Notes for MCM Teams-美國大學生數(shù)學建模競賽

NotesforMCMTeams

ByMichaelTuite,DepartmentofMathematicalPhysics,NationalUniversityofIreland,Galway

Thesenotesformadistillationofourobservationsandexperiencesoveranumberofyears.Theyaredesignedtobeanaidtoteamsbothintheirpreparationforthecompetitionandduringtheweekenditself.

Introduction

TheLayeredApproachtoModelBuilding

WritingStyle

Timetable

FinalGeneralRemarksandAdvice

Introduction

ThesuccessfulparticipationofateamintheMathematicalContestinModelinginvolvesanumberofkeyareas:

Logicalandclearanalysisoftheproblem.

Abilitytostateandsolvethemathematicalproblemclearly.

Computingskillsinnumericalandstatisticalanalysis,graphplottingandwordprocessing.

Linguisticskillsinreportwriting.

Goodcommunicationbetweenteammembers.

Teammemberscanofferskillsinthesedifferentareasindifferentwayssorememberthateachcontributionshouldbevalued.

TheLayeredApproachtoModelBuilding

TheproblemsgivenintheMCMareusuallyrealworldandopen-endedandareoftenstatedinanon-mathematicalway.Theaimofanymathematicalmodelistorestatethegivenproblem(orperhapssomeapproximationtoit)inaclearwayasamathematicalproblembasedonsomesetofassumptions.Thesolutiontothisproblemshouldthenmakesomesensibleandusefulpredictionsaboutthesystem.ThereisnotnecessarilyauniquemodelormethodofsolutiontoMCMproblems.Sometimesagreatdealofinformationisgivensomeofwhichcanbesuperfluous.Ontheotherhand,sometimesrelevantinformationcanalsobemissing!Itisnotusuallynecessarytoemployverysophisticatedmathematics-verycomplicatedlookingformulasareprobablynotmuchbetterthansomeappropriatesimplerones.Certainly,youshouldnotresorttolifting"magicformulas"fromsometextbookatleastnotwithoutstatingthemeaningandassumptionsassociatedwithsuchformulas.

Yourmodelshouldformaconsistentapproachtotheproblem.Tobuildanymathematicalmodel,alayeredapproachisusuallythebestonetotake.Itisimportanttoclearlydividethestatementandconsequentsolutionoftheproblemintovariousstagesofincreasingdetailandcomplexity.Beginwithasimplemodelbasedonarestatementoftheprobleminsomesimplifiedformwherestrong,clearlydefinedandwellunderstoodassumptionsaremade.Don'tbecomeboggeddownintechnicalities.Manyofthefinerdetailsoftheoriginalproblemaresetasideatthisstageforlaterdeliberation.Thissimplemodelshouldbesolvedandproperlyunderstoodwithclear(andperhapsobvious!)predictions.Anunderstandingofthelimitationsofthissimplemodelhelpstoclarifythinkingsothattheareasforfurtherimprovementshouldbeapparent.Themodelshouldalsobeanalysedforthestabilityofitspredictionsanderroranalysisi.e.iftheassumptionsareslightlymodifiedortheinputteddata,howdothepredictionsalsovary?Themodelthenprovidesthebackboneforfuturelayersofmodels.Theseshouldincludeanincreasingamountofdetailwithmorerealisticassumptionsorpredictions.Itisimportanttobeclearatallstagesofthemodel'sdevelopmentoftheexactassumptions,approximationsandinformationbeinginputtedandtheconsequentpredictions.Themodel'sstabilityshouldateachstageagainbeanalysed.Inthisway,anevermorerealisticmathematicalmodelisbuiltuplayerbylayer.

WritingStyle

ThemodelandresultsthatareobtainedarepresentedinaSummarySheetandthemainReport.TheSummarySheetisadocumentconsistingofonepageonlyandcontainsasummaryofyourassumptions,modelandpredictions.Itisvitalthatthesummarypagebeveryclearlyandwellpresentedwithaself-consistentdescriptionofyourwork.

ThemainReportisnotsupposedtobeamathematicaltextandsoitisnotappropriatetoemployaformal"Theorem-Lemma"formatofpresentation.RathertheReportshouldbewritteninclearlanguageinanessaystylewithconcisesentences.TheReportshouldavoidunnecessarytechnicaljargonandshouldalsobeacomprehensibledocumentasawholewithoutunnecessaryreferencestotextbooksorpapers.

Theoveralldiscussionshouldbepresentedinthefirstpersonpluralandinthepresenttensei.e."..wefindthefollowingpolynomialfitsthegivendata".ThebasicheadingstobecoveredintheReportmightinclude:

Arestatementoftheproblem.

Aclearstatementofassumptionsandapproximationsmade.

Themathematicalmodelproposedanditspredictions(howitcanbetested).

Erroranalysisandstabilityundermodificationstoassumptionsandinputteddata.

Ananalysisofthestrengthsandweaknesses.

References,figuresandprograms.

Timetable

Friday

TODAY'SPRIORITYTASKS:Formulateandsolveasimplemodelforoneoftheproblems

12.01a.m.Theproblemsarehandedout.Apreliminarydiscussiononbothproblemsshouldtakeplace.PostthisTimetableinaprominentplaceandkeeptoit!

Morning.Theteamworkstogetheronbothproblems.Takethetime(notlessthantwohourseach)toconsidertheproblemsproperlyandformulatepossibleapproachestothem.Undertakealiteraturesearchinthelibrary.

3p.m.Adecisionshouldbemadebythistimeonwhichproblemistobetackled.

Night.Afirstsimplemodelmustbeformulatedandsolved.

Saturday

TODAY'SPRIORITYTASKS:DecideonaPlanforremainderoftheweekendandwriteadraftreportforthesimplemodel.

Morning.BeginthedaybydecidingonaPlanfortherunningandtimetablingoftheremainderoftheweekend.PostthisPlaninaprominentplaceandkeeptoit!ThePlanshouldincludethefollowingitems:

Thestatementandwritingupofthesimplemodel-adraftReportmustbecompletedbySaturdaynight.

Displaytheassumptionscontainedinthesimplemodelsothatthisisclearlyunderstoodbyallteammembers.

Decideonnextstage(s)ofthemodel.

Delegatevarioustaskstodifferentteammemberse.g.writingup,computerprogramming,furthermodelbuildingetc.

Timetableconferencetimeswhenteammemberscometogethertodiscussprogresseverytwotothreehours.

Night.FirstdraftoftheReportonthesimplemodelmustbewrittenup.Furtherstagesofthemodelhavebeendeveloped.

Sunday

TODAY'SPRIORITYTASKS:CompletetheReportonthesimplemodelandwriteSummarysheet.Writeupdraftofimprovementstosimplemodel.

Morning.Consolidateyourwork.ThedraftReportshouldbetidiedup.Furtherrefinementscanthenbeaddedbasedonthenextlayer(s)ofthemodel.

Evening.FirstsimplemodelReportcompleted.BeginwritingSummarySheet-theSummarysheetisverysignificantinthejudgingprocesssoitisworthdoingcarefully.

Night.Summarysheetcompleted.MakemodificationstotheReportbasedonimprovedmodel.

Monday

TODAY'SPRIORITYTASK:FinishbyLunchtime

ConcentrateexclusivelyoncompletingtheReportandSummarysheet.Ensurealldiagramsandprogramsareincludedandareproperlylabeled.PhotocopyReportandSummarysheet.

Lunchtime.Everythingcompletedandreadytobeposted.

FinalGeneralRemarksandAdvice

TherulesoftheMCMallowteamsanyinanimatehelp.Youmustnotseekorobtainadviceontheproblemfromanyoneelse.

Sticktotheproposedtimetable.Inparticular,eachdayhasprioritytask(s)-donotbecomeside-trackedawayfromthesepriorities.

Writingupthereportisaverytimeconsumingbusiness.Makingthisanearlyprioritytakesthepressureoffandhelpsconsiderablyinthelayeredbuildingofthemodel.Theprocessofexplainingandorderingyourargumentsinthesimplemodelwillhelpyoutocriticiseyoursimplemodelandhenceclarifyyourthinkingonthenextstagesofimprovement.Furthermore,onceadraftversionisdownonpaper(ordisk),itiseasytomakemodificationsasrequiredoraddmoresections.

Itispossiblethataconflictofopinionmayariseonsomeissue.Werecommendthatnomorethanhalfanhourshouldbespentindiscussiononanyoneissueafte