数据模型与决策运筹学课后习题和案例答案007.docx

数据模型与决策运筹学课后习题和案例答案007.docx

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数据模型与决策运筹学课后习题和案例答案007

Chapter7

NetworkOptimizationProblems

ReviewQuestions

7.1-1Asupplynodeisanodewherethenetamountofflowgeneratedisafixedpositivenumber.Ademandnodeisanodewherethenetamountofflowgeneratedisafixednegativenumber.Atransshipmentnodeisanodewherethenetamountofflowgeneratedisfixedatzero.

7.1-2Themaximumamountofflowallowedthroughanarcisreferredtoasthecapacityofthatarc.

7.1-3Theobjectiveistominimizethetotalcostofsendingtheavailablesupplythroughthenetworktosatisfythegivendemand.

7.1-4Thefeasiblesolutionspropertyisnecessary.Itstatesthataminimumcostflowproblemwillhaveafeasiblesolutionifandonlyifthesumofthesuppliesfromitssupplynodesequalsthesumofthedemandsatitsdemandnodes.

7.1-5Aslongasallitssuppliesanddemandshaveintegervalues,anyminimumcostflowproblemwithfeasiblesolutionsisguaranteedtohaveanoptimalsolutionwithintegervaluesforallitsflowquantities.

7.1-6Networksimplexmethod.

7.1-7Applicationsofminimumcostflowproblemsincludeoperationofadistributionnetwork,solidwastemanagement,operationofasupplynetwork,coordinatingproductmixesatplants,andcashflowmanagement.

7.1-8Transportationproblems,assignmentproblems,transshipmentproblems,maximumflowproblems,andshortestpathproblemsarespecialtypesofminimumcostflowproblems.

7.2-1Oneofthecompany’smostimportantdistributioncenters（LosAngeles）urgentlyneedsanincreasedflowofshipmentsfromthecompany.

7.2-2Autoreplacementpartsareflowingthroughthenetworkfromthecompany’smainfactoryinEuropetoitsdistributioncenterinLA.

7.2-3TheobjectiveistomaximizetheflowofreplacementpartsfromthefactorytotheLAdistributioncenter.

7.3-1Ratherthanminimizingthecostoftheflow,theobjectiveistofindaflowplanthatmaximizestheamountflowingthroughthenetworkfromthesourcetothesink.

7.3-2Thesourceisthenodeatwhichallflowthroughthenetworkoriginates.Thesinkisthenodeatwhichallflowthroughthenetworkterminates.Atthesource,allarcspointawayfromthenode.Atthesink,allarcspointintothenode.

7.3-3Theamountismeasuredbyeithertheamountleavingthesourceortheamountenteringthesink.

7.3-41.Whereassupplynodeshavefixedsuppliesanddemandnodeshavefixeddemands,thesourceandsinkdonot.

2.Whereasthenumberofsupplynodesandthenumberofdemandnodesinaminimumcostflowproblemmaybemorethanone,therecanbeonlyonesourceandonlyonesinkinastandardmaximumflowproblem.

7.3-5Applicationsofmaximumflowproblemsincludemaximizingtheflowthroughadistributionnetwork,maximizingtheflowthroughasupplynetwork,maximizingtheflowofoilthroughasystemofpipelines,maximizingtheflowofwaterthroughasystemofaqueducts,andmaximizingtheflowofvehiclesthroughatransportationnetwork.

7.4-1Theoriginisthefirestationandthedestinationisthefarmcommunity.

7.4-2Flowcangoineitherdirectionbetweenthenodesconnectedbylinksasopposedtoonlyonedirectionwithanarc.

7.4-3Theoriginnowistheonesupplynode,withasupplyofone.Thedestinationnowistheonedemandnode,withademandofone.

7.4-4Thelengthofalinkcanmeasuredistance,cost,ortime.

7.4-5Sarahwantstominimizehertotalcostofpurchasing,operating,andmaintainingthecarsoverherfouryearsofcollege.

7.4-6When“realtravel”throughanetworkcanendatmorethatonenode,adummydestinationneedstobeaddedsothatthenetworkwillhavejustasingledestination.

7.4-7Quick’smanagementmustconsidertrade-offsbetweentimeandcostinmakingitsfinaldecision.

7.5-1Thenodesaregiven,butthelinksneedtobedesigned.

7.5-2Astate-of-the-artfiber-opticnetworkisbeingdesigned.

7.5-3Atreeisanetworkthatdoesnothaveanypathsthatbeginandendatthesamenodewithoutbacktracking.Aspanningtreeisatreethatprovidesapathbetweeneverypairofnodes.Aminimumspanningtreeisthespanningtreethatminimizestotalcost.

7.5-4Thenumberoflinksinaspanningtreealwaysisonelessthanthenumberofnodes.Furthermore,eachnodeisdirectlyconnectedbyasinglelinktoatleastoneothernode.

7.5-5Todesignanetworksothatthereisapathbetweeneverypairofnodesattheminimumpossiblecost.

7.5-6No,itisnotaspecialtypeofaminimumcostflowproblem.

7.5-7Agreedyalgorithmwillsolveaminimumspanningtreeproblem.

7.5-8Applicationsofminimumspanningtreeproblemsincludedesignoftelecommunicationnetworks,designofalightlyusedtransportationnetwork,designofanetworkofhigh-voltagepowerlines,designofanetworkofwiringonelectricalequipment,anddesignofanetworkofpipelines.

Problems

7.1a）

b）

c）

7.2a）