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

2003MCMProblems

PROBLEMA:

TheStuntPerson

Anexcitingactionsceneinamovieisgoingtobefilmed,andyouarethestuntcoordinator!

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

Yourjobisto:

∙determinewhatsizeboxestouse

∙determinehowmanyboxestouse

∙determinehowtheboxeswillbestacked

∙determineifanymodificationstotheboxeswouldhelp

∙generalizetodifferentbinedweights（stuntperson&motorcycle）anddifferentjumpheights

Notethat,in"TomorrowNeverDies",theJamesBondcharacteronamotorcyclejumpsoverahelicopter.

PROBLEMB:

GammaKnifeTreatmentPlanning

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

Thegammaknifeunitdeliversasinglehighdoseofionizingradiationemanatingfrom201cobalt-60unitsourcesthroughaheavyhelmet.All201beamssimultaneouslyintersectattheisocenter,resultinginaspherical（approximately）dosedistributionattheeffectivedoselevels.Irradiatingtheisocentertodeliverdoseistermeda“shot.”Shotscanberepresentedasdifferentspheres.Fourinterchangeableoutercollimatorhelmetswithbeamchanneldiametersof4,8,14,and18mmareavailableforirradiatingdifferentsizevolumes.Foratargetvolumelargerthanoneshot,multipleshotscanbeusedtocovertheentiretarget.Inpractice,mosttargetvolumesaretreatedwith1to15shots.Thetargetvolumeisabounded,three-dimensionaldigitalimagethatusuallyconsistsofmillionsofpoints.

Thegoalofradiosurgeryistodepletetumorcellswhilepreservingnormalstructures.Sincetherearephysicallimitationsandbiologicaluncertaintiesinvolvedinthistherapyprocess,atreatmentplanneedstoaccountforallthoselimitationsanduncertainties.Ingeneral,anoptimaltreatmentplanisdesignedtomeetthefollowingrequirements.

1.Minimizethedosegradientacrossthetargetvolume.

2.Matchspecifiedisodosecontourstothetargetvolumes.

3.Matchspecifieddose-volumeconstraintsofthetargetandcriticalorgan.

4.Minimizetheintegraldosetotheentirevolumeofnormaltissuesororgans.

5.Constraindosetospecifiednormaltissuepointsbelowtolerancedoses.

6.Minimizethemaximumdosetocriticalvolumes.

Ingammaunittreatmentplanning,wehavethefollowingconstraints:

1.Prohibitshotsfromprotrudingoutsidethetarget.

2.Prohibitshotsfromoverlapping（toavoidhotspots）.

3.Coverthetargetvolumewitheffectivedosageasmuchaspossible.Butatleast90%ofthetargetvolumemustbecoveredbyshots.

4.Useasfewshotsaspossible.

Yourtasksaretoformulatetheoptimaltreatmentplanningforagammaknifeunitasasphere-packingproblem,andproposeanalgorithmtofindasolution.Whiledesigningyouralgorithm,youmustkeepinmindthatyouralgorithmmustbereasonablyefficient.

2002ContestProblems

ProblemA

Authors:

TjallingYpma

Title:

WindandWaterspray

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

Theharderthewindblows,thelowerthewatervolumeandheighttowhichthewaterissquirted,hencethelesssprayfallsoutsidethepoolarea.

Yourtaskistodeviseanalgorithmwhichusesdataprovidedbytheanemometertoadjustthewater-flowfromthefountainasthewindconditionschange.

ProblemB

Authors:

BillFoxandRichWest

Title:

AirlineOverbooking

You'reallpackedandreadytogoonatriptovisityourbestfriendinNewYorkCity.Afteryoucheckinattheticketcounter,theairlineclerkannouncesthatyourflighthasbeenoverbooked.Passengersneedtocheckinimmediatelytodetermineiftheystillhaveaseat.

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

Airlinesdealwithbumpedpassengersinvariousways.Somearegivennothing,somearebookedonlaterflightsonotherairlines,andsomearegivensomekindofcashorairlineticketincentive.

Considertheoverbookingissueinlightofthecurrentsituation:

LessflightsbyairlinesfrompointAtopointB

Heightenedsecurityatandaroundairports

Passengers'fear

Lossofbillionsofdollarsinrevenuebyairlinestodate

Buildamathematicalmodelthatexaminestheeffectsthatdifferentoverbookingschemeshaveontherevenuereceivedbyanairlinepanyinordertofindanoptimaloverbookingstrategy,i.e.,thenumberofpeoplebywhichanairlineshouldoverbookaparticularflightsothatthepany'srevenueismaximized.Insurethatyourmodelreflectstheissuesabove,andconsideralternativesforhandling"bumped"passengers.Additionally,writeashortmemorandumtotheairline'sCEOsummarizingyourfindingsandanalysis.

MCM2000

ProblemAAirtrafficControl

Toimprovesafetyandreduceairtrafficcontrollerworkload,theFederalAviationAgency（FAA）isconsideringaddingsoftwaretotheairtrafficcontrolsystemthatwouldautomaticallydetectpotentialaircraftflightpathconflictsandalertthecontroller.Tothatend,ananalystattheFAArtrafficcontrolsystemthatwouldautomaticallydetectpotentialaircraftflightpathconflictsandalertthecontroller.Tothatend,ananalystattheFAAhasposedthefollowingproblems

RequirementA:

Giventwoairplanesflyinginspace,whenshouldtheairtrafficcontrollerldtheairtrafficcontrollerconsidertheobjectstobetoocloseandtorequireintervention?

RequirementB:

Anairspacesectoristhesectionofthree-dimensionalairspacethatoneairtrafficcontrollercontrols.Givenanyairspacesector,howwemeasurehowplexitisfromanairtrafficworkloadperspective?

Towhatextentisplexitydeterminedbythenumberofwemeasurehowplexitisfromanairtrafficworkloadperspective?

Towhatextentisplexitydeterminedbythenumberofaircraftsimultaneouslypassingthroughthatsector

（1）atanyoneinstant?

（2）Duringanygivenintervaloftime?

（3）Duringparticulartimeofday?

Howdoesthenumberofpotentialconflictsarisingduringthoseperiodsaffectplexity?

Doesthepresenceofadditionalsoftwaretoolstoautomaticallypredictconflictsandalertthecontrollerreduceoraddtothisplexity?

Inadditiontotheguidelinesforyourreport,writeasummary（nomorethantwopages）thattheFAAanalystcanpresenttoJaneGarvey,theFAAAdministrator,todefendyourconclusions

ProblemBRadioChannelAssignments

Weseektomodeltheassignmentofradiochannelstoasymmetricnetworkoftransmitterlocationsoveralargeplanararea,soastoavoidinterference.Onebasicapproachistopartitiontheregionintoregularhexagonsinagrid（honeyb-style）,asshowninFigure1,whereatransmitterislocatedatthecenterofeachhexagon.

Anintervalofthefrequencyspectrumistobeallottedfortransmitterfrequencies.Theintervalwillbedividedintoregularlyspacedchannels,whichwerepresentbyintegers1,2,3,....Eachtransmitterwillbeassignedonepositiveintegerchannel.Thesamechannelcanbeusedatmanylocations,providedthatinterferencefromnearbytransmittersisavoided.Ourgoalistominimizethewidthoftheintervalinthefrequencyspectrumthatisneededtoassignchannelssubjecttosomeconstraints.Thisisachievedwiththeconceptofaspan.Thespanistheminimum,overallassignmentssatisfyingtheconstraints,ofthelargestchannelusedatanylocation.Itisnotrequiredthateverychannelsmallerthanthespanbeusedinanassignmentthatattainsthespan.

Letsbethelengthofasideofoneofthehexagons.Weconcentrateonthecasethattherearetwolevelsofinterference

RequirementA:

Thereareseveralconstraintsonfrequencyassignments.First,notwotransmitterswithindistanceofeachothercanbegiventhesamechannel.Second,duetospectralspreading,transmitterswithindistance2sofeachothermustnotbegiventhesameoradjacentchannels:

Theirchannelsmustdifferbyatleast2.Undertheseconstraints,whatcanwesayaboutthespanin,

RequirementB:

RepeatRequirementA,assumingthegridintheexamplespreadsarbitrarilyfarinalldirections.

RequirementC:

RepeatRequirementsAandB,exceptassumenowmoregenerallythatchannelsfortransmitterswithindistancedifferbyatleastsomegivenintegerk,whilethoseatdistanceatmostmuststilldifferbyatleastone.Whatcanwesayaboutthespanandaboutefficientstrategiesfordesigningassignments,asafunctionofk?

RequirementD:

Considergeneralizationsoftheproblem,suchasseverallevelsofinterferenceorirregul