數(shù)據(jù)、模型與決策(運籌學)課后習題和案例答案

CHAPTER6

TRANSPORTATIONANDASSIGNMENTPROBLEMS

ReviewQuestions

6.1-1TheCEOisconcernedaboutescalatingcosts,inparticulartheshippingcostsforpeas.

6.1-2KimBakerisbeingaskedtolookatthecurrentshippingplanandseeiftheycandevelop

anewonethalwouldreducethelotaishippingcosttoanabsoluteminimum.

6.2-1Transportationproblemsingeneralareconcernedwithdistributinganycommodityfrom

anygroupofsupplycenters,calledsources,toanygroupofreceivingcenters,called

destinations,insuchawayastominimizethetotaldistributioncost.

6.2-2Thedataneededforatransportationproblemarcthesupplies,demands,andunitcosts.

6.2-3Formulatingaproblemasatransportationproblemonlyrequiresfillingoutatableinthe

formalofTable6.5.

6.2-4Atransportationproblemwillhavefeasiblesolutionsifandonlyifthesumofitssupplies

equalsthesumofitsdemands.

6.2-5Aslongasallitssuppliesanddemandshaveintegervalues,anytransportationproblem

withfeasiblesolutionsisguaranteedtohaveanoptimalsolutionwithintegervaluesfor

allitsdecisionvariables.

6.2-6Thetransportationsimplexmethodandnetworksimplexmethodcansolvethemfaster.

6.3-1A<signinsteadofan=signisusedintherespectivecellsandthecorresponding

constraintsintheSolverdialogbox.

6.3-2Insteadofademandrow,thereisbothaminimumrowandamaximumrow.Then

constraintsareenteredsothatShipped>MinimumandShipped<Maximum.

6.4-1Theareasofapplicationinthissectionaredistributingnaturalresources,production

scheduling,designingschoolattendancezones,meetingenergyneeds,andchoosinga

newsitelocation.

6.4-2Minimizethetotalcostofmeetingthewaterneedsofthefourcitiestheyserve.

6.4-3Thesourcesarctheproductionofjetenginesonregulartimeandovertimeineachofthe

fourmonths.Thedestinationsaretheirinstallationineachofthefourmonths.

6.4-4Distancesplaytheroleofunitcosts.

6.4-5Minimizethetotalcostofmeetingalltheenergyneeds.

6.5-1Thethreekeyfactorsare(1)thecostoftransportingtheoilfromitssourcestoallthe

refineries,includingthenewone;(2)thecostoftransportingfinishedproductfromallthe

refineries,includingthenewone,tothedistributioncenters;and(3)operatingcostsfor

thenewrefineries.

6.5-2Thenewrefinerywillhaveagreatimpactontheoperationoftheentiredistribution

system,includingdecisionsonhowmuchtoshiptoandfromeachrefinery(newandold).

6.5-3Threetransportationproblemsweresolvedtocomparetotalshippingcostsforcrudeoil

witheachpotentialchoiceofanewrefinerysite,andthreeweresolvedtocomparetotal

shippingcostsforfinishedproductwitheachpotentialchoiceofanewrefinerysite.

Thisresultedinsixtotalproblems.

6.5-4Managementmustconsidernon-financialfactorsaswell,suchasclosenesstocorporate

headquartersandwhetherthereareanycosttrendsortrendsinthemarketplacethatmight

alterthepictureinthefuture.

6.6-1Givenasetoftaskstobeperformedandasetofassigneesthatareavailabletoperform

thesetasks,theproblemistodeterminewhichassigneeshouldbeassignedtoeachtask.

6.6-2(1)Thenumberofassigneesandthenumberoftasksarethesame;(2)eachassigneeisto

beassignedtoexactlyonetask;(3)eachtaskistobeperformedbyexactlyoneassignee;

(4)thereisacostassociatedwitheachcombinationofanassigneeperformingatask;and

(5)theobjectiveistodeterminehowalltheassignmentshouldbemadetominimizethe

totalcost.

6.6-3(1)Identifytheassigneesandtasks,and(2)constructacosttablethatgivesthecost

associatedwitheachcombinationofanassigneeperformingatask.

6.6-4Whenanassignmentproblemisdescribedasatransportationproblem,thesourcesarc

assignments,thedestinationsaretasks,andthesuppliesanddemandsareallequalto1.

6.6-5TheHungarianmethodsolvesassignmentproblemswell.

6.7-1Whenformulatingthespreadsheetmodel,aconstraintisincludedintheSolverdialogbox

thatsetstheassignee/taskcombinationequalto0.

6.7-2Ifanassigneewillperformmorethanonetask,thesupplyischangefrom1tothegreater

amountthatcanbeperformed.

6.7-3Ifataskwillbeperformedbymorethanoneassignee,thedemandischangedfrom1to

thegreateramount.

Problems

6.1a)

b)

6.2a)

b)

IIII

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b)

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6.5

IIIIII

6.6

6.7

6.8

6.9

所岫嘛Mido曲眥

AllowableRange

Destination

SacramentoSaltLakeCityRapidCityAlbuquerque

SourceBellingham449tooo492to528570tooo516to888

Eugene-coto367401to437473tooo770to8

AlbertLea267tooo331tooo-coto472601to1,036

Theserangestellmanagementhowmucheachindividualcostcanchangewithout

changingtheoptimalsolution.

6.10

靦岫岫岫岫柵懶

a)Theoptimalsolutionwouldchangebecausethedecreaseof$3()millionisoutsidethe

allowabledecreaseof$20million.

b)Theoptimalsolutionwouldremainthesamesincetheallowableincreaseisoo.

PercentageofallowabledecreaseforCalorictoSanGo=(230-215)/20=75%.

PercentageofallowableincreaseforSacrontoLosDevils=(145-130)/co=0%.

Sum=75%<100%,sobythe100%-ruletheoptimalsolutionwillnotchange.

PercentageofallowabledecreaseforsupplyatSacron=(0.5/2.5)=20%.

PercentageofallowabledecreasefordemandatHollyglass=(0.5/1.5)=33.3%.

Sum=53.3%<100%,sobythe100%rulefbrsimultaneouschanges,theshadow

priceswillremainvalid.

6.11

6.12

6.13

6.14

6.15

1

6.16

6.17a)TheFeasibleSolutionsPropertyguaranteesthat,sincethesumofthesuppliesandthe

sumofthedemandsareequal,theproblemwillhavefeasiblesolutions.

b)TheIntegerSolutionPropertyguaranteesthat,sincesuppliesanddemandsare

integers,theresultingoptimalsolutionswillbeintegers.SuppliesandDemandsare

1,sotheonlypossiblevaluesofvariablesinanoptimalsolutionare0or1.

C)Thiscanbeinterpretedasanassignmentproblemsinceallthesuppliesanddemands

areequalto1.Eachsourceshipsoneunit(theassignee)toonedestination(thetask).

6.18

b)

2

6.19a)

b)

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6.20a)Thisproblemfitsasanassignmentproblemwithshipsasassigneesandportsas

assignments.

b)

IIIIII

7

6.21

6.22a)Theproblemfitsintotheformatforanassignmentproblemwithswimmersbeing

assignedtostrokes.Thetimesoftheswimmersreplacethecostsofatraditional

problem.

b)

_!_

三

6.23

6.24

Cases

6.1Option1(ShipDingbyRail):

—1-

RanaoNameColls

ShipmentQuanBrtO:F1

ShippingCostB3:F5

TotalAvailable110:112

TotalCost115

TotalReceivedB13:F1

TotalShippedG10:G1

IctallcrM［一815：卜1

_U.

_l_Ll

Oplion2(ShippingbyShip):

±

-

-

-

-

-

-

-

-

-

=-

1

RangeNameCells

CostFactor115

Oplion3(ShippingbyBeslAvailableforeachRoute):

IlIIIIII

Whencomparingthethreeoptions,ilisbesttousethecombinationplan,whileshipping

entirelybyrailleadstothehighestcosts.

Ifcostsofshippingbywaterareexpectedtoriseconsiderablymorethanforshippingby

rail,staywithrailanduseOption1.Ifthereverseistrue,thenuseOption2.Ifthecost

comparisonswillremainroughlythesame,useOption3.Option3isclearlythemost

feasiblebutmaynotbechosenifitistoologisticallycumbersome.Moreknovzlcdgcof

Ihesituationisnecessarytodeterminethis.

6.2a)$20millionissavedincomparisonwiththeresultsinFigure6.13byshipping20

millionfewerbarrelstoCharlestonand20millionmoretoSt.Louis.

b)S40millionissavedincomparisonwiththeresultsinFigure6.17.

Thecostofshippingbothcrudeoilandfinishedproductunderthisplanis$940

million+$1,390million=$2,330millionor$2.33billion—asavingsof$60million

comparedtotheoriginalresultsinTable6.20.

c)$35millionissavedincomparisonwiththeresultsinpart(b).

$75millionissavedincomparisonwiththeresultsinFigure6.17.

d)Thissolutioncosts$40millionmorethanthesolutioninpart(a).

Thissolutioncosts$20millionmorethanthesolutionisFigure6.13.

Thetotalcostofshippingbothcrudeoilandfinishedproductunderthisplanis$1,355

million+$980million=$2,335millionor$2,335billion.Thisis$5millionmore

thanthecostofthecombinedtotalobtainedinpart(b),but$55millionlessthanthe

totalinTable6.20.

e)Thetwotransportationproblems(shippingtorefineriesandshippingtodistributions

centers)arecombinedintoasinglemodel.Theamountshippedtotherefineriesis

constrainedtobenomorethancapacity:TotalReceived(D16:G16)<

Capacity(D18:G18).Thetotalshippedoutoftherefineriesisconstrainedtoequalthe

totalamountshippedin:ShippedOut(H31:H34)=ShippedIn(J31:J34).Thegoalisto

minimizethetotalcombinedcost(inJ45)whichisthesumofthetwointermediate

costs(inJ20andJ39).

Thetotalcombinedcostis$2,320millionor$2.32billion,whichisS10millionless

thaninpart(b),$15millionlessthaninpart(d),and$70millionlessthaninTable

6.20.

f)IftheLosAngelesrefineryischoseninstead,thenthecombinedshippingcostis

$2,450million.

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IftheGalvestonrefineryischoseninstead,thenthecombinedshippingcostisS2,470

million.

II_______I______I______I______I_______I_______LJ--------------1

TotalCostTotalCostOperatingCostTotal

ofShippingofShippingforNewVariable

SiteCrudeOilFinishedProductRefineryCost

LosAngeles$880million$1.57billion$620million$3.07billion

Galveston870million1.60billion570million3.12billion

St.Louis950million1.37billion530million2.92billion

g)Answerswillvary.

6.3a)Assignonescientisttoeachofthefiveprojectstomaximizethetotalnumberofbid

points.

TomaximizethescientistspreferencesyouwanttoassignDr.TsaitoleadprojectUp,

Dr.KvaaltoleadprojectStable,Dr.ZuncrtoleadprojectChoice,Dr.Mickeytolead

projectHope,andDr.RollinstoleadprojectRelease.

b)Dr.Rollinsisnotavailable,sohis“Supply“incell114isreducedtozero.Sincenow

mustallowaprojecttonotbedone,theconstraintsinrows15to17become

TotalAssigned(B15:F15)<Dcmand(B17:F17)ratherthan=.

_1_

ProjectUpwouldnotbedone.

c)SinceDr.ZoonerorDr.Mickeycanleadtwoprojects,their“Supply“incolumnIis

changedto2andthecorrespondingconstraintchangedto<(inordertoallowthemto

doeitheroneortwoprojects).

Dr.KvaalleadsprojectStable,Dr.ZunerleadsprojectChoice,Dr.Tsaileadsproject

Release,andDr.MickeyleadstheprojectsHopeandUp.

d)UnderthenewbidsofDr.Zunertheassignmentdoesnotchange:

e)CertainlyDr.ZunercouldbedisappointedthatsheisnotassignedtoprojectStable,

especiallywhensheexpressedahigherpreferenceforthatprojectthanthescientist

ass