2009年美國(guó)數(shù)學(xué)建模競(jìng)賽mcm試題

PAGEPAGE12009MCMJudges’Commentary—ProblemBByMarieVanisko,CarrollCollege,Helena,MontanaGeneralRemarksAsinpastyears,thediversebackgroundsoftheundergraduateparticipantsyieldedmanyinterestingmodelingapproachestothestatedproblem.Thejudgesassessedthepa breadthanddepthofysisforallmajorissuesraised,onthevalidityofproposedmodels,andontheoverallclarityandpresentationofsolutions.TheExecutiveSummaryisoftenstillbelo r;theSummaryshouldmotivatethereadertoreadthepaper.Itmustnotmerelyrestatetheproblem,butindicatehowitwasmodeledandwhatconclusionswerereached,withoutbeingundulytechnical.Assumptionsmustbeclearlystatedandjustifiedwhereappropriate.Someteamsoverlookimportantfactorsand/ormakeunrealisticassumptionswithnorationale.Itshouldbemadeclearinthemodelpreciselywherethoseassumptionsareused.Graphsneedlabelsand/orlegends.Theyshouldprovideinformationaboutwhatisreferredtointhepaper.Tablesandgraphsthataretakenfromothersourcesneedtohavespecificreferences.Failuretousereliableresourcesandtoproperlythoseresourceskeptsomepapersfromrisingtothetop.Thebestpapersnotonlylisttrustworthyresourcesbutalsotheirusethroughoutthepaper.TheProblemandSelectedModelingProblemBinvolvedthe“energy”consequencesoftherevolutionandfiveRequirementsweredelineated.ToreceiveanOutstandingorMeritoriousdesignation,teamshadtoaddressissuesraisedineachoftheseRequirements.Additionally,outstandingpapersconsideredbothwirelessandwiredlandlinesandtheinfrastructuretosupportsandRequirementTeamswerefirstaskedtoestimatethenumberofUShouseholdsinthepastthatwereservedbylandlinesandalsotoestimatetheaveragesizeofthosehouseholds.Theywerethentoconsidertheenergyconsequencesintermsofelectricityutilizationofacompletetransitionfromlandlinephonestos,withtheunderstandingthateaemberofeachhouseholdwouldgeta.Inordertoaddressthisproblem,theenergyusedbycurrentlandlineshadtobeconsidered.Whereascordedlandlinephonesuserelativelylittleelectricity,thesamecannotbeaboutcordlesslandlinephones.Thetoppapersresearchedthisissueandarrivedatedestimatesofthenumberofcordedversuscordlesslandlinephonesandtheaveragenumberofeachperhousehold.Thisledtoamorerealisticappraisaloftheenergyusedbythelandlinephonesystem.Withregardtos,teamsthatrosetothetopconsideredtheinfrastructurenecessary,forexample,thebuildingofnumerousadditionalcommunicationtowersifcellphonesaretocompleyrecelandlinephones.Thiswasofspecialimportanceintheysisofthetransitionalphase.Also,thevaryingamountofelectricityrequiredbydifferenttypesofswasaconsiderationinthetransitionalandsteadystateInterestingmodelswereconstructedforthetransitionalphaseofthe“takeover.”SometeamsconsideredthespreadofsasthespreadofadiseaseandusedtheVerhulstmodelforlogisticgrowth,usingthepopulationoftheUnitedStatesasthecarryingcapacityandestimatingtherateofgrowthofsfrompublishedreportsonthegrowthofuseintheUnitedStates.OtherteamsgeneralizedthistoanSIRmodelorusedtheLotkaVolterrapredator‐preymodel,withsasthepredatorsandlandlinephonesastheprey.Afewusedthecompetingspeciesmodel.Thejudgeslookedveryfavorablyuponmodelsforwhichsufficientrationalewasgivenastowhythatmodelmightbeappropriateinthiscircumstance.Interpretationoftheparametersandsolutionsastheyappliedtotheproblemathandwasessential.Manypapersignoredthetransitionphaseandonlyconsideredtheenergycomparisonforthesteadystateinordertodeterminetheenergyconsequence.Someteamsmerelytalkedtheirwaythroughtheissuesanddidnotconstructamathematicalmodel.Afterestimatingenergycostsassociatedwithlandlinephonesands,manyteamsusedlinearequationstomodelthetotalcostsassociatedwiththenumbersofphones.RequirementTeamswereaskedtoconsidera“PseudoUS”—acountrysimilartothecurrentUSinpopulationandeconomicstatus,butwithneitherlandlinesors.Theyweretodeterminetheoptimalwaytoprovidephoneservicetothiscountryfromanenergy.Theteamswerealsototakeintoaccountthesocialadvantagesthatsofferandthebroadconsequencesofhavingonlylandlinesoronlys.Onceagain,considerationoftheinfrastructureforphoneswasimportant.Inadditiontolandlineands,manyteamsconsideredtheVoIP(VoiceoverInternetProtocol)communicationoption.NoteveryteamthatconsideredVoIPtookintoaccountthecostsforlayingthecables.Someassumedthatsuchcableswerealreadyin ce,aquestionableassumption,butfailuretoconsidertheVoIPmethodofphoneservicemayhavekeptaMeritoriouspaperfrom inganOutstandingpaper.Ifoneweretoassumethathouseholdswouldalreadyhaveoneormorecomputerswithinternetaccess,theenergycostsassociatedwithVoIPwouldbequitesmall.Intermsoffindingtheoptimalwaytoprovidephoneservicefromanenergy,someteamsusedlinearprogramming,utilizingthecostsdeterminedinthefirstrequirementandfyinginvariouswaysthesocialadvantagesofs,aswellasthepreferenceforlandlinesincertaincircumstances.OtherteamsusedAHP(ogHierarchyProcess),whichworkedwelltogetparametersusedintheoptimizationroutine,butdidnotworkasanoptimizationtoolinitself.Teamsthatconsideredtheadvantagesanddisadvantagesofvariousphonetypesnotjustforindividuals,butforbusinessesalso,demonstratedathoroughnessthatwascommendable.Anotherfactorthatsometeamsconsideredwasthenumberofchildrenunder5whowouldhavenoneedllphones.RequirementThiswasanextensionofRequirement2,askingteamstoconsidertheelectricitywastedwhensarepluggedinthatdonotneedchargingandwhenchargersareleftpluggedinafterthephoneisremoved.TeamsweretocontinuetoassumethattheywereinthePseudoUSandweretointerpretenergywastedintermsofbarrelsofoilused.Todeterminetheamountofenergywasted,teamshadtofirstestimatethenumberofhoursthata“typical”chargerisinthestateofchargingaphone,leftpluggedintoaphonenotinneedofcharging,andleftpluggedinwhenthephoneisnotconnectedtoit.Someteamsdidtheirowninformalsurveys,butbetterestimateswerearrivedatfrompublicationsandsurveys.Insomepapers,probabilitydistributionswereusedtodescribethisbehavior,butuseofsuchdistributionswasnotalwaysjustified.Teamsthatweremorecomprehensivetookintoaccountthefactthatsomesandchargersuselesspowerthantodoothers,basedonbrands,age,andcapabilitiesofthephonesandchargers.Assumingthatallelectricalenergyisgeneratedbyoil,translatingkilowattsofenergyintobarrelsofoilusedwasastraightforwardtransformation.Requirement tendedtheconceptsinRequirement3andaskedteamstoestimatetheamountofenergywastedbyallelectronicchargers.Sincethisquestionwasveryopen‐ended,contestpapersshowedawidevarietyofestimatesfortheenergywastedintermsofbarrelsofoil.Thetopteamsestimatedtheaveragehoursthatappliancesareleftpluggedinchargingandnotchargingandalsothenumberofhourschargersarepluggedinwithouttheappliance.Morecomprehensivepapersconsideredmanyelectronicsand,bycomparison,showedthattheamountofenergywastedbysisrelativelysmallcomparedtomanyotherelectronicdevices.RequirementForthispart,studentsweretoconsiderthepopulationandeconomicgrowthofthePseudoUSforthenext50yearsandpredictenergyneedsforprovidingphoneservicebasedontheir ysisinthefirstthreeRequirements.Predictionsweretobeinterpretedintermsofbarrelsofoilused.Papersneededtoconsiderbotheconomicgrowthandpopulationgrowthinordertoestimateenergyneedsinthefuture.VarioustypesofregressionfitswereappliedtohistoricalpopulationdataandeconomicdatasuchasGDP.Usingearlierestimatesofenergyrequirements,coupledwiththeregressionequationsfromhistoricaldata,predictionsweremadefortheamountofenergyusedeverydecadeforthenext50years.Someteamspredictedgreaterefficiencyinfuturephonesandthedevelopmentofchargersthatwoulduselesselectricity.Papersshowedestimatesforthenumberofbarrelsofoilusedonaperdaybasisorperhapsonaperyearbasis.Therewasnoonerightanswerandanswersgivendependedonassumptionsmadeatthestart.Somepaperscontainedgraphsdisyingfuturevaluesbutdidnotgivetables.Atablewithagraphisabetterwaytodisyinformation.ConcludingMathematicalmodelingisanartthatrequiresconsiderableskillandpracticeinordertodevelopproficiency.Thebigproblemswefacenowandinthefuturewillbesolvedinlargepartbythosewiththetalent,theinsight,andthewilltomodeltheserealworldproblemsandcontinuouslyrefinethosemodels.ThejudgesareveryproudofallparticipantsinthisMathematicalContestinModeling,andwecommendyouforyourhardworkanddedication.MCM:TheMathematicalContestinModeling(2009).?Copyright2009byCOMAP,Inc.s Permissiontomakedigitalorhardcopiesofpartorallofthisworkfor alorclassroomuseisgrantedwithoutfeeprovidedthatcopiesarenotmadeordistributedforprofitorcommercialadvantageandthatcopiesbearthisnotice. ingwithcreditispermitted,butcopyrightsforcomponentsofthisworkownedbyothersthanCOMAPmustbehonored.Tocopyotherwise,torepublish,topostonservers,ortoredistributetolistsrequirespriorpermissionfromCOMAP.2009年MCM評(píng)委評(píng)論—問(wèn)題B作者：蒙大拿州學(xué)院的MarieVanisko。備注對(duì)所所有主要問(wèn)題進(jìn)行分析的廣度和深度、所提出模型的有效性以及解決方案的整體清晰度 圖表需要和/或圖例。他們應(yīng)該提供有關(guān) 無(wú)法躋身。 問(wèn)題和選定的建模方法問(wèn)題B涉及 的“能源”，并描述了五個(gè)要求。要獲得杰出或 及支持和固定 首先要求團(tuán)隊(duì)估計(jì)過(guò)去使用固定服務(wù)的家庭數(shù)量以及這些家庭的平均規(guī)模。然后，他們要考慮從固定完全過(guò)渡到的用電方面對(duì)能源的影響，前提是每個(gè)家庭的每個(gè)成員都會(huì)擁有一部。為了解決這個(gè)問(wèn)題，必須考慮當(dāng)前固定使用的能源。盡管有繩固定耗電量相對(duì)較少，但無(wú)繩固定卻不能同樣如此。熱門研究了這個(gè)問(wèn)題，并得出了有線與無(wú)繩固定數(shù)量以及每戶平均數(shù)量的記錄估計(jì)。這使得對(duì)固定系統(tǒng)所使用的能源進(jìn)行了更現(xiàn)實(shí)的評(píng)估。在方面，上升到最的團(tuán)隊(duì)考慮了必要的基礎(chǔ)設(shè)施，例如，如果要完全取代固定，就需要建造大量額外的通信塔。這對(duì)于過(guò)渡階段的分析特別重要。此外，不同類型的所需的電為“接管”的過(guò)渡階段構(gòu)建了有趣的模型。一些團(tuán)隊(duì)將的視為疾病的，并使用Verhulst模型進(jìn)行邏輯增長(zhǎng)，以 。其他團(tuán)隊(duì)將其推廣到SIR模型或使用LotkaVolterra捕 作為獵物。一些使用了競(jìng)爭(zhēng)物種模型。 和相關(guān)的能源成本后，許多團(tuán)隊(duì)使用線性方程對(duì)與2：團(tuán)隊(duì)被要求考慮一個(gè)“偽”——一個(gè)人口和經(jīng)濟(jì)狀況與當(dāng)前相似的國(guó)家，但既沒(méi)有固定電話也沒(méi)有。他們將從能源角度確定向該國(guó)提供 服務(wù)的最佳方式。這些團(tuán)隊(duì)還需要考慮 或的廣泛。再次強(qiáng)調(diào)，考慮 和之外，許多團(tuán)隊(duì)還考慮了VoIP（互聯(lián)網(wǎng)協(xié)議