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∙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
BillFoxandRichWest
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
Thereareseveralconstraintsonfrequencyassignments.First,notwotransmitterswithindistanceofeachothercanbegiventhesamechannel.Second,duetospectralspreading,transmitterswithindistance2sofeachothermustnotbegiventhesameoradjacentchannels:
Theirchannelsmustdifferbyatleast2.Undertheseconstraints,whatcanwesayaboutthespanin,
RepeatRequirementA,assumingthegridintheexamplespreadsarbitrarilyfarinalldirections.
RequirementC:
RepeatRequirementsAandB,exceptassumenowmoregenerallythatchannelsfortransmitterswithindistancedifferbyatleastsomegivenintegerk,whilethoseatdistanceatmostmuststilldifferbyatleastone.Whatcanwesayaboutthespanandaboutefficientstrategiesfordesigningassignments,asafunctionofk?
RequirementD:
Considergeneralizationsoftheproblem,suchasseverallevelsofinterferenceorirregul