丁永生数学建模竞赛技巧.ppt

数学建模竞赛技巧,丁永生东华大学信息学院,近年来获奖情况,全国二等奖和上海赛区一等奖，2004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，2003年全国大学生数学模型竞赛上海赛区二等奖，2002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，2001年全国大学生数学模型竞赛上海赛区二等奖，2001年全国大学生数学模型竞赛上海赛区二等奖，2000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全国一等奖和上海赛区一等奖，1998年全国大学生数学模型竞赛,近年来获奖情况,国际一等奖，2004年美国大学生数学模型竞赛国际特等奖，2003年美国大学生数学模型竞赛国际一等奖，2001年美国大学生数学模型竞赛全国一等奖，2004年全国部分高校研究生数学建模竞赛上海市精品课程，数学建模与数学实验，2004上海市优秀教学成果二等奖，工科数学课程的计算机实验教学改革与实践，2004上海赛区优秀教练员，2004年全国大学生数学建模竞赛,近年来获奖情况,国际特等奖，2003年美国大学生数学模型竞赛,自1989年以来，获得国际特等奖的中国高校未超过10所,数模小组的任务分工,三个人侧重点不同: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,竞赛时间的安排,第一天：上午：确定题目，并查阅文献下午：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：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的改进,竞赛时间的安排,第二天：晚上：得到第二个模型的初步结果12：00PM休息第三天：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，得到第三个数学模型，或对前二个模型的正确性进行验证晚上：得到最后结果，完成整篇论文,2004论文格式规范,论文（答卷）用白色A4纸，上下左右各留出2.5cm的页边距第一页为保证书，具体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的标志,2004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。论文中其他汉字一律采用小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重要权重，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均明确列出。正文引用处用“”标出，如13等。,2004论文格式规范,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：编号作者，书名，出版地：出版社，出版年期刊杂志论文的表述方式为：编号作者，论文名，杂志名，卷期号：起止页码，出版年网上资源的表述方式为：编号作者，资源标题，网址，访问时间（年月日）,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,论文评卷标准,1.假设的合理性2.建模的创造性3.结果的正确性4.文字清晰程度,一定要写好。主要写三个方面：1.解决什么问题（一句话）2.采取什么方法（引起阅卷老师的注意，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、必要时可采用小图表）,（一）摘要,正文10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重点解决的问题应着重说明，把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。,（二）问题重述,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1）假设太多，阅卷老师记不住。要归结出一些重要的假设，一般35条，有些不是很重要的假设在论文适当的地方提一下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,说明建模的思路有些简单的事情往往是最重要的东西，一定要说清楚刚刚开始的原始想法，很重要推导时，公式若很长，可放在附录中一般要求设计23个模型（一个简单的、再对模型进行改进，得到第二个模型，就会生动）,（四）建模,（1）模型的定性线性或非线性连续、离散或混合时变或非时变（2）模型求解利用现成的软件自己解出来，实际意义更清楚,（五）模型求解,（六）模型优缺点及改进,提出一些新的思路，使问题更精确、也使模型得到进一步优化。敢于讨论的学生，成绩会好。,举例说明,设某生物种群在其适应的环境下生存，试预测该种群的数量。,一般解法,记N(t)为t时刻该种群的数量，设该种群自然增长率为，则即假定初试时刻种群数量为N0，则有于是：,模型假设,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生成假定该种群的自然增长率与时刻t和时刻t时该种群的数量无关，记为因为种群数量很大，故可设种群个体N(t)是时间的连续可微函数假定初始时刻，种群的数量为N0,实验验证,美国战后10年的人口增长不同时间段的世界人口增长进一步改进（1）变参数（2）多种群共存（3）随机模型,飓风疏散问题建模StrategiesforEscapingaHurricanesWrath,2001年美国竞赛B题,AMonumentalTrafficJamin1999,TrafficslowedtoastandstillonInterstateI-26,whichistheprincipalroutegoinginlandfromCharlestontotherelativelysafehavenofColumbiainthecenterofthestate.Whatisnormallyaneasytwo-hourdrivetookupto18hourstocomplete.Manycarssimplyranoutofgasalongtheway.TrafficleavingColumbiagoingnorthwestwasmovingonlyveryslowly.,ReversaloftrafficonI-26,sothatbothsides,includingthecoastal-boundlanes,havetrafficheadedinlandfromCharlestontoColumbia.TrafficreversalonprincipalroadsleadinginlandfromMyrtleBeachandHiltonHeadisalsoplanned.,ThePrincipalProposal,Charlestonhasapproximately500,000peopleMyrtleBeachhasabout200,000people,andanother250,000peoplearespreadoutalongtherestofthecoastalstripColumbia,anothermetroareaofaround500,000people,PeopleinDifferentCities,Theinterstateshavetwolanesoftrafficineachdirectionexceptinthemetropolitanareaswheretheyhavethree.Columbiadoesnothavesufficienthotelspacetoaccommodatetheevacuees(includingsomecomingfromfarthernorthbyotherroutes),sosometrafficcontinuesoutboundonI-26towardsSpartanburg;onI-77northtoCharlotte;andonI-20easttoAtlanta.,OthersFactors,Thequestionsthatneedtobeaddressed:1.Underwhatconditionsdoestheplanforturningthetwocoastal-boundlanesofI-26intotwolanesofColumbia-boundtraffic,essentiallyturningtheentireI-26intoone-waytraffic,significantlyimproveevacuationtrafficflow?,ConstructaModel,2.In1999,thesimultaneousevacuationofthestatesentirecoastalregionwasordered.Wouldtheevacuationtrafficflowimproveunderanalternativestrategythatstaggerstheevacuation,perhapscounty-by-countyoversometimeperiodconsistentwiththepatternofhowhurricanesaffectthecoast?,ConstructaModel,3.SeveralsmallerhighwaysbesidesI-26extendinlandfromthecoast.Underwhatconditionswoulditimproveevacuationflowtoturnaroundtrafficonthese?4.WhateffectwouldithaveonevacuationflowtoestablishmoretemporarysheltersinColumbia,toreducethetrafficleavingColumbia?,ConstructaModel,5.In1999,manyfamiliesleavingthecoastbroughtalongtheirboats,campers,andmotorhomes.Manydrovealloftheircars.Underwhatconditionsshouldthereberestrictionsonvehicletypesornumbersofvehiclesbroughtinordertoguaranteetimelyevacuation?,ConstructaModel,6.Ithasbeensuggestedthatin1999someofthecoastalresidentsofGeorgiaandFlorida,whowerefleeingtheearlierpredictedlandfallsofHurricaneFloydtothesouth,cameupI-95andcompoundedthetrafficproblems.Howbiganimpactcantheyhaveontheevacuationtrafficflow?,ConstructaModel,Requires,Clearlyidentifywhatmeasuresofperformanceareusedtocomparestrategies.Prepareashortnewspaperarticle,nottoexceedtwopages,explainingtheresultsandconclusionsofyourstudytothepublic.,(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.,Assumptions,(3)Charlestonhasapproximately500,000people,MyrtleBeachhasabout200,000people,andanother250,000peoplearewell-distributedalongtherestofthecoastalstrip.Also,weshouldconsidersomeevacueesfromthecoastlineofGeorgiaandFloridainQuestion6.,Assumptions,(4)Columbiahasaround500,000people.And,accordingtoourroughsearchviaInternet,thetotalnumberofthehotelsandmotelsinColumbiaandbeyondColumbia,SC,isabout184,whichcanoccupyaround50,000people.Hence,Columbiadoesnothavesufficienthotelspacetoaccommodatetheevacuees(includingpeoplecomingfromfarthernorthbyotherroutes).So,sometrafficmustcontinueoutboundonI-26towardsSpartanburg;onI-77northtoCharlotte;andonI-20westtoAtlanta.,Assumptions,(5)ThereareseveraltropicalcycloneguidancemodelsavailabletotheNHCandCPHCforecastersforthehurricaneseason.Thesemodelsrangeincomplexityfromsimplestatisticalmodelstothree-dimensionalprimitiveequationmodels.Accordingtotheirmodels,weassumethattheNationalHurricaneCenter(NHC)inMiami,Floridacanissue72hourtropicalcyclonetrackandintensityforecasts.Also,hurricanesarepossibleinthespecifiedareaoftheWATCH,usuallywithin36hours,andareexpectedinthespecifiedareaoftheWARNING,usuallywithin24hours.,Assumptions,(6)EvacueestravelingonI-26inthenormalwestboundlanesoftravelwillbeallowedtoexitatallinterchangesbetweenCharlestonandColumbia.Evacueestravelinginthereversedlanesoftravelwillbeallowedtoexitatmost,butnotall,interchanges.,Assumptions,Fig.1.Thetrafficnetworkofthemainroutesofevacuation.,TheMaximumFlowProblem,TheobjectivefunctionforQuestions1to6exceptQuestion2:,Theconstraintconditions:,TheMaximumFlowProblem,TheobjectivefunctionforQuestions2:,Theconstraintconditions:,EvacueesfromSources(HiltonHead,Charleston,andMyrtleBeach),Weshouldfirstdealwiththe250,000peoplewelldistributingalongtherestofthecoastalstrip,anddividethemintothreesources.Thenwemustconsidertwocasesinouralgorithms:(1)Non-considerationofthesomeofthecoastalresidentsofGeorgiaandFloridacameupI-95;(2)ConsiderationofsomeofthecoastalresidentsofGeorgiaandFloridacameupI-95.,TheFamousGoldenSectionMethod,=500,000+125,000=625,000,=200,000+,I-95:,I-26:,US501:,(case2),TheImpactofTemporaryShelters,TheflowofenteringColumbiaandthatofstayingatandoutgoingColumbiashouldbeequalto:,-I-26westboundtraffictoColumbia-I-20westboundtraffictoColumbia-I-26westboundtrafficleavingColumbia-I-20westboundtrafficleavingColumbia-I-77northboundtrafficleavingColumbia-InhotelsandmotelsinColumbia-InestablishedtemporarysheltersinColumbia,ThePerformanceMeasureofTrafficFlow,Thetrafficflowtobeanimportantperformancemeasure,Flow=,SimulationResultsandModelTesting,FivedifferentstrategiesaccordingtoQuestions1to5consideringtwocases:(1)Non-considerationofthesomeofthecoastalresidentsofGeorgiaandFloridacameupI-95;(2)ConsiderationofsomeofthecoastalresidentsofGeorgiaandFloridacameupI-95,whichisQuestion6.,Q:Underwhatconditionsdoestheplanforturningthetwocoastal-boundlanesofI-26intotwolanesofColumbia-boundtraffic,essentiallyturningtheentireI-26intoone-waytraffic,significantlyimproveevacuationtrafficflow?A:,StrategyI,Q:SeveralsmallerhighwaysbesidesI-26extendinlandfromthecoast.Underwhatconditionswoulditimproveevacuationflowtoturnaroundtrafficonthese?A:,StrategyII,StrategyIII,Q:WhateffectwouldithaveonevacuationflowtoestablishmoretemporarysheltersinColumbia,toreducethetrafficleavingColumbia?,StrategyIV,Q:In1999,manyfamiliesleavingthecoastbroughtalongtheirboats,campers,andmotorhomes.Manydrovealloftheircars.Underwhatconditionsshouldthereberestrictionsonvehicletypesornumbersofvehiclesbroughtinordertoguaranteetimelyevacuation?,Table1.TheStrategiescomparisonleavingthecoasttoColumbia.,StrategyV,Q:In1999,thesimultaneousevacuationofthestatesentirecoastalregionwasordered.Wouldtheevacuationtrafficflowimproveunderanalternativestrategythatstaggerstheevacuation,perhapscounty-by-countyoversometimeperiodconsistentwiththepatternofhowhurricanesaffectthecoast?,Table2.ThecomparisonofevacuatingtimesbetweenusingStrategyVcombinedwithStrategyItoIVandusingonlyStrategyItoIV,SensitivityAnalysis,Twoimportantimpliedfactorstoaffecttheperformanceoftheentirenetwork:TheaccommodationcapacityinColumbia,andTheevacueesfromGeorgiaandFloridacameupI-95.,StrengthsandLimitationsDiscussionsandConclusions,Others,Fig.2.ThetrafficflowonI-26changingwiththetimeperiodofpredictedlandfallofhurricane,resultingindifferentregionsofthestrategies(ItoIV)tobecarriedout.,Fig.3.Theminimumtotalrequiredtrafficflowchangingwiththetimeperiodofpredictedlandfallofhurricane,resultingindifferentregionsofthestrategies(ItoIV)tobecarriedout.,Fig.4.Thecomparisonofmaximumtrafficflowbyevacuatingcounty-by-county(StrategyVcombinedwithStrategiesItoIV)andinasimultaneousway(usingonlyStrategiesItoIV).,Fig.5.TheinfluenceofevacueesaccommodatedinColumbiaonthetrafficflowonI-26,resultingindifferentregionsofthestrategies(ItoIII)tobecarriedout.,Fig.6.TheimpactsofevacueesfromGeorgiaandFloridaonthetrafficflowonI-26,resultingindifferentregionsofthestrategies(ItoIV)tobecarriedout.,Fig.7.TheimpactsofevacueesfromGeorgiaandFloridaonthetrafficflowonI-95,resultingindifferentregionsofthestrategies(ItoIV)tobecarriedout.,Thanks!,TheEnd,