数学建模讲座之十-数学建模竞赛论文写作省公开课一等奖全国示范课微课金奖课件

数学建模竞赛论文写作丁永生东南大学信息学院7/23/2024第1页近年来获奖情况年美国大学生数学模型竞赛,国际特等奖年全国大学生数学模型竞赛,上海赛区二等奖年美国大学生数学模型竞赛,国际一等奖年全国大学生数学模型竞赛,全国二等奖和上海赛区一等奖年全国大学生数学模型竞赛,上海赛区二等奖年全国大学生数学模型竞赛,上海赛区二等奖1999年全国大学生数学模型竞赛,上海赛区二等奖1998年全国大学生数学模型竞赛,全国一等奖和上海赛区一等奖7/23/2024第2页数模小组任务分工三个人侧重点不一样:建模:推导数学模型，数学能力强编程:计算机能力强论文写作:写作能力强7/23/2024第3页竞赛时间安排第一天:早晨:确定题目，并查阅文件下午:开始分析，建立初步模型晚上:编程，得到初步计算结果12:00PM休息第二天:早晨:得到第一个模型合理结果下午:开始写论文，并考虑对第一个模型改进7/23/2024第4页竞赛时间安排第二天:晚上:得到第二个模型初步结果12:00PM休息第三天:早晨:得到第二个模型合理结果下午:考虑对前二个模型深入优化，得到第三个数学模型，或对前二个模型正确性进行验证晚上:得到最终结果，完成整篇论文7/23/2024第5页论文格式规范论文（答卷）用白色A4纸,上下左右各留出2.5cm页边距第一页为确保书,详细格式按要求第二页为空白页,用于论文编号论文题目和摘要写在第三页上第四页开始是论文正文论文从第三页开始编写页码,页码必须位于每页页脚中部,从“1”开始连续编号论文不能有页眉,不能有任何可能显示答案人身份标志7/23/2024第6页论文格式规范论文题目用3号黑体字、一级标题用4号黑体字，并居中。论文中其它汉字一律采取小4号宋体字，行距用1.5倍行距注意:摘要在整篇论文中评阅中占有主要权重，请认真书写摘要引用他人结果或其它公开资料（包含网上查到资料）必须按照要求参考文件表述方式在正文引用处和参考文件中均明确列出。正文引用处用“[]”标出，如[1][3]等。7/23/2024第7页论文格式规范参考文件按正文中引用次序列出，其中书籍表述方式为:[编号]作者，书名，出版地:出版社，出版年期刊杂志论文表述方式为:[编号]作者，论文名，杂志名，卷期号:起止页码，出版年网上资源表述方式为:[编号]作者，资源标题，网址，访问时间（年月日）7/23/2024第8页论文学作及评卷标准论文组成部分:1.摘要2.问题重述3.假设4.建模5.求解6.讨论优缺点7.模型改进7/23/2024第9页论文评卷标准1.假设合理性2.建模创造性3.结果正确性4.文字清楚程度7/23/2024第10页一定要写好。主要写三个方面:1.处理什么问题（一句话）2.采取什么方法（引发阅卷老师注意，不能太粗，也不能太细）3.得到什么结果（简明扼要、生动、公式要简单、必要时可采取小图表）（一）摘要7/23/2024第11页正文10页左右,公式推导放在附录中将原问题用数学语言表示出来重点处理问题应着重说明,把阅卷老师引导到自己思绪中,把他们看成不懂本问题读者。（二）问题重述7/23/2024第12页最关键一步从假设开始。需要下很大功夫,简明扼要、准确清楚1）假设太多,阅卷老师记不住。要归结出一些主要假设,普通3~5条,有些不是很主要假设在论文适当地方提一下2）假设要数学化,重视逻辑性要求3）设计好符号,使人看起来清楚（三）假设7/23/2024第13页说明建模思绪有些简单事情往往是最主要东西,一定要说清楚刚才开始原始想法,很主要推导时,公式若很长,可放在附录中普通要求设计2~3个模型（一个简单、再对模型进行改进,得到第二个模型,就会生动）（四）建模7/23/2024第14页（1）模型定性线性或非线性连续、离散或混合时变或非时变（2）模型求解利用现成软件自己解出来,实际意义更清楚（五）模型求解7/23/2024第15页（六）模型优缺点及改进提出一些新思绪,使问题更准确、也使模型得到深入优化。勇于讨论学生,成绩会好。7/23/2024第16页举例说明设某生物种群在其适应环境下生存,试预测该种群数量。7/23/2024第17页普通解法记N(t)为t时刻该种群数量，设该种群自然增加率为，则即假定初试时刻种群数量为N0，则有于是:7/23/2024第18页模型假设假设该环境下只有一个生物群体,或者其它生物群体不影响此生物群体生成假定该种群自然增加率与时刻t和时刻t时该种群数量无关,记为因为种群数量很大,故可设种群个体N(t)是时间连续可微函数假定初始时刻,种群数量为N07/23/2024第19页试验验证美国战后10年人口增加不一样时间段世界人口增加深入改进（1）变参数（2）各种群共存（3）随机模型7/23/2024第20页飓风疏散问题建模StrategiesforEscapingaHurricane’sWrath

年美国竞赛B题7/23/2024第21页7/23/2024第22页AMonumentalTrafficJamin1999TrafficslowedtoastandstillonInterstateI-26,whichistheprincipalroutegoinginlandfromCharlestontotherelativelysafehavenofColumbiainthecenterofthestate.Whatisnormallyaneasytwo-hourdrivetookupto18hourstocomplete.Manycarssimplyranoutofgasalongtheway.TrafficleavingColumbiagoingnorthwestwasmovingonlyveryslowly.7/23/2024第23页ReversaloftrafficonI-26,sothatbothsides,includingthecoastal-boundlanes,havetrafficheadedinlandfromCharlestontoColumbia.TrafficreversalonprincipalroadsleadinginlandfromMyrtleBeachandHiltonHeadisalsoplanned.ThePrincipalProposal7/23/2024第24页Charlestonhasapproximately500,000peopleMyrtleBeachhasabout200,000people,andanother250,000peoplearespreadoutalongtherestofthecoastalstripColumbia,anothermetroareaofaround500,000peoplePeopleinDifferentCities7/23/2024第25页Theinterstateshavetwolanesoftrafficineachdirectionexceptinthemetropolitanareaswheretheyhavethree.Columbiadoesnothavesufficienthotelspacetoaccommodatetheevacuees(includingsomecomingfromfarthernorthbyotherroutes),sosometrafficcontinuesoutboundonI-26towardsSpartanburg;onI-77northtoCharlotte;andonI-20easttoAtlanta.OthersFactors7/23/2024第26页Thequestionsthatneedtobeaddressed:1.Underwhatconditionsdoestheplanforturningthetwocoastal-boundlanesofI-26intotwolanesofColumbia-boundtraffic,essentiallyturningtheentireI-26intoone-waytraffic,significantlyimproveevacuationtrafficflow?ConstructaModel7/23/2024第27页2.In1999,thesimultaneousevacuationofthestate'sentirecoastalregionwasordered.Wouldtheevacuationtrafficflowimproveunderanalternativestrategythatstaggerstheevacuation,perhapscounty-by-countyoversometimeperiodconsistentwiththepatternofhowhurricanesaffectthecoast?ConstructaModel7/23/2024第28页3.SeveralsmallerhighwaysbesidesI-26extendinlandfromthecoast.Underwhatconditionswoulditimproveevacuationflowtoturnaroundtrafficonthese?4.WhateffectwouldithaveonevacuationflowtoestablishmoretemporarysheltersinColumbia,toreducethetrafficleavingColumbia?ConstructaModel7/23/2024第29页5.In1999,manyfamiliesleavingthecoastbroughtalongtheirboats,campers,andmotorhomes.Manydrovealloftheircars.Underwhatconditionsshouldthereberestrictionsonvehicletypesornumbersofvehiclesbroughtinordertoguaranteetimelyevacuation?ConstructaModel7/23/2024第30页6.Ithasbeensuggestedthatin1999someofthecoastalresidentsofGeorgiaandFlorida,whowerefleeingtheearlierpredictedlandfallsofHurricaneFloydtothesouth,cameupI-95andcompoundedthetrafficproblems.Howbiganimpactcantheyhaveontheevacuationtrafficflow?ConstructaModel7/23/2024第31页RequiresClearlyidentifywhatmeasuresofperformanceareusedtocomparestrategies.Prepareashortnewspaperarticle,nottoexceedtwopages,explainingtheresultsandconclusionsofyourstudytothepublic.7/23/2024第32页(1)

Theinterstates(e.g.,I-26,I-77,I-20)havetwolanesoftrafficineachdirectionexceptinthemetropolitanareaswheretheyhavethree,whichmeansthetrafficenteringametropolitan(suchasColumbia)canbesmoothlyseparateintotheroutesleavingitinmostcase.(2)

ThenormaldrivetakestwohourstocompletefromCharlestontoColumbia.TheaveragedrivespeedforacarontheinterstateI-26,I-77,andI-20islimitedto60-80mph,whiletheoneforacaronprincipalroads(e.g.,US501)islimitedto50-70mph.Assumptions7/23/2024第33页(3)

Charlestonhasapproximately500,000people,MyrtleBeachhasabout200,000people,andanother250,000peoplearewell-distributedalongtherestofthecoastalstrip.Also,weshouldconsidersomeevacueesfromthecoastlineofGeorgiaandFloridainQuestion6.Assumptions7/23/2024第34页(4)Columbiahasaround500,000people.And,accordingtoourroughsearchviaInternet,thetotalnumberofthehotelsandmotelsinColumbiaandbeyondColumbia,SC,isabout184,whichcanoccupyaround50,000people.Hence,Columbiadoesnothavesufficienthotelspacetoaccommodatetheevacuees(includingpeoplecomingfromfarthernorthbyotherroutes).So,sometrafficmustcontinueoutboundonI-26towardsSpartanburg;onI-77northtoCharlotte;andonI-20westtoAtlanta.Assumptions7/23/2024第35页(5)

ThereareseveraltropicalcycloneguidancemodelsavailabletotheNHCandCPHCforecastersforthehurricaneseason.Thesemodelsrangeincomplexityfromsimplestatisticalmodelstothree-dimensionalprimitiveequationmodels.Accordingtotheirmodels,weassumethattheNationalHurricaneCenter(NHC)inMiami,Floridacanissue72hourtropicalcyclonetrackandintensityforecasts.Also,hurricanesarepossibleinthespecifiedareaoftheWATCH,usuallywithin36hours,andareexpectedinthespecifiedareaoftheWARNING,usuallywithin24hours.Assumptions7/23/2024第36页(6)EvacueestravelingonI-26inthenormalwestboundlanesoftravelwillbeallowedtoexitatallinterchangesbetweenCharlestonandColumbia.Evacueestravelinginthereversedlanesoftravelwillbeallowedtoexitatmost,butnotall,interchanges.Assumptions7/23/2024第37页10965432111TSx1x2x3x4x5x7x6x8x9x10x11x12x14x15x1387x16S—Supersource1(V1)—HiltonHead2(V2)—Charleston3(V3)—MyrtleBeach4(V4)—ThecrossoverpointbetweenI-95andI-265(V5)—ThecrossoverpointbetweenI-95andI-206(V6)—Columbia7(V7)—HotelsinColumbia8(V8)—TemporarysheltersinColumbia9(V9)—Atlanta10(V10)—Spartanburg11(V11)—CharlotteT—SupersinkFig.1.Thetrafficnetworkofthemainroutesofevacuation.7/23/2024第38页TheMaximumFlowProblem

TheobjectivefunctionforQuestions1to6exceptQuestion2:Theconstraintconditions:7/23/2024第39页TheMaximumFlowProblem

TheobjectivefunctionforQuestions2:Theconstraintconditions:7/23/2024第40页EvacueesfromSources

(HiltonHead,Charleston,andMyrtleBeach)Weshouldfirstdealwiththe250,000peoplewelldistributingalongtherestofthecoastalstrip,anddividethemintothreesources.Thenwemustconsidertwocasesinouralgorithms:(1)Non-considerationofthesomeofthecoastalresidentsofGeorgiaandFloridacameupI-95;(2)ConsiderationofsomeofthecoastalresidentsofGeorgiaandFloridacameupI-95.7/23/2024第41页TheFamous

GoldenSection

Method=500,000+125,000=625,000=200,000+I-95:I-26:US501:(case2)7/23/2024第42页TheImpactofTemporarySheltersTheflowofenteringColumbiaandthatofstayingatandoutgoingColumbiashouldbeequalto:--I-26westboundtraffictoColumbia--I-20westboundtraffictoColumbia--I-26westboundtrafficleavingColumbia--I-20westboundtrafficleavingColumbia--I-77northboundtrafficleavingColumbia--InhotelsandmotelsinColumbia--InestablishedtemporarysheltersinColumbia7/23/2024第43页ThePerformanceMeasureofTrafficFlow

ThetrafficflowtobeanimportantperformancemeasureFlow=7/23/2024第44页SimulationResultsandModelTesting

FivedifferentstrategiesaccordingtoQuestions1to5consideringtwocases:(1)Non-considerationofthesomeofthecoastalresidentsofGeorgiaandFloridacameupI-95;(2)ConsiderationofsomeofthecoastalresidentsofGeorgiaandFloridacameupI-95,whichisQuestion6.7/23/2024第45页Q:Underwhatconditionsdoestheplanforturningthetwocoastal-boundlanesofI-26intotwolanesofColumbia-boundtraffic,essentiallyturningtheentireI-26intoone-waytraffic,significantlyimproveevacuationtrafficflow?A:StrategyI7/23/2024第46页Q:SeveralsmallerhighwaysbesidesI-26extendinlandfromthecoast.Underwhatconditionswoulditimproveevacuationflowtoturnaroundtrafficonthese?A:StrategyII7/23/2024第47页StrategyIII

Q:WhateffectwouldithaveonevacuationflowtoestablishmoretemporarysheltersinColumbia,toreducethetrafficleavingColumbia?

7/23/2024第48页StrategyIV

Q:In1999,manyfamiliesleavingthecoastbroughtalongtheirboats,campers,andmotorhomes.Manydrovealloftheircars.Underwhatconditionsshouldthereberestrictionsonvehicletypesornumbersofvehiclesbroughtinordertoguaranteetimelyevacuation?7/23/2024第49页Table1.TheStrategiescomparisonleavingthecoasttoColumbia.Performancemeasuresoftrafficflow(cars/hour)StrategyIStrategyIIStrategyIIIStrategyIVTheFlowonI-26Case14327.54496.85048.85770.1Case23854.43823.15049.55770.8TheFlowonUS501toI-20Case11920224022402560Case21920224022402560TheFlowonI-95Case1330.6356.5385.7440.8Case2945.6992.91193.71364.27/23/2024第50页StrategyV

Q:In1999,thesimultaneousevacuationofthestate'sentirecoastalregionwasordered.Wouldtheevacuationtrafficflowimproveunderanalternativestrategythatstaggerstheevacuation,perhapscounty-by-countyoversometimeperiodconsistentwiththepatternofhowhurricanesaffectthecoast?7/23/2024第51页Table2.ThecomparisonofevacuatingtimesbetweenusingStrategyVcombinedwithStrategyItoIVandusingonlyStrategyItoIVStrategyIIIIIIIVCompletingsimultaneouslyCase1108.457172.216966.964961.8984Case2121.796878.136674.490563.8862Escapingcounty-by-countyCase1105.400866.191661.065755.8962Case2112.337972.769069.131956.52337/23/2024第52页SensitivityAnalysis

Twoimportantimpliedfactorstoaffecttheperformanceoft