数学建模竞赛技巧

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1、数学建模竞赛技巧数学建模竞赛技巧丁永生丁永生东华大学信息学院东华大学信息学院近年来获奖情况近年来获奖情况全国二等奖全国二等奖和和上海赛区一等奖上海赛区一等奖，2004年全国大学生数年全国大学生数学模型竞赛学模型竞赛全国一等奖全国一等奖和和上海赛区一等奖上海赛区一等奖，2003年全国大学生数年全国大学生数学模型竞赛学模型竞赛上海赛区二等奖上海赛区二等奖，2002年全国大学生数学模型竞赛年全国大学生数学模型竞赛全国二等奖全国二等奖和和上海赛区一等奖上海赛区一等奖，2001年全国大学生数学年全国大学生数学模型竞赛模型竞赛上海赛区二等奖上海赛区二等奖，2001年全国大学生数学模型竞赛年全国大学生数学模

2、型竞赛上海赛区二等奖上海赛区二等奖，2000年全国大学生数学模型竞赛年全国大学生数学模型竞赛上海赛区二等奖上海赛区二等奖，1999年全国大学生数学模型竞赛年全国大学生数学模型竞赛全国一等奖全国一等奖和和上海赛区一等奖上海赛区一等奖，1998年全国大学生数学年全国大学生数学模型竞赛模型竞赛近年来获奖情况近年来获奖情况国际一等奖国际一等奖，2004年美国大学生数学模型竞赛年美国大学生数学模型竞赛国际特等奖国际特等奖，2003年美国大学生数学模型竞赛年美国大学生数学模型竞赛国际一等奖国际一等奖，2001年美国大学生数学模型竞赛年美国大学生数学模型竞赛全国一等奖全国一等奖，2004年全国部分高校研究生

3、数学建模竞赛年全国部分高校研究生数学建模竞赛上海市精品课程上海市精品课程，数学建模与数学实验，数学建模与数学实验，2004上海市优秀教学成果二等奖上海市优秀教学成果二等奖，工科数学课程的计算机实，工科数学课程的计算机实验教学改革与实践，验教学改革与实践，2004上海赛区优秀教练员上海赛区优秀教练员，2004年全国大学生数学建模竞赛年全国大学生数学建模竞赛近年来获奖情况近年来获奖情况国际特等奖国际特等奖，2003年美国大学生数学模型竞赛年美国大学生数学模型竞赛 自自1989年以来，获得国际特等奖的中国高校未超过年以来，获得国际特等奖的中国高校未超过10所所 数模小组的任务分工数模小组的任务分工三

4、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 12：00 PM 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时间的安排竞赛时间

5、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 12：00 PM 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2004 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

6、求第一页为保证书，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2004 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

7、一律采用小小4号宋体字，行距用号宋体字，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，如标出，如13等。等。2004 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

8、：书籍的表述方式为：编号编号 作者，书名，出版地：出版社，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

9、.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、公式要简单、必要时可采用小图表）图表）（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

10、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，有些不是很重要的假设在论文适当的地方提一下当的地方提一下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）

11、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

12、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 设某生物种群在其适应的设某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ttNtNttN)()()()()()(lim0tNttNttNt)()(tNtN0)0()()(

13、NNtNtNteNtN0)(模型假设模型假设实验验证实验验证2001年美国竞赛年美国竞赛B题题A Monumental Traffic Jam in 1999 Traffic slowed to a standstill on Interstate I-26,which is the principal route going inland from Charleston to the relatively safe haven of Columbia in the center of the state.What is normally an easy two-hour drive took

14、 up to 18 hours to complete.Many cars simply ran out of gas along the way.Traffic leaving Columbia going northwest was moving only very slowly.Reversal of traffic on I-26,so that both sides,including the coastal-bound lanes,have traffic headed inland from Charleston to Columbia.Traffic reversal on p

15、rincipal roads leading inland from Myrtle Beach and Hilton Head is also planned.The Principal Proposal Charleston has approximately 500,000 people Myrtle Beach has about 200,000 people,and another 250,000 people are spread out along the rest of the coastal strip Columbia,another metro area of around

16、 500,000 peoplePeople in Different Cities The interstates have two lanes of traffic in each direction except in the metropolitan areas where they have three.Columbia does not have sufficient hotel space to accommodate the evacuees(including some coming from farther north by other routes),so some tra

17、ffic continues outbound on I-26 towards Spartanburg;on I-77 north to Charlotte;and on I-20 east to Atlanta.Others FactorsThe questions that need to be addressed:1.Under what conditions does the plan for turning the two coastal-bound lanes of I-26 into two lanes of Columbia-bound traffic,essentially

18、turning the entire I-26 into one-way traffic,significantly improve evacuation traffic flow?Construct a Model2.In 1999,the simultaneous evacuation of the states entire coastal region was ordered.Would the evacuation traffic flow improve under an alternative strategy that staggers the evacuation,perha

19、ps county-by-county over some time period consistent with the pattern of how hurricanes affect the coast?Construct a Model3.Several smaller highways besides I-26 extend inland from the coast.Under what conditions would it improve evacuation flow to turn around traffic on these?4.What effect would it

20、 have on evacuation flow to establish more temporary shelters in Columbia,to reduce the traffic leaving Columbia?Construct a Model5.In 1999,many families leaving the coast brought along their boats,campers,and motor homes.Many drove all of their cars.Under what conditions should there be restriction

21、s on vehicle types or numbers of vehicles brought in order to guarantee timely evacuation?Construct a Model6.It has been suggested that in 1999 some of the coastal residents of Georgia and Florida,who were fleeing the earlier predicted landfalls of Hurricane Floyd to the south,came up I-95 and compo

22、unded the traffic problems.How big an impact can they have on the evacuation traffic flow?Construct a ModelRequires Clearly identify what measures of performance are used to compare strategies.Prepare a short newspaper article,not to exceed two pages,explaining the results and conclusions of your st

23、udy to the public.(1)The interstates(e.g.,I-26,I-77,I-20)have two lanes of traffic in each direction except in the metropolitan areas where they have three,which means the traffic entering a metropolitan(such as Columbia)can be smoothly separate into the routes leaving it in most case.(2)The normal

24、drive takes two hours to complete from Charleston to Columbia.The average drive speed for a car on the interstate I-26,I-77,and I-20 is limited to 60-80 mph,while the one for a car on principal roads(e.g.,US501)is limited to 50-70 mph.Assumptions(3)Charleston has approximately 500,000 people,Myrtle

25、Beach has about 200,000 people,and another 250,000 people are well-distributed along the rest of the coastal strip.Also,we should consider some evacuees from the coastline of Georgia and Florida in Question 6.Assumptions(4)Columbia has around 500,000 people.And,according to our rough search via Inte

26、rnet,the total number of the hotels and motels in Columbia and beyond Columbia,SC,is about 184,which can occupy around 50,000 people.Hence,Columbia does not have sufficient hotel space to accommodate the evacuees(including people coming from farther north by other routes).So,some traffic must contin

27、ue outbound on I-26 towards Spartanburg;on I-77 north to Charlotte;and on I-20 west to Atlanta.Assumptions(5)There are several tropical cyclone guidance models available to the NHC and CPHC forecasters for the hurricane season.These models range in complexity from simple statistical models to three-

28、dimensional primitive equation models.According to their models,we assume that the National Hurricane Center(NHC)in Miami,Florida can issue 72 hour tropical cyclone track and intensity forecasts.Also,hurricanes are possible in the specified area of the WATCH,usually within 36 hours,and are expected

29、in the specified area of the WARNING,usually within 24 hours.Assumptions(6)Evacuees traveling on I-26 in the normal westbound lanes of travel will be allowed to exit at all interchanges between Charleston and Columbia.Evacuees traveling in the reversed lanes of travel will be allowed to exit at most

30、,but not all,interchanges.AssumptionsFig.1.The traffic network of the main routes of evacuation.The Maximum Flow Problem 31jjsvvff),(max),(),(:),(:),(:),(:),(:),(),(),(),(),(),(jijisssitijjsijjivvcvvfNNNNNvvfvvfvvfvvfvvfvvfvvftsivvfvvf087321867632111731 The objective function for Questions 1 to 6 ex

31、cept Question 2:The constraint conditions:The Maximum Flow Problem 31jjsvvff),(max The objective function for Questions 2:The constraint conditions:),(),(),(),(),(),(),(:),(:),(:),(),(),(),(),(),(jijissssitijjsijjivvcvvfvvtvvfNvvfNvvtvvfNNNNvvfvvfvvfvvfvvftsivvfvvf032332221118738676311731Evacuees fr

32、om Sources(Hilton Head,Charleston,and Myrtle Beach)We should first deal with the 250,000 people well distributing along the rest of the coastal strip,and divide them into three sources.Then we must consider two cases in our algorithms:(1)Non-consideration of the some of the coastal residents of Geor

33、gia and Florida came up I-95;(2)Consideration of some of the coastal residents of Georgia and Florida came up I-95.The Famous Golden Section Method011NN0001252151,)(2N=500,000+125,000=625,000 3N000125215,)(=200,000+1011NNNI-95:I-26:US501:(case 2)The Impact of Temporary Shelters 879101154NNNNNNN The

34、flow of entering Columbia and that of staying at and outgoing Columbia should be equal to:-I-26 westbound traffic to Columbia-I-20 westbound traffic to Columbia-I-26 westbound traffic leaving Columbia-I-20 westbound traffic leaving Columbia-I-77 northbound traffic leaving Columbia-In hotels and mote

35、ls in Columbia-In established temporary shelters in Columbia4N5N10N9N11N7N8NThe Performance Measure of Traffic Flow The traffic flow to be an important performance measure Flow=).(.20504671SSCSSimulation Results and Model Testing Five different strategies according to Questions 1 to 5 considering tw

36、o cases:(1)Non-consideration of the some of the coastal residents of Georgia and Florida came up I-95;(2)Consideration of some of the coastal residents of Georgia and Florida came up I-95,which is Question 6.Q:Under what conditions does the plan for turning the two coastal-bound lanes of I-26 into t

37、wo lanes of Columbia-bound traffic,essentially turning the entire I-26 into one-way traffic,significantly improve evacuation traffic flow?A:Strategy IHTNNNf3211maxQ:Several smaller highways besides I-26 extend inland from the coast.Under what conditions would it improve evacuation flow to turn aroun

38、d traffic on these?A:Strategy IIHTNNNf3212maxStrategy III Q:What effect would it have on evacuation flow to establish more temporary shelters in Columbia,to reduce the traffic leaving Columbia?Strategy IV Q:In 1999,many families leaving the coast brought along their boats,campers,and motor homes.Man

39、y drove all of their cars.Under what conditions should there be restrictions on vehicle types or numbers of vehicles brought in order to guarantee timely evacuation?Table 1.The Strategies comparison leaving the coast to Columbia.Strategy V Q:In 1999,the simultaneous evacuation of the states entire c

40、oastal region was ordered.Would the evacuation traffic flow improve under an alternative strategy that staggers the evacuation,perhaps county-by-county over some time period consistent with the pattern of how hurricanes affect the coast?Table 2.The comparison of evacuating times between using Strate

41、gy V combined with Strategy I to IV and using only Strategy I to IVSensitivity Analysis Two important implied factors to affect the performance of the entire network:(1)The accommodation capacity in Columbia,and(2)The evacuees from Georgia and Florida came up I-95.Strengths and Limitations Discussio

42、ns and Conclusions Others607080901001101200100020003000400050006000t(hours)flow in I-26(cars/hour)Strategy S t ra t e gy S t ra t e gy S t ra t e gy Fig.2.The traffic flow on I-26 changing with the time period of predicted landfall of hurricane,resulting in different regions of the strategies(I to I

43、V)to be carried out.607080901001101200.70.80.911.11.21.31.41.51.6t(hours)flow*103(cars/hour)Strategy Strategy Strategy Strategy Fig.3.The minimum total required traffic flow changing with the time period of predicted landfall of hurricane,resulting in different regions of the strategies(I to IV)to b

44、e carried out.20304050607080901001101200.511.522.53x 104t(hours)maximum flow of all sources(cars/hour)t h e st ra t e gy com p l e t in g si m u l ta neousl ythe strategy escaping county by countyFig.4.The comparison of maximum traffic flow by evacuating county-by-county(Strategy V combined with Str

45、ategies I to IV)and in a simultaneous way(using only Strategies I to IV).0246810121440004200440046004800500052005400560058006000evacuees accommodated in hotel*104(person)flow in I-26(cars/hour)Strategy Strategy Strategy Fig.5.The influence of evacuees accommodated in Columbia on the traffic flow on

46、I-26,resulting in different regions of the strategies(I to III)to be carried out.012345678910350040004500500055006000Strategy Strategy Strategy Strategy the numbers escaping from Georgia and Florida*104(person)maximum flow on I-26(cars/hour)Fig.6.The impacts of evacuees from Georgia and Florida on t

47、he traffic flow on I-26,resulting in different regions of the strategies(I to IV)to be carried out.012345678910200400600800100012001400S t ra t e gy S t ra t e gy S t ra t e gy Strategy the numbers escaping from Georgia and Florida*104(person)maximum flow on I-95(cars/hour)Fig.7.The impacts of evacuees from Georgia and Florida on the traffic flow on I-95,resulting in different regions of the strategies(I to IV)to be carried out.Thanks!The End