英文历届美国数学建模竞赛赛题.docx

英文历届美国数学建模竞赛赛题.docx

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英文历届美国数学建模竞赛赛题

2001MCMA:

ChoosingaBicycleWheel

Cyclistshavedifferenttypesofwheelstheycanuseontheirbicycles.Thetwobasictypesofwheelsarethoseconstructedusingwirespokesandthoseconstructedofasoliddisk（seeFigure1）Thespokedwheelsarelighter,butthesolidwheelsaremoreaerodynamic.Asolidwheelisneverusedonthefrontforaroadracebutcanbeusedontherearofthebike.

Professionalcyclistslookataracecourseandmakeaneducatedguessastowhatkindofwheelsshouldbeused.Thedecisionisbasedonthenumberandsteepnessofthehills,theweather,windspeed,thecompetition,andotherconsiderations.Thedirectorsportifofyourfavoriteteamwouldliketohaveabettersysteminplaceandhasaskedyourteamforinformationtohelpdeterminewhatkindofwheelshouldbeusedforagivencourse.

Figure1:

Asolidwheelisshownontheleftandaspokedwheelisshownontheright.

Thedirectorsportifneedsspecificinformationtohelpmakeadecisionandhasaskedyourteamtoaccomplishthetaskslistedbelow.Foreachofthetasksassumethatthesamespokedwheelwillalwaysbeusedonthefrontbutthereisachoiceofwheelsfortherear.

∙Task1.Provideatablegivingthewindspeedatwhichthepowerrequiredforasolidrearwheelislessthanforaspokedrearwheel.Thetableshouldincludethewindspeedsfordifferentroadgradesstartingfromzeropercenttotenpercentinonepercentincrements.（Roadgradeisdefinedtobetheratioofthetotalriseofahilldividedbythelengthoftheroad.Ifthehillisviewedasatriangle,thegradeisthesineoftheangleatthebottomofthehill.）Ariderstartsatthebottomofthehillataspeedof45kph,andthedecelerationoftheriderisproportionaltotheroadgrade.Ariderwillloseabout8kphforafivepercentgradeover100meters.

∙Task2.Provideanexampleofhowthetablecouldbeusedforaspecifictimetrialcourse.

∙Task3.Determineifthetableisanadequatemeansfordecidingonthewheelconfigurationandofferothersuggestionsastohowtomakethisdecision.

2001MCMB:

EscapingaHurricane'sWrath（AnIllWind...）

EvacuatingthecoastofSouthCarolinaaheadofthepredictedlandfallofHurricaneFloydin1999ledtoamonumentaltrafficjam.TrafficslowedtoastandstillonInterstateI-26,whichistheprincipalroutegoinginlandfromCharlestontotherelativelysafehavenofColumbiainthecenterofthestate.Whatisnormallyaneasytwo-hourdrivetookupto18hourstocomplete.Manycarssimplyranoutofgasalongtheway.Fortunately,Floydturnednorthandsparedthestatethistime,butthepublicoutcryisforcingstateofficialstofindwaystoavoidarepeatofthistrafficnightmare.

TheprincipalproposalputforthtodealwiththisproblemisthereversaloftrafficonI-26,sothatbothsides,includingthecoastal-boundlanes,havetrafficheadedinlandfromCharlestontoColumbia.Planstocarrythisouthavebeenprepared（andpostedontheWeb）bytheSouthCarolinaEmergencyPreparednessDivision.TrafficreversalonprincipalroadsleadinginlandfromMyrtleBeachandHiltonHeadisalsoplanned.

AsimplifiedmapofSouthCarolinaisshown.Charlestonhasapproximately500,000people,MyrtleBeachhasabout200,000people,andanother250,000peoplearespreadoutalongtherestofthecoastalstrip.（Moreaccuratedata,ifsought,arewidelyavailable.）

Theinterstateshavetwolanesoftrafficineachdirectionexceptinthemetropolitanareaswheretheyhavethree.Columbia,anothermetroareaofaround500,000people,doesnothavesufficienthotelspacetoaccommodatetheevacuees（includingsomecomingfromfarthernorthbyotherroutes）,sosometrafficcontinuesoutboundonI-26towardsSpartanburg;onI-77northtoCharlotte;andonI-20easttoAtlanta.In1999,trafficleavingColumbiagoingnorthwestwasmovingonlyveryslowly.Constructamodelfortheproblemtoinvestigatewhatstrategiesmayreducethecongestionobservedin1999.Herearethequestionsthatneedtobeaddressed:

1.Underwhatconditionsdoestheplanforturningthetwocoastal-boundlanesofI-26intotwolanesofColumbia-boundtraffic,essentiallyturningtheentireI-26intoone-waytraffic,significantlyimproveevacuationtrafficflow?

2.In1999,thesimultaneousevacuationofthestate'sentirecoastalregionwasordered.Wouldtheevacuationtrafficflowimproveunderanalternativestrategythatstaggerstheevacuation,perhapscounty-by-countyoversometimeperiodconsistentwiththepatternofhowhurricanesaffectthecoast?

3.SeveralsmallerhighwaysbesidesI-26extendinlandfromthecoast.Underwhatconditionswoulditimproveevacuationflowtoturnaroundtrafficonthese?

4.WhateffectwouldithaveonevacuationflowtoestablishmoretemporarysheltersinColumbia,toreducethetrafficleavingColumbia?

5.In1999,manyfamiliesleavingthecoastbroughtalongtheirboats,campers,andmotorhomes.Manydrovealloftheircars.Underwhatconditionsshouldthereberestrictionsonvehicletypesornumbersofvehiclesbroughtinordertoguaranteetimelyevacuation?

6.Ithasbeensuggestedthatin1999someofthecoastalresidentsofGeorgiaandFlorida,whowerefleeingtheearlierpredictedlandfallsofHurricaneFloydtothesouth,cameupI-95andcompoundedthetrafficproblems.Howbiganimpactcantheyhaveontheevacuationtrafficflow?

Clearlyidentifywhatmeasuresofperformanceareusedtocomparestrategies.Required:

Prepareashortnewspaperarticle,nottoexceedtwopages,explainingtheresultsandconclusionsofyourstudytothepublic.

Clearlyidentifywhatmeasuresofperformanceareusedtocomparestrategies.

Required:

Prepareashortnewspaperarticle,nottoexceedtwopages,explainingtheresultsandconclusionsofyourstudytothepublic.

2002MCMA:

WindandWaterspray

Anornamentalfountaininalargeopenplazasurroundedbybuildingssquirtswaterhighintotheair.Ongustydays,thewindblowssprayfromthefountainontopassersby.Thewater-flowfromthefountainiscontrolledbyamechanismlinkedtoananemometer（whichmeasureswindspeedanddirection）locatedontopofanadjacentbuilding.Theobjectiveofthiscontrolistoprovidepassersbywithanacceptablebalancebetweenanattractivespectacleandasoaking:

Theharderthewindblows,thelowerthewatervolumeandheighttowhichthewaterissquirted,hencethelesssprayfallsoutsidethepoolarea.

Yourtaskistodeviseanalgorithmwhichusesdataprovidedbytheanemometertoadjustthewater-flowfromthefountainasthewindconditionschange.

2002MCMB:

AirlineOverbooking

You'reallpackedandreadytogoonatriptovisityourbestfriendinNewYorkCity.Afteryoucheckinattheticketcounter,theairlineclerkannouncesthatyourflighthasbeenoverbooked.Passengersneedtocheckinimmediatelytodetermineiftheystillhaveaseat.

Historically,airlinesknowthatonlyacertainpercentageofpassengerswhohavemadereservationsonaparticularflightwillactuallytakethatflight.Consequently,mostairlinesoverbook-thatis,theytakemorereservationsthanthecapacityoftheaircraft.Occasionally,morepassengerswillwanttotakeaflightthanthecapacityoftheplaneleadingtooneormorepassengersbeingbumpedandthusunabletotaketheflightforwhichtheyhadreservations.

Airlinesdealwithbumpedpassengersinvariousways.Somearegivennothing,somearebookedonlaterflightsonotherairlines,andsomearegivensomekindofcashorairlineticketincentive.

Considertheoverbookingissueinlightofthecurrentsituation:

LessflightsbyairlinesfrompointAtopointBHeightenedsecurityatandaroundairportsPassengers'fearLossofbillionsofdollarsinrevenuebyairlinestodate

Buildamathematicalmodelthatexaminestheeffectsthatdifferentoverbookingschemeshaveontherevenuereceivedbyanairlinecompanyinordertofindanoptimaloverbookingstrategy,i.e.,thenumberofpeoplebywhichanairlineshouldoverbookaparticularflightsothatthecompany'srevenueismaximized.Insurethatyourmodelreflectstheissuesabove,andconsideralternativesforhandling“bumped”passengers.Additionally,writeashortmemorandumtotheairline'sCEOsummarizingyourfindingsandanalysis.

2003MCMA:

TheStuntPerson

Anexcitingactionsceneinamovieisgoingtobefilmed,andyouarethestuntcoordinator!

Astuntpersononamotorcyclewilljumpoveranelephantandlandinapileofcardboardboxestocushiontheirfall.Youneedtoprotectthestuntperson,andalsouserelativelyfewcardboardboxes（lowercost,notseenbycamera,etc.）.

Yourjobisto:

∙determinewhatsizeboxestouse

∙determinehowmanyboxestouse

∙determinehowtheboxeswillbestacked

∙determineifanymodificationstotheboxeswouldhelp

∙generalizetodifferentcombinedweights（stuntperson&motorcycle）anddifferentjumpheights

Notethat,in“TomorrowNeverDies”,theJamesBondcharacteronamotorcyclejumpsoverahelicopter.

2003MCMB:

GammaKnifeTreatmentPlanning

Stereotacticradiosurgerydeliversasinglehighdoseofionizingradiationtoaradiographicallywell-defined,smallintracranial3Dbraintumorwithoutdeliveringanysignificantfractionoftheprescribeddosetothesurroundingbraintissue.Threemodalitiesarecommonlyusedinthisarea;theyarethegammaknifeunit,heavychargedparticlebeams,andexternalhigh-energyphotonbeamsfromlinearaccelerators.

Thegammaknifeunitdeliversasinglehighdoseofionizingradiationemanatingfrom201cobalt-60unitsourcesthroughaheavyhelmet.All201beamssimultaneouslyintersectattheisocenter,resultingin