竞赛赛前讲座材料_数学建模讲座 slides.ppt

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1、数模数模小组的任务分工小组的任务分工三个人侧重点不同三个人侧重点不同:建模：推导数学模型，数学能力建模：推导数学模型，数学能力强强编程：计算机能力强编程：计算机能力强论文写作：写作能力强论文写作：写作能力强竞赛时间的安排竞赛时间的安排第一天：第一天：上午：确定题目，并查阅文献上午：确定题目，并查阅文献下午：开始分析，建立初步模型下午：开始分析，建立初步模型晚上：编程，得到初步计算结果晚上：编程，得到初步计算结果12：00PM休息休息第二天：第二天：上午：得到第一个模型的合理结果上午：得到第一个模型的合理结果下午：开始写论文，并考虑对第一个模型下午：开始写论文，并考虑对第一个模型的改进的改进竞赛

2、时间的安排竞赛时间的安排第二天：第二天：晚上：得到第二个模型的初步结果晚上：得到第二个模型的初步结果12：00PM休息休息第三天：第三天：上午：得到第二个模型的合理结果上午：得到第二个模型的合理结果下午：考虑对前二个模型的进一步优化，下午：考虑对前二个模型的进一步优化，得到第三个数学模型，或对前二个模型的得到第三个数学模型，或对前二个模型的正确性进行验证正确性进行验证晚上：得到最后结果，完成整篇论文晚上：得到最后结果，完成整篇论文论文格式规范论文格式规范论文（答卷）用白色论文（答卷）用白色A4纸，上下左右各留纸，上下左右各留出出2.5cm的页边距的页边距第一页为保证书，具体格式按要求第一页为保

3、证书，具体格式按要求第二页为空白页，用于论文编号第二页为空白页，用于论文编号论文题目和摘要写在第三页上论文题目和摘要写在第三页上第四页开始是论文正文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，从于每页页脚中部，从“1”开始连续编号开始连续编号论文不能有页眉，不能有任何可能显示答论文不能有页眉，不能有任何可能显示答案人身份的标志案人身份的标志论文格式规范论文格式规范论文题目用论文题目用3号黑体字、一级标题用号黑体字、一级标题用4号黑号黑体字，并居中。论文中其他汉字一律采用体字，并居中。论文中其他汉字一律采用小小4号宋体字，行距用号宋

4、体字，行距用1.5倍行距倍行距注意：摘要在整篇论文中评阅中占有重要注意：摘要在整篇论文中评阅中占有重要权重，请认真书写摘要权重，请认真书写摘要引用别人的成果或其他公开的资料（包括引用别人的成果或其他公开的资料（包括网上查到的资料）必须按照规定的参考文网上查到的资料）必须按照规定的参考文献的表述方式在正文引用处和参考文献中献的表述方式在正文引用处和参考文献中均明确列出。正文引用处用均明确列出。正文引用处用“”标出，标出，如如13等。等。论文格式规范论文格式规范参考文献按正文中的引用次序列出，其中参考文献按正文中的引用次序列出，其中书籍的表述方式为：书籍的表述方式为：编号编号作者，书名，出版地：出

5、版社，出作者，书名，出版地：出版社，出版年版年期刊杂志论文的表述方式为：期刊杂志论文的表述方式为：编号编号作者，论文名，杂志名，卷期号：作者，论文名，杂志名，卷期号：起止页码，出版年起止页码，出版年网上资源的表述方式为：网上资源的表述方式为：编号编号作者，资源标题，网址，访问时间作者，资源标题，网址，访问时间（年月日）（年月日）论文学作及评卷标准论文学作及评卷标准论文组成部分论文组成部分:1.摘要摘要2.问题重述问题重述3.假设假设4.建模建模5.求解求解6.讨论优缺点讨论优缺点7.模型改进模型改进论文评卷标准论文评卷标准1.假设的合理性假设的合理性2.建模的创造性建模的创造性3.结果的正确性

6、结果的正确性4.文字清晰程度文字清晰程度一定要写好。主要写三个方面：一定要写好。主要写三个方面：1.解决什么问题（一句话）解决什么问题（一句话）2.采取什么方法（引起阅卷老师的采取什么方法（引起阅卷老师的注意，不能太粗，也不能太细）注意，不能太粗，也不能太细）3.得到什么结果（简明扼要、生动、得到什么结果（简明扼要、生动、公式要简单、必要时可采用小图公式要简单、必要时可采用小图表）表）（一）摘（一）摘 要要正正文文10页左右，公式推导放在附录中页左右，公式推导放在附录中将原问题用数学的语言表达出来将原问题用数学的语言表达出来重点解决的问题应着重说明，把重点解决的问题应着重说明，把阅卷老师引导到

7、自己的思路中，阅卷老师引导到自己的思路中，把他们看成不懂本问题的读者。把他们看成不懂本问题的读者。（二）问题重述（二）问题重述最关键的一步从假设开始。需要下最关键的一步从假设开始。需要下很大功夫，简明扼要、准确清楚很大功夫，简明扼要、准确清楚1）假设太多，阅卷老师记不住。要归）假设太多，阅卷老师记不住。要归结出一些重要的假设，一般结出一些重要的假设，一般35条，条，有些不是很重要的假设在论文适当有些不是很重要的假设在论文适当的地方提一下的地方提一下2）假设要数学化，重视逻辑性要求）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）设计好符号，使人看起来清楚（三）假（三）假 设设说明建

8、模的思路说明建模的思路有些简单的事情往往是最重要的东西，有些简单的事情往往是最重要的东西，一定要说清楚一定要说清楚刚刚开始的原始想法，很重要刚刚开始的原始想法，很重要推导时，公式若很长，可放在附录中推导时，公式若很长，可放在附录中一般要求设计一般要求设计23个模型（一个简单个模型（一个简单的、再对模型进行改进，得到第二个的、再对模型进行改进，得到第二个模型，就会生动）模型，就会生动）（四）建（四）建模模（1）模型的定性）模型的定性线性或非线性线性或非线性连续、离散或混合连续、离散或混合时变或非时变时变或非时变（2）模型求解）模型求解利用现成的软件利用现成的软件自己解出来，实际意义更清楚自己解出

9、来，实际意义更清楚（五）（五）模型求解模型求解（六）模型优缺点及改进（六）模型优缺点及改进提出一些新的思路，使问题更提出一些新的思路，使问题更精确、也使模型得到进一步优精确、也使模型得到进一步优化。化。敢于讨论的学生，成绩会好。敢于讨论的学生，成绩会好。举例说明举例说明 设设某某生物种群在其适应的生物种群在其适应的环境下生存，试预测该种群环境下生存，试预测该种群的数量。的数量。一般解法一般解法记记N(t)为为t 时刻该种群的数量，设时刻该种群的数量，设该种群自然增长率为该种群自然增长率为，则，则即即假定初试时刻种群数量为假定初试时刻种群数量为N0，则有则有于是：于是：模型假设模型假设假设该环境

10、下只有一种生物群体，或假设该环境下只有一种生物群体，或者其它生物群体不影响此生物群体的者其它生物群体不影响此生物群体的生成生成假定该种群的自然增长率与时刻假定该种群的自然增长率与时刻t和和时刻时刻t时该时该种群的数量无关，记为种群的数量无关，记为因为种群数量很大，故可设种群个体因为种群数量很大，故可设种群个体N(t)是时间的连续可微函数是时间的连续可微函数假定初始时刻，种群的数量为假定初始时刻，种群的数量为N0实验验证实验验证 美国战后美国战后1010年的人口增长年的人口增长 不同时间段的世界人口增长不同时间段的世界人口增长 进一步改进进一步改进 （1 1）变参数）变参数 （2 2）多种群共存

11、）多种群共存 （3 3）随机模型）随机模型飓风疏散问题建模飓风疏散问题建模StrategiesforEscapingaHurricanesWrath2001年美国竞赛年美国竞赛B题题AMonumentalTrafficJamin1999TrafficslowedtoastandstillonInterstateI-26,whichistheprincipalroutegoinginlandfromCharlestontotherelativelysafehavenofColumbiainthecenterofthestate.Whatisnormallyaneasytwo-hourdrivet

12、ookupto18hourstocomplete.Manycarssimplyranoutofgasalongtheway.TrafficleavingColumbiagoingnorthwestwasmovingonlyveryslowly.ReversaloftrafficonI-26,sothatbothsides,includingthecoastal-boundlanes,havetrafficheadedinlandfromCharlestontoColumbia.TrafficreversalonprincipalroadsleadinginlandfromMyrtleBeach

13、andHiltonHeadisalsoplanned.ThePrincipalProposalCharlestonhasapproximately500,000peopleMyrtleBeachhasabout200,000people,andanother250,000peoplearespreadoutalongtherestofthecoastalstripColumbia,anothermetroareaofaround500,000peoplePeopleinDifferentCitiesTheinterstateshavetwolanesoftrafficineachdirecti

14、onexceptinthemetropolitanareaswheretheyhavethree.Columbiadoesnothavesufficienthotelspacetoaccommodatetheevacuees(includingsomecomingfromfarthernorthbyotherroutes),sosometrafficcontinuesoutboundonI-26towardsSpartanburg;onI-77northtoCharlotte;andonI-20easttoAtlanta.OthersFactorsThequestionsthatneedtob

15、eaddressed:1.Underwhatconditionsdoestheplanforturningthetwocoastal-boundlanesofI-26intotwolanesofColumbia-boundtraffic,essentiallyturningtheentireI-26intoone-waytraffic,significantlyimproveevacuationtrafficflow?ConstructaModel2.In1999,thesimultaneousevacuationofthestatesentirecoastalregionwasordered

16、.Wouldtheevacuationtrafficflowimproveunderanalternativestrategythatstaggerstheevacuation,perhapscounty-by-countyoversometimeperiodconsistentwiththepatternofhowhurricanesaffectthecoast?ConstructaModel3.SeveralsmallerhighwaysbesidesI-26extendinlandfromthecoast.Underwhatconditionswoulditimproveevacuati

17、onflowtoturnaroundtrafficonthese?4.WhateffectwouldithaveonevacuationflowtoestablishmoretemporarysheltersinColumbia,toreducethetrafficleavingColumbia?ConstructaModel5.In1999,manyfamiliesleavingthecoastbroughtalongtheirboats,campers,andmotorhomes.Manydrovealloftheircars.Underwhatconditionsshouldthereb

18、erestrictionsonvehicletypesornumbersofvehiclesbroughtinordertoguaranteetimelyevacuation?ConstructaModel6.Ithasbeensuggestedthatin1999someofthecoastalresidentsofGeorgiaandFlorida,whowerefleeingtheearlierpredictedlandfallsofHurricaneFloydtothesouth,cameupI-95andcompoundedthetrafficproblems.Howbiganimp

19、actcantheyhaveontheevacuationtrafficflow?ConstructaModelRequiresClearlyidentifywhatmeasuresofperformanceareusedtocomparestrategies.Prepareashortnewspaperarticle,nottoexceedtwopages,explainingtheresultsandconclusionsofyourstudytothepublic.(1)Theinterstates(e.g.,I-26,I-77,I-20)havetwolanesoftrafficine

20、achdirectionexceptinthemetropolitanareaswheretheyhavethree,whichmeansthetrafficenteringametropolitan(suchasColumbia)canbesmoothlyseparateintotheroutesleavingitinmostcase.(2)ThenormaldrivetakestwohourstocompletefromCharlestontoColumbia.TheaveragedrivespeedforacarontheinterstateI-26,I-77,andI-20islimi

21、tedto60-80mph,whiletheoneforacaronprincipalroads(e.g.,US501)islimitedto50-70mph.Assumptions(3)Charlestonhasapproximately500,000people,MyrtleBeachhasabout200,000people,andanother250,000peoplearewell-distributedalongtherestofthecoastalstrip.Also,weshouldconsidersomeevacueesfromthecoastlineofGeorgiaand

22、FloridainQuestion6.Assumptions(4)Columbiahasaround500,000people.And,accordingtoourroughsearchviaInternet,thetotalnumberofthehotelsandmotelsinColumbiaandbeyondColumbia,SC,isabout184,whichcanoccupyaround50,000people.Hence,Columbiadoesnothavesufficienthotelspacetoaccommodatetheevacuees(includingpeoplec

23、omingfromfarthernorthbyotherroutes).So,sometrafficmustcontinueoutboundonI-26towardsSpartanburg;onI-77northtoCharlotte;andonI-20westtoAtlanta.Assumptions(5)ThereareseveraltropicalcycloneguidancemodelsavailabletotheNHCandCPHCforecastersforthehurricaneseason.Thesemodelsrangeincomplexityfromsimplestatis

24、ticalmodelstothree-dimensionalprimitiveequationmodels.Accordingtotheirmodels,weassumethattheNationalHurricaneCenter(NHC)inMiami,Floridacanissue72hourtropicalcyclonetrackandintensityforecasts.Also,hurricanesarepossibleinthespecifiedareaoftheWATCH,usuallywithin36hours,andareexpectedinthespecifiedareao

25、ftheWARNING,usuallywithin24hours.Assumptions(6)EvacueestravelingonI-26inthenormalwestboundlanesoftravelwillbeallowedtoexitatallinterchangesbetweenCharlestonandColumbia.Evacueestravelinginthereversedlanesoftravelwillbeallowedtoexitatmost,butnotall,interchanges.Assumptions10109 96 65 54 43 32 21 11111

26、T TS Sx x1 1x x2 2x x3 3x x4 4x x5 5x x7 7x x6 6x x8 8x x9 9x x1010 x x1111x x1212x x1414x x1515x x13138 87 7x x1616SSupersourceSSupersource1(1(V V1 1)HiltonHead)HiltonHead2(2(V V2 2)Charleston)Charleston3(3(V V3 3)MyrtleBeach)MyrtleBeach4(4(V V4 4)Thecrossoverpoint)ThecrossoverpointbetweenI-95andI-

27、26betweenI-95andI-265(5(V V5 5)Thecrossoverpoint)ThecrossoverpointbetweenI-95andI-20betweenI-95andI-206(6(V V6 6)Columbia)Columbia7(7(V V7 7)HotelsinColumbia)HotelsinColumbia8(8(V V8 8)TemporarysheltersinColumbia)TemporarysheltersinColumbia9(9(V V9 9)Atlanta)Atlanta10(10(V V1010)Spartanburg)Spartanb

28、urg11(11(V V1111)Charlotte)CharlotteTSupersinkTSupersinkFig.1.Thetrafficnetworkofthemainroutesofevacuation.TheMaximumFlowProblem TheobjectivefunctionforQuestions1to6exceptQuestion2:Theconstraintconditions:TheMaximumFlowProblem TheobjectivefunctionforQuestions2:Theconstraintconditions:EvacueesfromSou

29、rces(HiltonHead,Charleston,andMyrtleBeach)Weshouldfirstdealwiththe250,000peoplewelldistributingalongtherestofthecoastalstrip,anddividethemintothreesources.Thenwemustconsidertwocasesinouralgorithms:(1)Non-considerationofthesomeofthecoastalresidentsofGeorgiaandFloridacameupI-95;(2)Considerationofsomeo

30、fthecoastalresidentsofGeorgiaandFloridacameupI-95.TheFamousGoldenSectionMethod=500,000+125,000=625,000=200,000+I-95:I-26:US501:(case2)TheImpactofTemporarySheltersTheflowofenteringColumbiaandthatofstayingatandoutgoingColumbiashouldbeequalto:-I-26westboundtraffictoColumbia-I-20westboundtraffictoColumb

31、ia-I-26westboundtrafficleavingColumbia-I-20westboundtrafficleavingColumbia-I-77northboundtrafficleavingColumbia-InhotelsandmotelsinColumbia-InestablishedtemporarysheltersinColumbiaThePerformanceMeasureofTrafficFlow ThetrafficflowtobeanimportantperformancemeasureFlow=SimulationResultsandModelTesting

32、FivedifferentstrategiesaccordingtoQuestions1to5consideringtwocases:(1)Non-considerationofthesomeofthecoastalresidentsofGeorgiaandFloridacameupI-95;(2)ConsiderationofsomeofthecoastalresidentsofGeorgiaandFloridacameupI-95,whichisQuestion6.Q:Underwhatconditionsdoestheplanforturningthetwocoastal-boundla

33、nesofI-26intotwolanesofColumbia-boundtraffic,essentiallyturningtheentireI-26intoone-waytraffic,significantlyimproveevacuationtrafficflow?A:StrategyIQ:SeveralsmallerhighwaysbesidesI-26extendinlandfromthecoast.Underwhatconditionswoulditimproveevacuationflowtoturnaroundtrafficonthese?A:StrategyIIStrate

34、gyIII Q:WhateffectwouldithaveonevacuationflowtoestablishmoretemporarysheltersinColumbia,toreducethetrafficleavingColumbia?StrategyIV Q:In1999,manyfamiliesleavingthecoastbroughtalongtheirboats,campers,andmotorhomes.Manydrovealloftheircars.Underwhatconditionsshouldthereberestrictionsonvehicletypesornu

35、mbersofvehiclesbroughtinordertoguaranteetimelyevacuation?Table1.TheStrategiescomparisonleavingthecoasttoColumbia.PerformancePerformancemeasuresoftrafficmeasuresoftrafficflow(cars/hour)flow(cars/hour)StrategyIStrategyIStrategyIIStrategyIIStrategyIIIStrategyIIIStrategyIVStrategyIVTheTheFlowonFlowonI-2

36、6I-26Case1Case14327.54327.54496.84496.85048.85048.85770.15770.1Case2Case23854.43854.43823.13823.15049.55049.55770.85770.8TheTheFlowonFlowonUS501US501toI-20toI-20Case1Case119201920224022402240224025602560Case2Case219201920224022402240224025602560TheTheFlowonFlowonI-95I-95Case1Case1330.6330.6356.5356.

37、5385.7385.7440.8440.8Case2Case2945.6945.6992.9992.91193.71193.71364.21364.2StrategyV Q:In1999,thesimultaneousevacuationofthestatesentirecoastalregionwasordered.Wouldtheevacuationtrafficflowimproveunderanalternativestrategythatstaggerstheevacuation,perhapscounty-by-countyoversometimeperiodconsistentw

38、iththepatternofhowhurricanesaffectthecoast?Table2.ThecomparisonofevacuatingtimesbetweenusingStrategyVcombinedwithStrategyItoIVandusingonlyStrategyItoIVStrategyStrategyI IIIIIIIIIIIIVIVCompletingCompletingsimultaneouslysimultaneouslyCase1Case1108.4571108.457172.216972.216966.964966.964961.898461.8984

39、Case2Case2121.7968121.796878.136678.136674.490574.490563.886263.8862EscapingEscapingcounty-by-county-by-countycountyCase1Case1105.4008105.400866.191666.191661.065761.065755.896255.8962Case2Case2112.3379112.337972.769072.769069.131969.131956.523356.5233SensitivityAnalysis Twoimportantimpliedfactorsto

40、affecttheperformanceoftheentirenetwork:(1)TheaccommodationcapacityinColumbia,and(2)TheevacueesfromGeorgiaandFloridacameupI-95.StrengthsandLimitationsDiscussionsandConclusions OthersFig.2.ThetrafficflowonI-26changingwiththetimeperiodofpredictedlandfallofhurricane,resultingindifferentregionsofthestrat

41、egies(ItoIV)tobecarriedout.Fig.3.Theminimumtotalrequiredtrafficflowchangingwiththetimeperiodofpredictedlandfallofhurricane,resultingindifferentregionsofthestrategies(ItoIV)tobecarriedout.Fig.4.Thecomparisonofmaximumtrafficflowbyevacuatingcounty-by-county(StrategyVcombinedwithStrategiesItoIV)andinasi

42、multaneousway(usingonlyStrategiesItoIV).Fig.5.TheinfluenceofevacueesaccommodatedinColumbiaonthetrafficflowonI-26,resultingindifferentregionsofthestrategies(ItoIII)tobecarriedout.Fig.6.TheimpactsofevacueesfromGeorgiaandFloridaonthetrafficflowonI-26,resultingindifferentregionsofthestrategies(ItoIV)tobecarriedout.Fig.7.TheimpactsofevacueesfromGeorgiaandFloridaonthetrafficflowonI-95,resultingindifferentregionsofthestrategies(ItoIV)tobecarriedout.Thanks!TheEnd