1985美国大学生数学建模竞赛题目集锦.docx

1985美国大学生数学建模竞赛题目集锦.docx

《1985美国大学生数学建模竞赛题目集锦.docx》由会员分享，可在线阅读，更多相关《1985美国大学生数学建模竞赛题目集锦.docx（50页珍藏版）》请在冰豆网上搜索。

1985美国大学生数学建模竞赛题目集锦

1985~2015年美国大学生数学建模竞赛题目集锦

目录

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,havebeenidentifiedbybiologistsGroganandWirthonthebasisofantennaandwinglength（seeFigure1）.ItisimportanttobeabletoclassifyaspecimenasAfofApf,giventheantennaandwinglength.

1.GivenamidgethatyouknowisspeciesAforApf,howwouldyougoaboutclassifyingit?

2.Applyyourmethodtothreespecimenswith（antenna,wing）lengths（1.24,1.80）,（1.28,1.84）,（1.40,2.04）.

3.AssumethatthespeciesisavaluablepollinatorandspeciesApfisacarrierofadebilitatingdisease.Wouldyoumodifyyourclassificationschemeandifso,how?

1989MCMB:

AircraftQueueing

Acommonprocedureatairportsistoassignaircraft（A/C）torunwaysonafirst-come-first-servedbasis.Thatis,assoonasanA/Cisreadytoleavethegate（“push-back”）,thepilotcallsgroundcontrolandisaddedtothequeue.SupposethatacontroltowerhasaccesstoafastonlinedatabasewiththefollowinginformationforeachA/C:

▪thetimeitisscheduledforpushback;

▪thetimeitactuallypushesback;thenumberofpassengerswhoarescheduledtomakeaconnectionatthenextstop,aswellasthetimetomakethatconnection;and

▪thescheduletimeofarrivalatitsnextstopAssumethatthereareseventypesofA/Cwithpassengercapacitiesvaryingfrom100to400instepsof50.Developandanalyzeamathematicalmodelthattakesintoaccountboththetravelers'andairlines'satisfaction.

1990MCMA:

TheBrain-DrugProblem

Researchesonbraindisorderstesttheeffectsofthenewmedicaldrugs–forexample,dopamineagainstParkinson'sdisease–withintracerebralinjections.Tothisend,theymustestimatethesizeandthesapeofthespatialdistributionofthedrugaftertheinjection,inordertoestimateaccuratelytheregionofthebrainthatthedrughasaffected.

Theresearchdataconsistofthemeasurementsoftheamountsofdrugineachof50cylindricaltissuesamples（seeFigure1andTable1）.Eachcylinderhaslength0.76mmanddiameter0.66mm.Thecentersoftheparallelcylinderslieonagridwithmesh1mmX0.76mmX1mm,sothatthesylinderstouchoneanotherontheircircularbasesbutnotalongtheirsides,asshownintheaccompanyingfigure.Theinjectionwasmadenearthecenterofthecylinderwiththehighestscintillationcount.Naturally,oneexpectsthatthereisadrugalsobetweenthecylindersan