数学建模竞赛论文写作 丁永生 东华大学信息学院

近年来获奖情况 2003 年美国大学生数学模型竞赛，国际特等奖 2002 年全国大学生数学模型竞赛，上海赛区二等奖 2001 年美国大学生数学模型竞赛，国际一等奖 2001 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 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，上海赛区二等奖 1998 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

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 ： 00 PM 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 ： 00 PM 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

2003 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

2003 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 [1][3] 等。

2003 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 3~5 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 2~3 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 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 N 0 ， 则有 于是： 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 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 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 N 0 假定初始时刻，种群的数量为 N 0

实验验证 美国战后 10 年的人口增长 美国战后 10 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 ）变参数 （ 1 ）变参数 （ 2 ）多种群共存 （ 2 ）多种群共存 （ 3 ）随机模型 （ 3 ）随机模型

飓风疏散问题建模 Strategies for Escaping a Hurricane’s Wrath 2001 年美国竞赛 B 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 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 principal 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 500,000 people People 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 traffic continues outbound on I-26 towards Spartanburg; on I-77 north to Charlotte; and on I-20 east to Atlanta. Others Factors

The 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 turning the entire I-26 into one-way traffic, significantly improve evacuation traffic flow? Construct a Model

2. In 1999, the simultaneous evacuation of the state's entire coastal 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? Construct a Model

3. 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 have on evacuation flow to establish more temporary shelters in Columbia, to reduce the traffic leaving Columbia? Construct a Model

5. 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 restrictions on vehicle types or numbers of vehicles brought in order to guarantee timely evacuation? Construct a Model

6. 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 compounded the traffic problems. How big an impact can they have on the evacuation traffic flow? Construct a Model

Requires 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 study 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 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 mph, while the one for a car on principal roads (e.g., US501) is limited to mph. Assumptions

(3) Charleston has approximately 500,000 people, Myrtle 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 Internet, 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 continue 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- 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 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, but not all, interchanges. Assumptions

T S x1x1x1x1 x2x2x2x2 x3x3x3x3 x4x4x4x4 x5x5x5x5 x7x7x7x7 x6x6x6x6 x8x8x8x8 x9x9x9x9 x 10 x 11 x 12 x 14 x 15 x x 16 S — Super source 1(V 1 )—Hilton Head 2(V 2 )—Charleston 3(V 3 )—Myrtle Beach 4(V 4 )—The crossover point between I-95 and I-26 between I-95 and I-26 5(V 5 )—The crossover point between I-95 and I-20 between I-95 and I-20 6(V 6 )—Columbia 7(V 7 )—Hotels in Columbia 8(V 8 )—Temporary shelters in Columbia 9(V 9 )—Atlanta 10(V 10 )— Spartanburg 11(V 11 )— Charlotte T — Super sink Fig. 1. The traffic network of the main routes of evacuation.

The Maximum Flow Problem The objective function for Questions 1 to 6 except Question 2: The constraint conditions:

The Maximum Flow Problem The objective function for Questions 2: The constraint conditions:

Evacuees from 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 Georgia 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 Method = 500, ,000=625,000 =200,000+ I-95 : I-26: US501: (case 2)

The Impact of Temporary Shelters The 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 motels in Columbia -- In established temporary shelters in Columbia

The Performance Measure of Traffic Flow The traffic flow to be an important performance measure Flow =

Simulation Results and Model Testing Five different strategies according to Questions 1 to 5 considering two 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 two lanes of Columbia-bound traffic, essentially turning the entire I-26 into one- way traffic, significantly improve evacuation traffic flow? A: Strategy I

Q: 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? A: Strategy II

Strategy 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. Many 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. Performance measures of traffic flow (cars/hour) Strategy I Strategy II Strategy III Strategy IV The Flow on I-26 Case Case The Flow on US501 to I-20 Case Case The Flow on I-95 Case Case

Strategy V Q: In 1999, the simultaneous evacuation of the state's entire coastal 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 Strategy V combined with Strategy I to IV and using only Strategy I to IV StrategyIIIIIIIV Completing simultaneously Case Case Escaping county-by- county Case Case

Sensitivity 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 Discussions and Conclusions Others

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 IV) to be carried out.

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 be carried out.

Fig. 4. The comparison of maximum traffic flow by evacuating county-by-county (Strategy V combined with Strategies I to IV) and in a simultaneous way (using only Strategies I to IV).

Fig. 5. The influence of evacuees accommodated in Columbia on the traffic flow on I-26, resulting in different regions of the strategies (I to III) to be carried out.

Fig. 6. The impacts of evacuees from Georgia and Florida on the traffic flow on I-26, resulting in different regions of the strategies (I to IV) to be carried out.

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