美国数学建模比赛历年试题Word格式文档下载.docx
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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
AnimalPopulations
Chooseafishormammalforwhichappropriatedataareavailabletomodelitaccurately.Modeltheanimal'
snaturalinteractionswithitsenvironmentbyexpressingpopulationlevelsofdifferentgroupsintermsofthesignificantparametersoftheenvironment.Thenadjustthemodeltoaccountforharvestinginaformconsistentwiththeactualmethodbywhichtheanimalisharvested.Includeanyoutsideconstraintsimposedbyfoodorspacelimitationsthataresupportedbythedata.
Considerthevalueofthevariousquantitiesinvolved,thenumberharvested,andthepopulationsizeitself,inordertodeviseanumericalquantitythatrepresentstheoverallvalueoftheharvest.Findaharvestingpolicyintermsofpopulationsizeandtimethatoptimizesthevalueoftheharvestoveralongperiodoftime.Checkthatthepolicyoptimizesthatvalueoverarealisticrangeofenvironmentalconditions.
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.
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
185.5
22.5
195.0
137.5
105.5
85.5
157.5
-6.5
9
107.5
-81.0
77.0
3.0
162.0
-66.5
84.0
117.5
-35.5
Emergency-FacilitiesLocation
ThetownshipofRioRanchohashithertonothaditsownemergencyfacilities.Ithassecuredfundstoerecttwoemergencyfacilitiesin1986,eachofwhichwillcombineambulance,fire,andpoliceservices.Figure1indicatesthedemand[figureomitted],ornumberofemergenciespersquareblock,for1985.The“L”regioninthenorthisanobstacle,whiletherectangleinthesouthisapartwithshallowpond.Ittakesanemergencyvehicleanaverageof15secondstogooneblockintheN-Sdirectionand20secondsintheE-Wdirection.Yourtaskistolocatethetwofacilitiessoastominimizethetotalresponsetime.
▪Assumethatthedemandisconcentratedatthecenteroftheblockandthatthefacilitieswillbelocatedoncorners.
▪Assumethatthedemandisuniformlydistributedonthestreetsborderingeachblockandthatthefacilitiesmaybelocatedanywhereonthestreets.
TheSaltStorageProblem
Forapproximately15years,aMidwesternstatehasstoredsaltusedonroadsinthewinterincirculardomes.Figure1showshowsalthasbeenstoredinthepast.Thesaltisbroughtintoandremovedfromthedomesbydrivingfront-endloadersuprampsofsaltleadingintothedomes.Thesaltispiled25to30fthigh,usingthebucketsonthefront-endloaders.
Recently,apaneldeterminedthatthispracticeisunsafe.Ifthefront-endloadergetstooclosetotheedgeofthesaltpile,thesaltmightshift,andtheloadercouldbethrownagainsttheretainingwallsthatreinforcethedome.Thepanelrecommendedthatifthesaltistobepiledwiththeuseoftheloaders,thenthepilesshouldberestrictedtoamatimumheightof15ft.
Constructamathematicalmodelforthissituationandfindarecommendedmaximumheightforsaltinthedomes.
ParkingLotDesign
Theownerofapaved,100'
by200'
cornerparkinglotinaNewEnglandtownhiresyoutodesignthelayout,thatis,todesignhowthe``linesaretobepainted.
Yourealizethatsqueezingasmanycarsintothelotaspossibleleadstoright-angleparkingwiththecarsalignedsidebyside.However,inexperienceddrivershavedifficultyparkingtheircarsthisway,whichcangiverisetoexpensiveinsuranceclaims.Toreducethelikelihoodofdamagetoparkedvehicles,theownermightthenhavetohireexpertdriversfor``valetparking.
Ontheotherhand,mostdriversseemtohavelittledifficultyinparkinginoneattemptifthereisalargeenough``turningradius'
'
fromtheaccesslane.Ofcourse,thewidertheaccesslane,thefewercarscanbeaccommodatedinthelot,leadingtolessrevenuefortheparkinglotowner.
TheDrugRunnerProblem
Twolisteningposts5.43milesapartpickupabriefradiosignal.Thesensingdeviceswereorientedat110degreesand119degrees,respectively,whenthesignalwasdetected;
andtheyareaccuratetowithin2degrees.Thesignalcamefromaregionofactivedrugexchange,anditisinferredthatthereisapowerboatwaitingforsomeonetopickupdrugs.itisdusk,theweatheriscalm,andtherearenocurrents.AsmallhelicopterleavesfromPost1andisabletoflyaccuratelyalongthe110degreeangledirection.Thehelicopter'
sspeedisthreetimesthespeedoftheboat.Thehelicopterwillbeheardwhenitgetswithin500ftoftheboat.Thishelicopterhasonlyonedetectiondevice,asearchlight.At200ft,itcanjustilluminateacircularregionwitharadiusof25ft.
▪Developanoptimalsearchmethodforthehelicopter.
▪Usea95%confidencelevelinyourcalculations.
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].
TheMidgeClassificationProblem
Twospeciesofmidges,AfandApf,havebeenidentifiedbybiologistsGroganandWirthonthebasisofantenna