美国数学建模MCM1985题目汇总.docx

美国数学建模MCM1985题目汇总.docx

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美国数学建模MCM1985题目汇总

1985

ProblemAAnimalPopulations

Chooseafishormammalforwhichappropriatedataareavailabletomodelitaccurately.Modeltheanimal'snaturalinteractionswithitsenvironmentbyexpressingpopulationlevelsofdifferentgroupsintermsofthesignificantparametersoftheenvironment.Thenadjustthemodeltoaccountforharvestinginaformconsistentwiththeactualmethodbywhichtheanimalisharvested.Includeanyoutsideconstraintsimposedbyfoodorspacelimitationsthataresupportedbythedata.Considerthevalueofthevariousquantitiesinvolved,thenumberharvested,andthepopulationsizeitself,inordertodeviseanumericalquantitythatrepresentstheoverallvalueoftheharvest.Findaharvestingpolicyintermsofpopulationsizeandtimethatoptimizesthevalueoftheharvestoveralongperiodoftime.Checkthatthepolicyoptimizesthatvalueoverarealisticrangeofenvironmentalconditions.

ProblemBStrategicReserveManagement

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?

1986

ProblemAHydrographicData

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

ProblemBEmergency-FacilitiesLocation

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

Assumethatthedemandisconcentratedatthecenteroftheblockandthatthefacilitieswillbelocatedoncorners.

Assumethatthedemandisuniformlydistributedonthestreetsborderingeachblockandthatthefacilitiesmaybelocatedanywhereonthestreets.

MCM1987A题TheSaltStorageProblem

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

Recently,apaneldeterminedthatthispracticeisunsafe.Ifthefront-endloadergetstooclosetotheedgeofthesaltpile,thesaltmightshift,andtheloadercouldbethrownagainsttheretainingwallsthatreinforcethedome.Thepanelrecommendedthatifthesaltistobepiledwiththeuseoftheloaders,thenthepilesshouldberestrictedtoamaximumheightof15ft.

Constructamathematicalmodelforthissituationandfindarecommendedmaximumheightforsaltinthedomes.

盐的存贮

美国中西部一个州把冬天用来洒在马路上的盐存贮在一个球顶仓库里大约有15年了。

图87A-1表示在过去15年中盐是怎么存贮的*通过驾驶铲斗车在由盐铺成的坡道上进出仓里并利用铲斗车上的铲子把盐装进仓里或从仓里取出来。

最近，一个小组确定这种做法是不安全的。

如果铲斗车太靠近盐堆的顶端，盐就要滑动，而铲斗车就耍翻到为加固仓库而筑的拥壁上去。

小组建议，如果盐堆是用铲斗车堆起来的，那么盐堆的最高高度不要超过15英尺。

对这种情况建立一个数学模型并求得在仓库中的盐堆的最大高度。

图中仓高50英尺，拥壁高4英尺，仓的外直径103英尺，门的净空高l9英尺9英寸，铲斗车高10英尺9英寸。

ProblemBParkingLotDesign

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

MCM1988A题TheDrugRunnerProblem

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

Developanoptimalsearchmethodforthehelicopter.

Usea95%confidencelevelinyourcalculations.

确定毒品走私船的位置

相距5.43哩的监听站收听到一个短暂的无线电讯号。

收听到讯早的时候测向仪分别定位在111°和119°处〔见图88A-1），测向仪的精度为±2°，该讯号来自一个毒品交换活跃的地方，据推测该处有一只机动船正等着有人来取毒品。

当时正值黄昏、无风、无潮流。

一架小型直升飞机离开监听站①的简易机场并能精确地沿111°角方向飞行。

直升飞机的飞行速度是走私船的三倍。

在离船500英尺时船上能听到直升飞机的声音。

直升飞机只有一种侦察仪器--探照订。

在200英尺远的地方探照灯只能照明半径为25英尺的圆域。

1.说明飞行员能找到正等着的毒品船的（最小）区域。

2.研究一种直升飞机的最佳搜索方法。

在你的计算中要有95％的精度。

本题是由加州ClaremontMcKenna学院的J.A.Ferling提供的。

这是一个分类（分组问题）的修正简化形式。

原问题和现在简化的问题都还没有一种已知的最化解法。

ProblemPackingRailroadFlatcars

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].

MCM1989A题TheMidgeClassificationProblem

Twospeciesofmidges,AfandApf,havebeenidentifiedbybiologistsGroganandWirthonthebasisofantennaandwinglength（seeFigure1）.ItisimportanttobeabletoclassifyaspecimenasAfofApf,giventheantennaandwinglength.

GivenamidgethatyouknowisspeciesAforApf,howwouldyougoaboutclassifyingit?

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

AssumethatthespeciesisavaluablepollinatorandspeciesApfisacarrierofadebilitatingdisease.Wouldyoumodifyyourclassificationschemeandifso,how?

蠓的分类

两种蠓Af和Apf己由生物学家W.L.Grongan和W.W.Wirth（1981年）根据它们的触角长度和翼长加以区分（见图89A-1），9只Af蠓用圆圈标记，6只Apf蠓用黑点标记。

根据给出的触角长度和翼长识别出一只标本是Af还是Apf是重要的。

1.给定一只Af或者Apf族的蝶，你如何正确地区分它属于哪一族?

2.将你的方法用于触角长和翼长分别为（1.24，1.80）、（1.28，1.84）、（1.40，2.04）的三个标本。

3.设Af是宝贵的传粉益虫，Apf是某种疾病的载体，是否应该修改你的分类方法，若需修改，怎么改?

ProblemAircraftQueuing

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.

MCM1991B题TheSteinerTreeProblem

Thecostforacommunicationlinebetweentwostationsisproportionaltothelengthoftheline.Thecostforconventionalminimalspanningtreesofasetofstationscanoftenbecutbyintroducing"phantom"stationsandthenconstructinganewSteinertree.Thisdeviceallowscoststobecutbyupto13.4%（=1-sqrt（3/4））.Moreover,anetworkwithnstationsneverrequiresmorethann-2pointstoconstructthecheapestSteinertree.TwosimplecasesareshowninFigure1.

Forlocalnetworks,itoftenisnecessarytouserectilinearor"checker-board"distances,insteadofstraightEuclideanlines.Distancesinthismetricare