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

數(shù)學(xué)建模競(jìng)賽論文寫作丁永生東南大學(xué)信息學(xué)院7/23/2024第1頁(yè)近年來(lái)獲獎(jiǎng)情況年美國(guó)大學(xué)生數(shù)學(xué)模型競(jìng)賽,國(guó)際特等獎(jiǎng)年全國(guó)大學(xué)生數(shù)學(xué)模型競(jìng)賽,上海賽區(qū)二等獎(jiǎng)年美國(guó)大學(xué)生數(shù)學(xué)模型競(jìng)賽,國(guó)際一等獎(jiǎng)年全國(guó)大學(xué)生數(shù)學(xué)模型競(jìng)賽,全國(guó)二等獎(jiǎng)和上海賽區(qū)一等獎(jiǎng)年全國(guó)大學(xué)生數(shù)學(xué)模型競(jìng)賽,上海賽區(qū)二等獎(jiǎng)年全國(guó)大學(xué)生數(shù)學(xué)模型競(jìng)賽,上海賽區(qū)二等獎(jiǎng)1999年全國(guó)大學(xué)生數(shù)學(xué)模型競(jìng)賽,上海賽區(qū)二等獎(jiǎng)1998年全國(guó)大學(xué)生數(shù)學(xué)模型競(jìng)賽,全國(guó)一等獎(jiǎng)和上海賽區(qū)一等獎(jiǎng)7/23/2024第2頁(yè)數(shù)模小組任務(wù)分工三個(gè)人側(cè)重點(diǎn)不一樣:建模:推導(dǎo)數(shù)學(xué)模型，數(shù)學(xué)能力強(qiáng)編程:計(jì)算機(jī)能力強(qiáng)論文寫作:寫作能力強(qiáng)7/23/2024第3頁(yè)競(jìng)賽時(shí)間安排第一天:早晨:確定題目，并查閱文件下午:開(kāi)始分析，建立初步模型晚上:編程，得到初步計(jì)算結(jié)果12:00PM休息第二天:早晨:得到第一個(gè)模型合理結(jié)果下午:開(kāi)始寫論文，并考慮對(duì)第一個(gè)模型改進(jìn)7/23/2024第4頁(yè)競(jìng)賽時(shí)間安排第二天:晚上:得到第二個(gè)模型初步結(jié)果12:00PM休息第三天:早晨:得到第二個(gè)模型合理結(jié)果下午:考慮對(duì)前二個(gè)模型深入優(yōu)化，得到第三個(gè)數(shù)學(xué)模型，或?qū)η岸€(gè)模型正確性進(jìn)行驗(yàn)證晚上:得到最終結(jié)果，完成整篇論文7/23/2024第5頁(yè)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(yè)邊距第一頁(yè)為確保書,詳細(xì)格式按要求第二頁(yè)為空白頁(yè),用于論文編號(hào)論文題目和摘要寫在第三頁(yè)上第四頁(yè)開(kāi)始是論文正文論文從第三頁(yè)開(kāi)始編寫頁(yè)碼,頁(yè)碼必須位于每頁(yè)頁(yè)腳中部,從“1”開(kāi)始連續(xù)編號(hào)論文不能有頁(yè)眉,不能有任何可能顯示答案人身份標(biāo)志7/23/2024第6頁(yè)論文格式規(guī)范論文題目用3號(hào)黑體字、一級(jí)標(biāo)題用4號(hào)黑體字，并居中。論文中其它漢字一律采取小4號(hào)宋體字，行距用1.5倍行距注意:摘要在整篇論文中評(píng)閱中占有主要權(quán)重，請(qǐng)認(rèn)真書寫摘要引用他人結(jié)果或其它公開(kāi)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(biāo)出，如[1][3]等。7/23/2024第7頁(yè)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(hào)]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(hào)]作者，論文名，雜志名，卷期號(hào):起止頁(yè)碼，出版年網(wǎng)上資源表述方式為:[編號(hào)]作者，資源標(biāo)題，網(wǎng)址，訪問(wèn)時(shí)間（年月日）7/23/2024第8頁(yè)論文學(xué)作及評(píng)卷標(biāo)準(zhǔn)論文組成部分:1.摘要2.問(wèn)題重述3.假設(shè)4.建模5.求解6.討論優(yōu)缺點(diǎn)7.模型改進(jìn)7/23/2024第9頁(yè)論文評(píng)卷標(biāo)準(zhǔn)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(yè)一定要寫好。主要寫三個(gè)方面:1.處理什么問(wèn)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(xì)）3.得到什么結(jié)果（簡(jiǎn)明扼要、生動(dòng)、公式要簡(jiǎn)單、必要時(shí)可采取小圖表）（一）摘要7/23/2024第11頁(yè)正文10頁(yè)左右,公式推導(dǎo)放在附錄中將原問(wèn)題用數(shù)學(xué)語(yǔ)言表示出來(lái)重點(diǎn)處理問(wèn)題應(yīng)著重說(shuō)明,把閱卷老師引導(dǎo)到自己思緒中,把他們看成不懂本問(wèn)題讀者。（二）問(wèn)題重述7/23/2024第12頁(yè)最關(guān)鍵一步從假設(shè)開(kāi)始。需要下很大功夫,簡(jiǎn)明扼要、準(zhǔn)確清楚1）假設(shè)太多,閱卷老師記不住。要?dú)w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(jì)好符號(hào),使人看起來(lái)清楚（三）假設(shè)7/23/2024第13頁(yè)說(shuō)明建模思緒有些簡(jiǎn)單事情往往是最主要東西,一定要說(shuō)清楚剛才開(kāi)始原始想法,很主要推導(dǎo)時(shí),公式若很長(zhǎng),可放在附錄中普通要求設(shè)計(jì)2~3個(gè)模型（一個(gè)簡(jiǎn)單、再對(duì)模型進(jìn)行改進(jìn),得到第二個(gè)模型,就會(huì)生動(dòng)）（四）建模7/23/2024第14頁(yè)（1）模型定性線性或非線性連續(xù)、離散或混合時(shí)變或非時(shí)變（2）模型求解利用現(xiàn)成軟件自己解出來(lái),實(shí)際意義更清楚（五）模型求解7/23/2024第15頁(yè)（六）模型優(yōu)缺點(diǎn)及改進(jìn)提出一些新思緒,使問(wèn)題更準(zhǔn)確、也使模型得到深入優(yōu)化。勇于討論學(xué)生,成績(jī)會(huì)好。7/23/2024第16頁(yè)舉例說(shuō)明設(shè)某生物種群在其適應(yīng)環(huán)境下生存,試預(yù)測(cè)該種群數(shù)量。7/23/2024第17頁(yè)普通解法記N(t)為t時(shí)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(shí)刻種群數(shù)量為N0，則有于是:7/23/2024第18頁(yè)模型假設(shè)假設(shè)該環(huán)境下只有一個(gè)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(shí)刻t和時(shí)刻t時(shí)該種群數(shù)量無(wú)關(guān),記為因?yàn)榉N群數(shù)量很大,故可設(shè)種群個(gè)體N(t)是時(shí)間連續(xù)可微函數(shù)假定初始時(shí)刻,種群數(shù)量為N07/23/2024第19頁(yè)試驗(yàn)驗(yàn)證美國(guó)戰(zhàn)后10年人口增加不一樣時(shí)間段世界人口增加深入改進(jìn)（1）變參數(shù)（2）各種群共存（3）隨機(jī)模型7/23/2024第20頁(yè)颶風(fēng)疏散問(wèn)題建模StrategiesforEscapingaHurricane’sWrath

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

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

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

TheobjectivefunctionforQuestions1to6exceptQuestion2:Theconstraintconditions:7/23/2024第39頁(yè)TheMaximumFlowProblem

TheobjectivefunctionforQuestions2:Theconstraintconditions:7/23/2024第40頁(yè)EvacueesfromSources

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

GoldenSection

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

ThetrafficflowtobeanimportantperformancemeasureFlow=7/23/2024第44頁(yè)SimulationResultsandModelTesting

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

Q:WhateffectwouldithaveonevacuationflowtoestablishmoretemporarysheltersinColumbia,toreducethetrafficleavingColumbia?

7/23/2024第48頁(yè)StrategyIV

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

Q:In1999,thesimultaneousevacuationofthestate'sentirecoastalregionwasordered.Wouldtheevacuationtrafficflowimproveunderanalternativestrategythatstaggerstheevacuation,perhapscounty-by-countyoversometimeperiodconsistentwiththepatternofhowhurricanesaffectthecoast?7/23/2024第51頁(yè)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(yè)SensitivityAnalysis

Twoimportantimpliedfactorstoaffecttheperformanceoft