美国数学建模比赛历年试题.docx

美国数学建模比赛历年试题.docx

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美国数学建模比赛历年试题

1985~2014年美国大学生数学建模竞赛题目集锦

目录

1985MCMA:

AnimalPopulations3

1985MCMB:

StrategicReserveManagement3

1986MCMA:

HydrographicData4

1986MCMB:

Emergency-FacilitiesLocation4

1987MCMA:

TheSaltStorageProblem5

1987MCMB:

ParkingLotDesign5

1988MCMA:

TheDrugRunnerProblem5

1988MCMB:

PackingRailroadFlatcars6

1989MCMA:

TheMidgeClassificationProblem6

1989MCMB:

AircraftQueueing6

1990MCMA:

TheBrain-DrugProblem6

1990MCMB:

SnowplowRouting7

1991MCMA:

WaterTankFlow8

1991MCMB:

TheSteinerTreeProblem8

1992MCMA:

Air-Traffic-ControlRadarPower8

1992MCMB:

EmergencyPowerRestoration9

1993MCMA:

OptimalComposting10

1993MCMB:

Coal-TippleOperations11

1994MCMA:

ConcreteSlabFloors11

1994MCMB:

NetworkDesign12

1995MCMA:

HelixConstruction13

1995MCMB:

FacultyCompensation13

1996MCMA:

SubmarineTracking13

1996MCMB:

PaperJudging13

1997MCMA:

TheVelociraptorProblem14

1997MCMB:

MixWellforFruitfulDiscussions15

1998MCMA:

MRIScanners16

1998MCMB:

GradeInflation17

1999MCMA:

DeepImpact17

1999MCMB:

UnlawfulAssembly18

2000MCMA:

AirTrafficControl18

2000MCMB:

RadioChannelAssignments19

2001MCMA:

ChoosingaBicycleWheel20

2001MCMB:

EscapingaHurricane'sWrath（AnIllWind...）21

2002MCMA:

WindandWaterspray23

2002MCMB:

AirlineOverbooking23

2003MCMA:

TheStuntPerson24

2003MCMB:

GammaKnifeTreatmentPlanning24

2004MCMA:

AreFingerprintsUnique?

25

2004MCMB:

AFasterQuickPassSystem25

2005MCMA:

FloodPlanning26

2005MCMB:

Tollbooths26

2006MCMA:

PositioningandMovingSprinklerSystemsforIrrigation27

2006MCMB:

WheelChairAccessatAirports27

2007MCMA:

Gerrymandering29

2007MCMB:

TheAirplaneSeatingProblem29

2008MCMA:

TakeaBath30

2008MCMB:

CreatingSudokuPuzzles30

2009MCMA:

DesigningaTrafficCircle30

2009MCMB:

EnergyandtheCellPhone30

2010MCMA:

TheSweetSpot32

2010MCMB:

Criminology32

2011MCMA:

SnowboardCourse33

2011MCMB:

RepeaterCoordination33

2012MCMA:

TheLeavesofaTree33

2012MCMB:

CampingalongtheBigLongRiver34

2013MCMA:

TheUltimateBrowniePan34

2013MCMB:

Water,Water,Everywhere35

2014MCMA:

TheKeep-Right-Except-To-PassRule35

2014MCMB:

CollegeCoachingLegends36

1985MCMA:

AnimalPopulations

Chooseafishormammalforwhichappropriatedataareavailabletomodelitaccurately.Modeltheanimal'snaturalinteractionswithitsenvironmentbyexpressingpopulationlevelsofdifferentgroupsintermsofthesignificantparametersoftheenvironment.Thenadjustthemodeltoaccountforharvestinginaformconsistentwiththeactualmethodbywhichtheanimalisharvested.Includeanyoutsideconstraintsimposedbyfoodorspacelimitationsthataresupportedbythedata.

Considerthevalueofthevariousquantitiesinvolved,thenumberharvested,andthepopulationsizeitself,inordertodeviseanumericalquantitythatrepresentstheoverallvalueoftheharvest.Findaharvestingpolicyintermsofpopulationsizeandtimethatoptimizesthevalueoftheharvestoveralongperiodoftime.Checkthatthepolicyoptimizesthatvalueoverarealisticrangeofenvironmentalconditions.

1985MCMB:

StrategicReserveManagement

Cobalt,whichisnotproducedintheUS,isessentialtoanumberofindustries.（Defenseaccountedfor17%ofthecobaltproductionin1979.）MostcobaltcomesfromcentralAfrica,apoliticallyunstableregion.TheStrategicandCriticalMaterialsStockpilingActof1946requiresacobaltreservethatwillcarrytheUSthroughathree-yearwar.Thegovernmentbuiltupastockpileinthe1950s,soldmostofitoffintheearly1970s,andthendecidedtobuilditupagaininthelate1970s,withastockpilegoalof85.4millionpounds.Abouthalfofthisstockpilehadbeenacquiredby1982.

Buildamathematicalmodelformanagingastockpileofthestrategicmetalcobalt.Youwillneedtoconsidersuchquestionsas:

▪Howbigshouldthestockpilebe?

▪Atwhatrateshoulditbeacquired?

▪Whatisareasonablepricetopayforthemetal?

Youwillalsowanttoconsidersuchquestionsas:

▪Atwhatpointshouldthestockpilebedrawndown?

▪Atwhatrateshoulditbedrawndown?

▪Atwhatpriceisitreasonabletosellthemetal?

▪Howshoulditbeallocated?

UsefulInformationonCobalt

Thegovernmenthasprojectedaneedot25millionpoundsofcobaltin1985.

TheU.S.hasabout100millionpoundsofprovencobaltdeposits.Productionbecomeseconomicallyfeasiblewhenthepricereaches$22/lb（asoccurredin1981）.Ittakesfouryearstogetoperationsrolling,andthsnsixmillionpoundsperyearcanbeproduced.

In1980,1.2millionpoundsofcobaltwererecycled,7%oftotalconsumption.

1986MCMA:

HydrographicData

ThetablebelowgivesthedepthZofwaterinfeetforsurfacepointswithrectangularcoordinatesX,Yinyards[tableof14datapointsomitted].Thedepthmeasurementsweretakenatlowtide.Yourshiphasadraftoffivefeet.Whatregionshouldyouavoidwithintherectangle（75,200）x（-50,150）?

X

Y

Z

129.0

7.5

4

140.0

141.5

8

108.5

28.0

6

88.0

147.0

8

185.5

22.5

6

195.0

137.5

8

105.5

85.5

8

157.5

-6.5

9

107.5

-81.0

9

77.0

3.0

8

162.0

-66.5

9

162.0

84.0

4

117.5

-35.5

9

1986MCMB:

Emergency-FacilitiesLocation

ThetownshipofRioRanchohashithertonothaditsownemergencyfacilities.Ithassecuredfundstoerecttwoemergencyfacilitiesin1986,eachofwhichwillcombineambulance,fire,andpoliceservices.Figure1indicatesthedemand[figureomitted],ornumberofemergenciespersquareblock,for1985.The“L”regioninthenorthisanobstacle,whiletherectangleinthesouthisapartwithshallowpond.Ittakesanemergencyvehicleanaverageof15secondstogooneblockintheN-Sdirectionand20secondsintheE-Wdirection.Yourtaskistolocatethetwofacilitiessoastominimizethetotalresponsetime.

▪Assumethatthedemandisconcentratedatthecenteroftheblockandthatthefacilitieswillbelocatedoncorners.

▪Assumethatthedemandisuniformlydistributedonthestreetsborderingeachblockandthatthefacilitiesmaybelocatedanywhereonthestreets.

1987MCMA:

TheSaltStorageProblem

Forapproximately15years,aMidwesternstatehasstoredsaltusedonroadsinthewinterincirculardomes.Figure1showshowsalthasbeenstoredinthepast.Thesaltisbroughtintoandremovedfromthedomesbydrivingfront-endloadersuprampsofsaltleadingintothedomes.Thesaltispiled25to30fthigh,usingthebucketsonthefront-endloaders.

Recently,apaneldeterminedthatthispracticeisunsafe.Ifthefront-endloadergetstooclosetotheedgeofthesaltpile,thesaltmightshift,andtheloadercouldbethrownagainsttheretainingwallsthatreinforcethedome.Thepanelrecommendedthatifthesaltistobepiledwiththeuseoftheloaders,thenthepilesshouldberestrictedtoamatimumheightof15ft.

Constructamathematicalmodelforthissituationandfindarecommendedmaximumheightforsaltinthedomes.

1987MCMB:

ParkingLotDesign

Theownerofapaved,100'by200',cornerparkinglotinaNewEnglandtownhiresyoutodesignthelayout,thatis,todesignhowthe``linesaretobepainted. Yourealizethatsqueezingasmanycarsintothelotaspossibleleadstoright-angleparkingwiththecarsalignedsidebyside.However,inexperienceddrivershavedifficultyparkingtheircarsthisway,whichcangiverisetoexpensiveinsuranceclaims.Toreducethelikelihoodofdamagetoparkedvehicles,theownermightthenhavetohireexpertdriversfor``valetparking. Ontheotherhand,mostdriversseemtohavelittledifficultyinparkinginoneattemptifthereisalargeenough``turningradius''fromtheaccesslane.Ofcourse,thewidertheaccesslane,thefewercarscanbeaccommodatedinthelot,leadingtolessrevenuefortheparkinglotowner.

1988MCMA:

TheDrugRunnerProblem

Twolisteningposts5.43milesapartpickupabriefradiosignal.Thesensingdeviceswereorientedat110degreesand119degrees,respectively,whenthesignalwasdetected;andtheyareaccuratetowithin2degrees.Thesignalcamefromaregionofactivedrugexchange,anditisinferredthatthereisapowerboatwaitingforsomeonetopickupdrugs.itisdusk,theweatheriscalm,andtherearenocurrents.AsmallhelicopterleavesfromPost1andisabletoflyaccuratelyalongthe110degreeangledirection.Thehelicopter'sspeedisthreetimesthespeedoftheboat.Thehelicopterwillbeheardwhenitgetswithin500ftoftheboat.Thishelicopterhasonlyonedetectiondevice,asearchlight.At200ft,itcanjustilluminateacircularregionwitharadiusof25ft.

▪Developanoptimalsearchmethodforthehelicopter.

▪Usea95%confidencelevelinyourcalculations.

1988MCMB:

PackingRailroadFlatcars

Tworailroadflatcarsaretobeloadedwithseventypesofpackingcrates.Thecrateshavethesamewidthandheightbutvaryingthickness（t,incm）andweight（w,inkg）.Table1gives,foreachcrate,thethickness,weight,andnumberavailable[tableomitted].Eachcarhas10.2metersoflengthavailableforpackingthecrates（likeslicesoftoast）andcancarryupto40metrictons.ThereisaspecialconstraintonthetotalnumberofC_5,C_6,andC_7cratesbecauseofasubsequentlocaltruckingrestriction:

Thetotalspace（thickness）occupiedbythesecratesmustnotexceed302.7cm.Loadthetwoflatcars（seeFigure1）soastominimizethewastedfloorspace[figureomitted].

1989MCMA:

TheMidgeClassificationProblem

Twospeciesofmidges,AfandApf,havebeenidentifiedbybiologistsGroganandWirthonthebasisofantenna