數(shù)學(xué)建模講座之十-數(shù)學(xué)建模競賽論文寫作省公開課一等獎全國示范課微課金獎?wù)n件

數(shù)學(xué)建模競賽論文寫作丁永生東南大學(xué)信息學(xué)院7/23/2024第1頁近年來獲獎情況年美國大學(xué)生數(shù)學(xué)模型競賽,國際特等獎年全國大學(xué)生數(shù)學(xué)模型競賽,上海賽區(qū)二等獎年美國大學(xué)生數(shù)學(xué)模型競賽,國際一等獎年全國大學(xué)生數(shù)學(xué)模型競賽,全國二等獎和上海賽區(qū)一等獎年全國大學(xué)生數(shù)學(xué)模型競賽,上海賽區(qū)二等獎年全國大學(xué)生數(shù)學(xué)模型競賽,上海賽區(qū)二等獎1999年全國大學(xué)生數(shù)學(xué)模型競賽,上海賽區(qū)二等獎1998年全國大學(xué)生數(shù)學(xué)模型競賽,全國一等獎和上海賽區(qū)一等獎7/23/2024第2頁數(shù)模小組任務(wù)分工三個人側(cè)重點不一樣:建模:推導(dǎo)數(shù)學(xué)模型，數(shù)學(xué)能力強編程:計算機能力強論文寫作:寫作能力強7/23/2024第3頁競賽時間安排第一天:早晨:確定題目，并查閱文件下午:開始分析，建立初步模型晚上:編程，得到初步計算結(jié)果12:00PM休息第二天:早晨:得到第一個模型合理結(jié)果下午:開始寫論文，并考慮對第一個模型改進7/23/2024第4頁競賽時間安排第二天:晚上:得到第二個模型初步結(jié)果12:00PM休息第三天:早晨:得到第二個模型合理結(jié)果下午:考慮對前二個模型深入優(yōu)化，得到第三個數(shù)學(xué)模型，或?qū)η岸€模型正確性進行驗證晚上:得到最終結(jié)果，完成整篇論文7/23/2024第5頁論文格式規(guī)范論文（答卷）用白色A4紙,上下左右各留出2.5cm頁邊距第一頁為確保書,詳細格式按要求第二頁為空白頁,用于論文編號論文題目和摘要寫在第三頁上第四頁開始是論文正文論文從第三頁開始編寫頁碼,頁碼必須位于每頁頁腳中部,從“1”開始連續(xù)編號論文不能有頁眉,不能有任何可能顯示答案人身份標志7/23/2024第6頁論文格式規(guī)范論文題目用3號黑體字、一級標題用4號黑體字，并居中。論文中其它漢字一律采取小4號宋體字，行距用1.5倍行距注意:摘要在整篇論文中評閱中占有主要權(quán)重，請認真書寫摘要引用他人結(jié)果或其它公開資料（包含網(wǎng)上查到資料）必須按照要求參考文件表述方式在正文引用處和參考文件中均明確列出。正文引用處用“[]”標出，如[1][3]等。7/23/2024第7頁論文格式規(guī)范參考文件按正文中引用次序列出，其中書籍表述方式為:[編號]作者，書名，出版地:出版社，出版年期刊雜志論文表述方式為:[編號]作者，論文名，雜志名，卷期號:起止頁碼，出版年網(wǎng)上資源表述方式為:[編號]作者，資源標題，網(wǎng)址，訪問時間（年月日）7/23/2024第8頁論文學(xué)作及評卷標準論文組成部分:1.摘要2.問題重述3.假設(shè)4.建模5.求解6.討論優(yōu)缺點7.模型改進7/23/2024第9頁論文評卷標準1.假設(shè)合理性2.建模創(chuàng)造性3.結(jié)果正確性4.文字清楚程度7/23/2024第10頁一定要寫好。主要寫三個方面:1.處理什么問題（一句話）2.采取什么方法（引發(fā)閱卷老師注意，不能太粗，也不能太細）3.得到什么結(jié)果（簡明扼要、生動、公式要簡單、必要時可采取小圖表）（一）摘要7/23/2024第11頁正文10頁左右,公式推導(dǎo)放在附錄中將原問題用數(shù)學(xué)語言表示出來重點處理問題應(yīng)著重說明,把閱卷老師引導(dǎo)到自己思緒中,把他們看成不懂本問題讀者。（二）問題重述7/23/2024第12頁最關(guān)鍵一步從假設(shè)開始。需要下很大功夫,簡明扼要、準確清楚1）假設(shè)太多,閱卷老師記不住。要歸結(jié)出一些主要假設(shè),普通3~5條,有些不是很主要假設(shè)在論文適當(dāng)?shù)胤教嵋幌?）假設(shè)要數(shù)學(xué)化,重視邏輯性要求3）設(shè)計好符號,使人看起來清楚（三）假設(shè)7/23/2024第13頁說明建模思緒有些簡單事情往往是最主要東西,一定要說清楚剛才開始原始想法,很主要推導(dǎo)時,公式若很長,可放在附錄中普通要求設(shè)計2~3個模型（一個簡單、再對模型進行改進,得到第二個模型,就會生動）（四）建模7/23/2024第14頁（1）模型定性線性或非線性連續(xù)、離散或混合時變或非時變（2）模型求解利用現(xiàn)成軟件自己解出來,實際意義更清楚（五）模型求解7/23/2024第15頁（六）模型優(yōu)缺點及改進提出一些新思緒,使問題更準確、也使模型得到深入優(yōu)化。勇于討論學(xué)生,成績會好。7/23/2024第16頁舉例說明設(shè)某生物種群在其適應(yīng)環(huán)境下生存,試預(yù)測該種群數(shù)量。7/23/2024第17頁普通解法記N(t)為t時刻該種群數(shù)量，設(shè)該種群自然增加率為，則即假定初試時刻種群數(shù)量為N0，則有于是:7/23/2024第18頁模型假設(shè)假設(shè)該環(huán)境下只有一個生物群體,或者其它生物群體不影響此生物群體生成假定該種群自然增加率與時刻t和時刻t時該種群數(shù)量無關(guān),記為因為種群數(shù)量很大,故可設(shè)種群個體N(t)是時間連續(xù)可微函數(shù)假定初始時刻,種群數(shù)量為N07/23/2024第19頁試驗驗證美國戰(zhàn)后10年人口增加不一樣時間段世界人口增加深入改進（1）變參數(shù)（2）各種群共存（3）隨機模型7/23/2024第20頁颶風(fēng)疏散問題建模StrategiesforEscapingaHurricane’sWrath

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

Charlestonhasapproximately500,000people,MyrtleBeachhasabout200,000people,andanother250,000peoplearewell-distributedalongtherestofthecoastalstrip.Also,weshouldconsidersomeevacueesfromthecoastlineofGeorgiaandFloridainQuestion6.Assumptions7/23/2024第34頁(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頁(5)

ThereareseveraltropicalcycloneguidancemodelsavailabletotheNHCandCPHCforecastersforthehurricaneseason.Thesemodelsrangeincomplexityfromsimplestatisticalmodelstothree-dimensionalprimitiveequationmodels.Accordingtotheirmodels,weassumethattheNationalHurricaneCenter(NHC)inMiami,Floridacanissue72hourtropicalcyclonetrackandintensityforecasts.Also,hurricanesarepossibleinthespecifiedareaoftheWATCH,usuallywithin36hours,andareexpectedinthespecifiedareaoftheWARNING,usuallywithin24hours.Assumptions7/23/2024第36頁(6)EvacueestravelingonI-26inthenormalwestboundlanesoftravelwillbeallowedtoexitatallinterchangesbetweenCharlestonandColumbia.Evacueestravelinginthereversedlanesoftravelwillbeallowedtoexitatmost,butnotall,interchanges.Assumptions7/23/2024第37頁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頁TheMaximumFlowProblem

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

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

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

GoldenSection

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

ThetrafficflowtobeanimportantperformancemeasureFlow=7/23/2024第44頁SimulationResultsandModelTesting

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

Q:WhateffectwouldithaveonevacuationflowtoestablishmoretemporarysheltersinColumbia,toreducethetrafficleavingColumbia?

7/23/2024第48頁StrategyIV

Q:In1999,manyfamiliesleavingthecoastbroughtalongtheirboats,campers,andmotorhomes.Manydrovealloftheircars.Underwhatconditionsshouldthereberestrictionsonvehicletypesornumbersofvehiclesbroughtinordertoguaranteetimelyevacuation?7/23/2024第49頁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頁StrategyV

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

Twoimportantimpliedfactorstoaffecttheperformanceoft