美国数学建模辅导课

一、美国数学建模竞赛介绍

MCM:MathematicalContestinModelingICM:InterdisciplinaryContestinModeling主办：美国运筹学和管理科学协会、工业及应用数学协会、美国数学协会以及国家安全局赞助，美国数学及应用协会承办时间（美国东部）：每年二月份的第二个周末，中国时间：周四晚上8：00开始，周一晚上8：00结束

题目：MCM竞赛一般有两个问题，都是根据各行各业的专家建议精选出来的实际问题。

问题的答案不是唯一的，完全是开放的，

主要考察参赛者处理问题的巧妙性，灵活性，新颖性。ICM只有一道题，涵盖数学、环境科学、环境工程及资源管理等概念的建模问题。评审标准及评审程序四个等级：Outstanding(特等奖），Meritorious(一等奖)，HonorableMention(二等奖)，SuccessfulParticipation(成功参赛奖).三轮审稿Initialround：至少两人审稿重点看summary：文字通顺，语言精练、准确；包括解题的方法和结论；摘要和全文结论要一致。不合格者被淘汰。Screeninground：由第二组人审稿

重点检查是否满足进入下一轮的最低要求：文字通顺，语法正确仍然是必须的；文章前后的一致性更加重要；清楚地说明哪些是自己做的，哪些是引用的，引用资源的出处要明确。Finalrounds：由第三组人审稿，每篇文章多人阅读，每次审阅时间延长；按照加分元素筛选进入下一轮的参赛队，被留下时间越长，得最高奖的可能性越高。

重点除了summary以外，全篇的一致性更加重要，每一解题部分都要经过审查。可以加分的元素1、文字优美，流畅，语言丰富，表达清晰，方程、标点引用格式正确。2、文摘的要求：包括问题的简单概括，建模使用的模型和方法，所得的结论。3、合理的假设以及必要的解释。4、多个模型由浅入深地对问题求解5、对模型使用的方法条件及结论进行分析6、模型的敏感度和稳定度分析7、当可引用的资源很多时，要对你引用的资源进行分析8、理论分析和数字模拟相结合，对于理论结果要有数字模拟加以说明9、图表的合理使用，让阅卷人对图表的解释和讨论一目了然10、模型的优势、弱势分析二、论文的组成部分*Summary/Abstract1、简单的背景介绍2、建立和使用的模型由浅入深地介绍；求解模型的方法，包括解析解、数值解或者图解。3、主要的结论和建议*Overview/Analysisoftheproblem主要是对问题的理解做一解释，对问题的解答建立在哪些前提和背景下*Assumptions要对假设做些合理性的解释*Designofthemodels1、Effectsof……2、Determinationof……3、Equationsof……*Sensitivityandstabilityanalysis1、Sensitivityoftheequation2、Stabilityofthemodeltotheparameter*Conclusions总结建立模型的条件、方法；使用模型的注意事项；模型推广的可能性，模型存在的问题。*Strengthsandweaknesses1、参数不是任意给出的，而是用….方法得到的2、模型运算时间很短，具有使用性*References顺序与文章中引用的顺序要一致三、论文目录举例Example1Contents

Introduction................................................................22.ThePlan........................................................................2Objectives....................................................................34.Definingthe“SweetSpot”........................................45.ModelA-CenterofPercussion:Physics’SweetSpot.......55-b.ExampleI.......................................................................56.ModelB:BrodyPowerModel..........................................66-a.FactsandAssumptions..................................................66-b.GeneralForm.................................................................76-c.ExampleII........................................................................96-d.Torque.............................................................................97.Discussion..........................................................................107-a.“Corking”........................................................……….107-b.CorkModelAugmentation..........................................117-c.Aluminumvs.Wood.....................................................137-d.AluminumModelAugmentation.................................138.SensitivityAnalysis.............................................................159.Conclusion..........................................................................1910.References........................................................................22Example2ContentsAbstract.............................................................................22.Statementoftheproblemandapproach.............……….52.1SurveyofPreviousResearch:EnvironmentalCriminology....................................................................62.2Assumptions.................................................................62.3PropositionsandFoundation.....................................8Methods........................................................................113.1ConstructionoftheMap–GeographicalMethod....113.2StaticandDynamic–RiskIntensityMethod..........18SimulationResultsandDiscussion................................304.1ResultsoftheGeographicalMethod...........................314.2ResultsoftheRiskIntensity,“Static”and“Dynamic”Method............................................................................364.3Discussion................................................................374.3.1SensitivityandRobustnessTesting.....................374.3.2AccuracyofthePrediction.................................404.3.3CombinationoftheTwoMethods.....................40StrengthandWeaknessoftheModel...........................416.ConclusionandRecommendation..............................41Appendices..........................................43ABibliography........................................43BData.......................................................45CCode......................................................46

Example3Contents1Introduction………..………12TheModels…………………32.1ASimplifiedModel...................………....32.2AnIntermediateModel...................……...32.3ACongestionModel..................…...…….42.4ExtendingtheModelUsingComputerSimulation....42.4.1SimulationAssumptions........................52.4.2Limitations....................……….....73AnalyzingtheModels……..………….83.1TheSimplestModel........................83.2IntermediateModel........................83.3CongestionModel.........................93.4SimulationResults........................124Conclusion…………...15四、摘要举例Abstract1Theadventofcomputerandtechnologicalprogresshasintroducedanewstageinthedevelopmentofcriminology.Investigatorscannowusecomputationaltechniquesofgeographicprofilinginordertodeterminethepatternsofmovementoftheirsuspects.

Weproposeamodelthataimstopredictareaswithhighprobabilityofbeingthenextonthecriminal‘stargetlist.Wehaveassumedthatserialcrimesareinstrumentalratherthanexpressive,thusensuringthatthecriminalfollowsapredictablepatternofmovement.

Wealsoassumethatthispatternischaracterizedbyacertainstabilityandcontinuitywhichfacilitatesacorrelationwiththeactionsofothercriminalsinthatarea.Ourmodelfirstusesaninitial“geographicalmethod”whichreducestheareasunderconsiderationbasedonparameterssuchaslocationcoordinates,area,populationandcriminalrate,aswellasthehistoryandpsychologicalvalueThisinputisusedtodeterminetheshapeofaGaussian2Dfunctionshowingthedistributionoftheareaswiththehighestprobabilityofbecomingthelocationofafuturecrime.Weimprovetheseresultsbyusingtheriskintensitymethod,acombinationoftwoschemes,a“static”anda“dynamic”.Thestaticmethodconsistsoffirstgeneratingtheriskintensitiesofdifferentlocationsbasedonvariablessuchascrimeratesanddistancesfromtheanchorpoint,byusingtoolssuchasthedistance-decayfunction.Wethenassigncrimecoefficients,whichindicatetheextenttowhichthecrimecanbecategorizedasmurder,rape,arsonorrobbery.Inthedynamicmodel,wecategorizethestaticparametersintohomotypic,heterotypicandcumulativetypesbycomputingthemeanandcovariancematrixoftheseparameters.Weapplydifferentalgorithms:logisticregression,linearregressionandnearestneighboralgorithmrespectivelytothesetypesandthenweighthemdifferentlytoobtainaparameterprobability.Thisisthencombinedwiththeresultsofthestaticprocesstogeneratetheprobabilityofacrimeatacertainlocation.Wetestedourmodelusingexamplesfromdifferentcategoriesofserialcrimes:robbery,murder,arson,whichdemonstrateddistinctcriminalpatterns.Thesurfacesgeneratedusingthegeographicmethodandthefinalpredictedprobabilitiesgenerallyagreedwithourexpectationsofareaswherethecriminalwillattackagain.Thetestforsensitivitysuggestedthatparameterssuchascrimerateorpopulationdensity(areaandlocation)arewelltakenintoconsiderationbyourmodel.Smallchangesinlocation,however,affectedtoasignificantextentourresults,probablybecausethedifferencesincoordinatesofthelocationswerenotlargetobeginwith.Summary2Ourgoalwastodesignamodelthatcouldaccountforthedynamicsofvehiclesinatrafficcircle.Wemainlyfocusedontherateofentryintothecircletodeterminethebestwaytoregulatetraffic.Weassumedthatvehiclescirculateinasinglelaneandthatonlyincomingtrafficcanberegulated.Forourmodel,theadjustableparametersweretherateofentryintothequeue,therateofentryintothecircle(servicerate),themaximumcapacityofthetrafficcircle,andtherateofdeparturefromthecircle(departurerate).Vehiclesfirstenterthequeuefromtheoutsideworld,thenenterthetrafficcirclefromthequeue,andlastlyexitthetrafficcircletotheoutsideworld.Wemodeledboththeservicerateandthedeparturerateasdependentonthenumberofvehiclesinsidethetrafficcircle.Inaddition,werancomputersimulationstohaveavisualrepresentationofwhathappensintrafficcirclesduringdifferentsituations.Inmanycases,wefoundthatafastserviceratewastheoptimalwaytomaintaintrafficflow.However,whenthecirclebecamemoreHeavilytrafficked,aslowerserviceratebetteraccommodatedthetraffic,indicatingthatatrafficlightshouldbeused.withvariabletimingdependingontheexpectedamountoftraffic.Themainadvantageofourapproachwasthatthemodelwasverysimpleandallowedustoclearlyseethedynamicsofthesystem.Also,thecomputersimulationsthatweranprovidedmorein-depthinformationabouttrafficflowunderconditionsthatthemodelcouldnoteasilyshowvisualobservationofthetraffic.Somedisadvantagestoourapproachwerethatwecouldnotanalyzetheaffectsofmultiplelanesnorstoplightsthatcontrolledtheflowoftrafficinsidethecircle.Inaddition,wehadnowayofanalyzingsingularitiesinthesituation,suchasvehiclesthatdrivefasterorslowerthantherestofthetrafficcircleandpedestrians.Summary3Ourbasicmodelcanbedividedintotwoparts.Thefirstpartistofindahalf-pipeshapewhichcanmaximize“VerticalAir”;inthesecondpart,weadapttheshapetomaximizethepossibletotalangleofrotation.Intheextendedmodel,weanalyzethesnowboarder’ssubjectiveinfluenceon“VerticalAir”andthetotaldegreeofrotation.Finally,wediscussthefeasibilityandthetrade-offofbuildingapracticalcourse.Inthefirstpartofourbasicmodel,weobtainadifferentialequationofElostbasedonforceanalysisandEnergyConservation.ThenwederivetherepresentationofElostbysolvingtheequation,whichisafunctional.Inthesecondpartofthebasicmodel,wederivetherepresentationoftheinitialangularmomentumbeforethefly,anddiscussthefactorsinfluencingit.Inourextendedmodel,wecalculatethe“VerticalAir”whentakingthesnowboarder’ssubjectiveinfluenceintoaccount.Weusenumericalmethodtosolvethemodelandcompareanalyticalresultswithrealityandvalidateourmethodtobecorrectandrobust.Weanalyzedtheeffectsoffactorssuchaswidth,heightandgradientangleofhalf-pipeon“verticalAir”bycontrollingotherfactors.UsingGeneAlgorithm,wegloballyoptimizedthecourseshapetoprovidelargest“verticalair”andtotaldegreeofrotation.Implementingahybridscoringsystemastheobjectivefunction,weoptimizethecourseshapetoa“half-blood”shape.Yet,themodelhasn'tprovidedanalyticsolutionofoptimalcourse.五、写作练习1、本文是