyV.P.wouldbeassignedtoaplant.Inreality,thisplantwouldnotbeaudited.AssignmentModelUnequalSupplyandDemand:Inthesolution,thedummyV.P32InthisAssignmentmodel,theresponsefromeachassignmentisaprofitratherthanacost.MaximizationModelsForexample,AutoPowermustnowassignfournewsalespeopletothreeterritoriesinordertomaximizeprofit.Theeffectofassigninganysalespersontoaterritoryismeasuredbytheanticipatedmarginalincreaseinprofitcontributionduetotheassignment.AssignmentModelInthisAssignmentmodel,the33Hereistheprofitmatrixforthismodel.

NUMBEROF TERRITORY SALESPEOPLE SALESPERSON 1 2 3AVAILABLE

A 40 30 20 1

B 18 28 22 1

C 12 16 20 1

D 25 24 27 1No.of 4

Salespeople 1 1 1 3

Required

ThisvaluerepresentstheprofitcontributionifAisassignedtoTerritory3.AssignmentModelHereistheprofitmatrixfor34TheAssignmentModelCertainassignmentsinthemodelmaybeunacceptableforvariousreasons.SituationswithUnacceptableAssignmentsInthiscase,youcanassignanarbitrarilylargeunitcost(orsmallunitprofit)numbertothatassignment.ThiswillforceSolvertoeliminatetheuseofthatassignmentsince,forexample,thecostofmakingthatassignmentwouldbemuchlargerthanthatofanyotherfeasiblealternative.AssignmentModelTheAssignmentModelCertainas35NetworkModelsTransportationandassignmentmodelsaremembersofamoregeneralclassofmodelscallednetworkmodels.Networkmodelsinvolvefrom-tosourcesanddestinations.Appliedtomanagementlogisticsanddistribution,networkmodelsareimportantbecause:Theycanbeappliedtoawidevarietyofrealworldmodels.Flowsmayrepresentphysicalquantities,Internetdatapackets,cash,airplanes,cars,ships,products,…NetworkModelsTransportationa36ZigwellInc.isAutoPower’slargestUSdistributorofUPSgeneratorsinfiveMidwesternstates.NetworkModelsACapacitatedTransshipmentModelZigwellhas10BigGen’satsite1Thesegeneratorsmustbedeliveredtoconstructionsitesintwocitiesdenotedand343BigGen’sarerequiredatsiteand7arerequiredatsite34NetworkModelsZigwellInc.isAutoPower’sla371+102543-3-7Thisisanetworkdiagramornetworkflowdiagram.Eacharrowiscalledanarcorbranch.

Eachsiteistermedanode.SupplyDemandNetworkModels1+102543-3-7Thisisanetwork38cij thecosts(perunit)oftraversingthe

routesuij thecapacitiesalongtheroutesCostsareprimarilyduetofuel,tolls,andthecostofthedriverfortheaveragetimeittakestotraversethearc.Becauseofpre-establishedagreementswiththeteamsters,Zigwellmustchangedriversateachsiteitencountersonaroute.Becauseoflimitationsonthecurrentavailabilityofdrivers,thereisanupperbound,uij,onthenumberofgeneratorsthatmaytraverseanarc.NetworkModelscij thecosts(perunit)oftr391+102543-3-7c12c23c24c25c34c43c53u12u23u24u25u34u43u53NetworkModels1+102543-3-7c12c23c24c25c34c4340LPFormulationoftheModelNetworkModelsACapacitatedTransshipmentModelThegoalistofindashipmentplanthatsatisfiesthedemandsatminimumcost,subjecttothecapacityconstraints.Thecapacitatedtransshipmentmodelisbasicallyidenticaltothetransportationmodelexceptthat:1.

Anyplantorwarehousecanshiptoanyother

plantorwarehouse2.

Therecanbeupperand/orlowerbounds

(capacities)oneachshipment(branch)NetworkModelsLPFormulationoftheModelNet41Thedecisionvariablesare:xij=totalnumberofBigGen’ssentonarc(i,j)=flowfromnodeitonodejThemodelbecomes:Minc12x12+c23x23+c24x24+c25x25+c34x34+

c43x43+c53x53+c54x54s.t.+x12=10-x12+x23+x24+x25=0-x23–x43–x53+x34=-3-x24+x43–x34–x54=-7-x25+x53+x54=00<

xij<uijallarcs(i,j)inthenetworkThedecisionvariablesare:xij42PropertiesoftheModel1.

xijisassociatedwitheachofthe8arcsinthe

network.Therefore,thereare8corresponding

variables:x12,x23,x24,x25,x34,x43,x53,andx54Theobjectiveistominimizetotalcost.2.

Thereisonematerialflowbalanceequation

associatedwitheachnodeinthenetwork.For

example:Totalflowoutofnodeis10units1Totalflowoutofnodeminusthetotalflowintonodeiszero(i.e.,totalflowoutmustequaltotalflowintonode).222Totalflowoutofnodemustbe3unitslessthanthetotalflowintonode.33PropertiesoftheModel1.xij43Intermediatenodesthatareneithersupplypointsnordemandpointsareoftentermedtransshipmentnodes.3.

Thepositiveright-handsidescorrespondto

nodesthatarenetsuppliers(origins).Thesumofallright-hand-sidetermsiszero(i.e.,totalsupplyinthenetworkequalstotaldemand).Thezeroright-handsidescorrespondtonodesthathaveneithersupplynordemand.Thenegativeright-handsidescorrespondtonodesthatarenetdestinations.Intermediatenodesthatarene44Ingeneral,flowbalanceforagivennode,j,is:Totalflowoutofnodej–totalflowintonodej=supplyatnodejNegativesupplyisarequirement.Nodeswithnegativesupplyarecalleddestinations,

sinks,

or

demandpoints.Nodeswithpositivesupplyarecalledorigins,sources,orsupplypoints.Nodeswithzerosupplyarecalledtransshipmentpoints.4.

AsmallmodelcanbeoptimizedwithSolver.Ingeneral,flowbalancefora45IntegerOptimalSolutionsNetworkModelsACapacitatedTransshipmentModelTheintegerpropertyofthenetworkmodelcanbestatedthus:IfalltheRHStermsandarccapacities,uij,areintegersinthecapacitatedtransshipmentmodel,therewillalwaysbeaninteger-valuedoptimalsolutiontothismodel.NetworkModelsIntegerOptimalSolutionsNetwo46ThestructureofthismodelmakesitpossibletoapplyspecialsolutionmethodsandsoftwarethatoptimizethemodelmuchmorequicklythanthemoregeneralsimplexmethodusedbySolver.EfficientSolutionProceduresNetworkModelsACapacitatedTransshipmentModelThismakesitpossibletooptimizeverylargescalenetworkmodelsquicklyandcheaply.NetworkModelsThestructureofthismodelma47Theshortest-routemodelreferstoanetworkforwhicheacharc(i,j)hasanassociatednumber,cij,whichisinterpretedasthedistance(orcost,ortime)fromnodeitonodej.NetworkModelsAShortest-RouteModelArouteorpathbetweentwonodesisanysequenceofarcsconnectingthetwonodes.Theobjectiveistofindtheshortest(orleast-costorleast-time)routesfromaspecificnodetoeachoftheothernodesinthenetwork.NetworkModelsTheshortest-routemodelrefer48Inthisexample,AaronDrunnermakesfrequentwinedeliveriesto7differentsites:874H12347651611213323Notethatthearcsareundirected(flowispermittedineitherdirection).Distancebetweennodes.HomeBaseInthisexample,AaronDrunner49Thegoalistominimizeoverallcostsbymakingsurethatanyfuturedeliverytoanygivensiteismadealongtheshortestroutetothatsite.ThegoalistominimizeoverallcostsbyfindingtheshortestroutefromnodeHtoanyoftheother7nodes.Notethatinthismodel,thetaskistofindanoptimalroute,notoptimalxij’s.NetworkModelsThegoalistominimizeoveral50Inthisexample,MichaelCarrisresponsibleforobtainingahighspeedprintingpressforhisnewspapercompany.NetworkModelsAnEquipmentReplacementModelInagivenyearhemustchoosebetweenpurchasing:NewPrintingPressOldPrintingPresshighannual

acquisitioncostlowinitial

maintenancecostnoannual

acquisitioncosthighinitial

maintenancecostNetworkModelsInthisexample,MichaelCarr51Assumea4-yeartimehorizon.Let:cijdenotethecostofbuyingnewequipment

atthebeginningofyeari,i=1,2,3,4

andmaintainingittothebeginningofyear

j,j=2,3,4,5Threealternativefeasiblepoliciesare:1.

Buyingnewequipmentatthebeginningof

eachyear.Total(acquisition+maintenance)cost=

c12+c23+c34+c452.

Buynewequipmentonlyatthebeginningof

year1andmaintainitthroughallsuccessive

years.Total(buying+maintenance)cost=c15Assumea4-yeartimehorizon.523.

Buynewequipmentatthebeginningofyears

1and4.Totalcost=c14+c45Thesolutiontothismodelisobtainedbyfindingtheshortest(i.e.,minimumcost)routefromnode1tonode5ofthenetwork.Eachnodeontheshortestroutedenotesareplacement,thatis,ayearatwhichnewequipmentshouldbebought.3.Buynewequipmentattheb53Hereisthenetworkmodelforthisproblem.Assumethefollowingcosts:$1,600,000purchasecost$500,000maintenancecostinpurchaseyear$1,000,000,$1,500,000,and$2,200,000foreachadditionalyeartheequipmentiskept12345c12c23c34c45c13c14c15c24c25c35Hereisthenetworkmodelfor54Inthemaximal-flowmodel,thereisasinglesourcenode(theinputnode)andasinglesinknode(theoutputnode).NetworkModelsAMaximalFlowModelThegoalistofindthemaximumamountoftotalflowthatcanberoutedthroughaphysicalnetwork(fromsourcetosink)inaunitoftime.Theamountofflowperunittimeoneacharcislimitedbycapacityrestrictions.Theonlyrequirementisthatforeachnode(otherthanthesourceorthesink):flowoutofthenode=flowintothenodeNetworkModelsInthemaximal-flowmodel,the55TheUDPC(UrbanDevelopmentPlanningCommission)isanadhocspecialintereststudygroup.AnApplicationofMaximal-Flow:

TheUrbanDevelopmentPlanningCommissionNetworkModelsAMaximalFlowModelThegroup’scurrentresponsibilityistocoordinatetheconstructionofthenewsubwaysystemwiththestate’shighwaymaintenancedepartment.Becausethenewsubwaysystemisbeingbuiltnearthecity’sbeltway,theeastboundtrafficonthebeltwaymustbedetoured.NetworkModelsTheUDPC(UrbanDevelopmentPl56Theproposednetworkofalternativeroutesandthedifferentspeedlimitsandtrafficpatterns(producingdifferentflowcapacities)aregivenbelow:154326406003402060016002DetourBeginsDetourEndsIndicatesacapacityof6000vehiclesperhourinthedirectionofthearrow.Indicates0capacityinthedirectionofthearrow.Theproposednetworkofaltern57TranslatingtheExcelsolutiontotheoriginalnetworkdiagramgivesthefollowingtrafficpattern:154326442266288TranslatingtheExcelsolution58第五讲TransportationandNetworkModels第五讲59IntroductionSeveralspecificmodels(whichcanbeusedastemplatesforreal-lifeproblems)willbeintroduced.TRANSPORTATIONMODEL

ASSIGNMENTMODEL

NETWORKMODELS

IntroductionSeveralspecificm60IntroductionTRANSPORTATIONMODEL

ASSIGNMENTMODEL

Determinehowtosendproductsfromvarioussourcestovariousdestinationsinordertosatisfyrequirementsatthelowestpossiblecost.Allocatingfixed-sizedresourcestodeterminetheoptimalassignmentofsalespeopletodistricts,jobstomachines,taskstocomputers…NETWORKMODELS

Involvethemovementorassignmentofphysicalentities(e.g.,money).

IntroductionTRANSPORTATIONMOD61TransportationModelAnexample,theAutoPowerCompanymakesavarietyofbatteryandmotorizeduninterruptibleelectricpowersupplies(UPS’s).AutoPowerhas4finalassemblyplantsinEuropeandthedieselmotorsusedbytheUPS’sareproducedintheUS,shippedto3harborsandthensenttotheassemblyplants.Productionplansforthethirdquarter(July–Sept.)havebeenset.Therequirements(demandatthedestination)andtheavailablenumberofmotorsatharbors(supplyatorigins)areshownonthenextslide:TransportationModelAnexample62DemandSupplyAssemblyPlant

No.ofMotorsRequiredLeipzig 400(2)Nancy

900(3)Liege 200(4)Tilburg 500 2000

Harbor

No.ofMotorsAvailable(A)Amsterdam 500(B)Antwerp

700(C)LeHavre 800 2000BalancedDemandSupplyAssemblyPlant N63GraphicalpresentationofLeHavre(C)800Antwerp(B)700Amsterdam(A)500SupplyLiege(3)200Tilburg(4)500Leipzig(1)