美国（国际）大学生数学建模竞赛，是一项国际级的权威竞赛项目，其中，2009年国际建模竞赛题被当作经典样题，和该期获奖论文一起，为后来者参考学习。

竞赛介绍(时间 内容 我国参赛情况)

竞赛详情(竞赛题目 获奖论文选登)

竞赛介绍

时间

2009年美国（国际）大学生数学建模竞赛于2009年2月5日至2月9日举行。

内容

美国大学生数学建模竞赛（MCM/ICM），是一项国际级的竞赛项目，更是现今各类数学建模竞赛之鼻祖。MCM/ICM 是 Mathematical Contest in Modeling 和 Interdisciplinary Contest in Modeling 的缩写，即“数学建模竞赛”和“交叉学科建模竞赛”。 MCM 始于 1985 年，ICM 始于 2000 年，由竞赛的主持者是美国数学及其应用联合会COMAP，并得到美国运筹及工业和应用数学协会、美国工业与应用数学学会、美国数学协会等多个组织的赞助。美国赛着重强调研究问题、解决方案的原创性、团队合作、交流以及结果的合理性。 美国国际大学生数学建模竞赛从1985年开始举办，英文全称“Mathematical Contest in Modeling”，缩写为“MCM”。MCM的每个参赛队由3名队员和1名指导教师组成，比赛为期四天，每次只有两个考题，每队只需任选一题。在四天的参赛时间内参赛者可以使用包括计算机、软件包、教科书、杂志和手册等资源。比赛时要求就选定的赛题每个队在连续四天的时间里写出论文，它包括：问题的适当阐述；合理的假设；模型的分析、建立、求解、验证；结果的分析；模型优缺点讨论等。数学建模竞赛宗旨是鼓励大学师生对范围并不固定的各种实际问题予以阐明、分析并提出解法，通过这样一种方式鼓励师生积极参与并强调实现完整的模型构造的过程。以竞赛的方式培养学生应用数学进行分析、推理、证明和计算的能力；用数学语言表达实际问题及用普通人能理解的语言表达数学结果的能力；应用计算机及相应数学软件的能力；独立查找文献，自学的能力，组织、协调、管理的能力；创造力、想象力、联想力和洞察力。他还可以培养学生不怕吃苦、敢于战胜困难的坚强意志，培养自律、团结的优秀品质，培养正确的数学观。

我国参赛情况

我国一些著名大学从1989年起开始参加这项赛事，并经常在该项赛事中取得令人瞩目的成绩，大大提升了学校的国际知名度，如上海交通大学、浙江大学等。该竞赛每年一届，一般在二月份的一个周末举行，历时四天。每个参赛学校最多报四支队，其中每个系最多报两支队。由于比赛时间正值中国农历新年，为我国大学师生参赛带来了一定困难。

竞赛详情

竞赛题目

PROBLEM A: Designing a Traffic Circle

Many cities and communities have traffic circles-from large ones with many lanes in the circle (such as at the Arc de Triomphe in Paris and the Victory Monument in Bangkok) to small ones with one or two lanes in the circle. Some of these traffic circles position a stop sign or a yield sign on every incoming road that gives priority to traffic already in the circle; some position a yield sign in the circle at each incoming road to give priority to incoming traffic; and some position a traffic light on each incoming road (with no right turn allowed on a red light). Other designs may also be possible.

The goal of this problem is to use a model to determine how best to control traffic flow in, around, and out of a circle. State clearly the objective(s) you use in your model for making the optimal choice as well as the factors that affect this choice. Include a Technical Summary of not more than two double-spaced pages that explains to a Traffic Engineer how to use your model to help choose the appropriate flow-control method for any specific traffic circle. That is, summarize the conditions under which each type of traffic-control method should be used. When traffic lights are recommended, explain a method for determining how many seconds each light should remain green (which may vary according to the time of day and other factors). Illustrate how your model works with specific examples.

PROBLEM B: Energy and the Cell Phone

This question involves the “energy” consequences of the cell phone revolution. Cell phone usage is mushrooming, and many people are using cell phones and giving up their landline telephones. What is the consequence of this in terms of electricity use? Every cell phone comes with a battery and a recharger.

Requirement 1

Consider the current US, a country of about 300 million people. Estimate from available data the number H of households, with m members each, that in the past were serviced by landlines. Now, suppose that all the landlines are replaced by cell phones; that is, each of the m members of the household has a cell phone. Model the consequences of this change for electricity utilization in the current US, both during the transition and during the steady state. The analysis should take into account the need for charging the batteries of the cell phones, as well as the fact that cell phones do not last as long as landline phones (for example, the cell phones get lost and break).

Requirement 2

Consider a second “Pseudo US”-a country of about 300 million people with about the same economic status as the current US. However, this emerging country has neither landlines nor cell phones. What is the optimal way of providing phone service to this country from an energy perspective? Of course, cell phones have many social consequences and uses that landline phones do not allow. A discussion of the broad and hidden consequences of having only landlines, only cell phones, or a mixture of the two is welcomed.

Requirement 3

Cell phones periodically need to be recharged. However, many people always keep their recharger plugged in. Additionally, many people charge their phones every night, whether they need to be recharged or not. Model the energy costs of this wasteful practice for a Pseudo US based upon your answer to Requirement 2. Assume that the Pseudo US supplies electricity from oil. Interpret your results in terms of barrels of oil.

Requirement 4

Estimates vary on the amount of energy that is used by various recharger types (TV, DVR, computer peripherals, and so forth) when left plugged in but not charging the device. Use accurate data to model the energy wasted by the current US in terms of barrels of oil per day.

Requirement 5

Now consider population and economic growth over the next 50 years. How might a typical Pseudo US grow? For each 10 years for the next 50 years, predict the energy needs for providing phone service based upon your analysis in the first three requirements. Again, assume electricity is provided from oil. Interpret your predictions in term of barrels of oil.

PROBLEM A: Designing a Traffic Circle

Many cities and communities have traffic circles—from large ones with many lanes in the circle (such as at the Arc de Triomphe in Paris and the Victory Monument in Bangkok) to small ones with one or two lanes in the circle. Some of these traffic circles position a stop sign or a yield sign on every incoming road that gives priority to traffic already in the circle; some position a yield sign in the circle at each incoming road to give priority to incoming traffic; and some position a traffic light on each incoming road (with no right turn allowed on a red light). Other designs may also be possible.

The goal of this problem is to use a model to determine how best to control traffic flow in, around, and out of a circle. State clearly the objective(s) you use in your model for making the optimal choice as well as the factors that affect this choice. Include a Technical Summary of not more than two double-spaced pages that explains to a Traffic Engineer how to use your model to help choose the appropriate flow-control method for any specific traffic circle. That is, summarize the conditions under which each type of traffic-control method should be used. When traffic lights are recommended, explain a method for determining how many seconds each light should remain green (which may vary according to the time of day and other factors). Illustrate how your model works with specific examples.

PROBLEM B: Energy and the Cell Phone

This question involves the “energy” consequences of the cell phone revolution. Cell phone usage is mushrooming, and many people are using cell phones and giving up their landline telephones. What is the consequence of this in terms of electricity use? Every cell phone comes with a battery and a recharger.

Requirement 1

Consider the current US, a country of about 300 million people. Estimate from available data the number H of households, with m members each, that in the past were serviced by landlines. Now, suppose that all the landlines are replaced by cell phones; that is, each of the m members of the household has a cell phone. Model the consequences of this change for electricity utilization in the current US, both during the transition and during the steady state. The analysis should take into account the need for charging the batteries of the cell phones, as well as the fact that cell phones do not last as long as landline phones (for example, the cell phones get lost and break).

Requirement 2

Consider a second “Pseudo US”—a country of about 300 million people with about the same economic status as the current US. However, this emerging country has neither landlines nor cell phones. What is the optimal way of providing phone service to this country from an energy perspective? Of course, cell phones have many social consequences and

A题 设计一个交通环岛

在许多城市和社区都建立有交通环岛，既有多条行车道的大型环岛（例如巴黎的凯旋门和曼谷的胜利纪念碑路口），又有一至两条行车道的小型环岛。有些环岛在进入口设有“停车”标志或者让行标志，其目的是给已驶入环岛的车辆提供行车优先权；而在一些环岛的进入口的逆向一侧设立的让行标志是为了向即将驶入环岛的车辆提供行车优先权；还有一些环岛会在入口处设立交通灯（红灯会禁止车辆右转）；也可能会有其他的设计方案。

这一设计的目的在于利用一个模型来决定如何最优地控制环岛内部，周围以及外部的交通流。该设计的目的在于可利用模型做出最佳的方案选择以及分析影响选择的众多因素。解决方案中需要包括一个不超过两页纸，双倍行距打印的技术摘要，它可以指导交通工程师利用你们模型对任何特殊的环岛进行适当的流量控制。该模型可以总结出在何种情况之下运用哪一种交通控制法为最优。当考虑使用红绿灯的时候，给出一个绿灯的时长的控制方法（根据每日具体时间以及其他因素进行协调）。找一些特殊案例，展示你的模型的实用性。

B题

如题～有PROBLEM B: Energy and the Cell Phone

This question involves the “energy” consequences of the cell phone revolution. Cell phone usage is mushrooming, and many people are using cell phones and giving up their landline telephones. What is the consequence of this in terms of electricity use? Every cell phone comes with a battery and a recharger.

这个问题涉及到手机革命的能源问题。手机使用率迅速增加，许多人使用手机并放弃了固定电话。这方面的用电能使用会带来什么后果？每个手机都配备了电池和充电器。

Requirement 1

Consider the current US, a country of about 300 million people. Estimate from available data the number H of households, with m members each, that in the past were serviced by landlines. Now, suppose that all the landlines are replaced by cell phones; that is, each of the m members of the household has a cell phone. Model the consequences of this change for electricity utilization in the current US, both during the transition and during the steady state. The analysis should take into account the need for charging the batteries of the cell phones, as well as the fact that cell phones do not last as long as landline phones (for example, the cell phones get lost and break).

要求1

目前认为美国是一个人口约为3亿人的国家，从现有数据估计家庭号为h，每个家庭有M个成员，以前是使用座机电话的。现在，假设所有的座机被手机取代，也就是说每个家庭成员都有手机。在当前美国这种用电模型即过渡又是稳定的，分析应该考虑到对移动电话充电的需要，同时移动电话不能像固定电话那样持续使用也是一个现实问题（比如说移动电话可能会丢失或者损坏）

Requirement 2

Consider a second “Pseudo US”-a country of about 300 million people with about the same economic status as the current US. However, this emerging country has neither landlines nor cell phones. What is the optimal way of providing phone service to this country from an energy perspective? Of course, cell phones have many social consequences and uses that landline phones do not allow. A discussion of the broad and hidden consequences of having only landlines, only cell phones, or a mixture of the two is welcomed.

考虑到第二个“伪美国”--一个约3亿人口、跟当前美国具有相同的经济地位的国家，然而，这个新兴国家既没有固定电话也没有移动电话，从这个国家的能源角度看，用什么最佳方式为这个国家提供电话服务，当然，手机有很多固定电话所不具有的用处和社会影响。这个讨论是关于单独使用固定电话或者单独使用移动电话，或者混合使用两种电话带来的广泛或潜在的影响。

Requirement 3

Cell phones periodically need to be recharged. However, many people always keep their recharger plugged in. Additionally, many people charge their phones every night, whether they need to be recharged or not. Model the energy costs of this wasteful practice for a Pseudo US based upon your answer to Requirement 2. Assume that the Pseudo US supplies electricity from oil. Interpret your results in terms of barrels of oil.

手机需要定期充电。但是许多人在不考虑手机是否要充电的情况下，总是将充电器一直插在电器插槽上。在前两个假设成立的基础上，建立能源浪费的数学模型。另外，假定美国以石油作为电力来源，计算、解释浪费的石油量。

Requirement 4

Estimates vary on the amount of energy that is used by various recharger types (TV, DVR, computer peripherals, and so forth) when left plugged in but not charging the device. Use accurate data to model the energy wasted by the current US in terms of barrels of oil per day.

在估计不同电器设备（电视、DVR、电脑外围设备等）所使用能源数量时，电器特性：没有充电设备 要求用精确的数据建立模型 模型是关于当前美国每天所浪费的能源 以原油（桶/天）计量

Requirement 5

Now consider population and economic growth over the next 50 years. How might a typical Pseudo US grow? For each 10 years for the next 50 years, predict the energy needs for providing phone service based upon your analysis in the first three requirements. Again, assume electricity is provided from oil. Interpret your predictions in term of barrels of oil.

现在考虑人口及经济增长在未来的50年内的情况。如何使这个假设中的美国发展壮大。 对于每一个10年的今后50年内进行能源的需求预测，前提是在你前三次的分析基础上而进行的提供的电话服务。另外还有一个假设是：电力来自石油。解释你预测到的石油桶数目。

获奖论文选登

Team #4094

Round and Round We Go

February 9, 2009

Abstract

The study of traffic flow and control has been a fruitful area of mathematical research

for decades. Here we attempt to analyze and model the traffic flow that occurs in a

traffic circle. We use a powerful macroscopic approach developed by B. Piccoli that

uses network analysis and a wave-front tracking algorithm to produce powerful the-

oretical results with regards to right of way parameters in arbitrarily large networks

of traffic circles. We then follow the classical approach of Lighthill and Whitham

in order to model the effects of our proposed control system on the dynamics of a

traffic circle, employing the Runge-Kutta algorithm to process multiple solutions to

the famous conservation PDE with high-accuracy and speed. We found that prior-

itizing the right of way of the cars inside the circle optimizes the efficiency of the

circle, and developed a control system that responds to incoming traffic density in

real time and keeps the outgoing flux at a maximum regardless of the number of roads

leading to the junction. Next we developed a far more descriptive discrete model,

revising the older, standard car-following model, and reaffirmed our original control.

We conclude with simulations of our models on specific examples, a reflection of our

methods, and a technical description to a Traffic Engineer outlining how and when

to use our methods in traffic control development for traffic circles。

Team#4329

Modeling Roundabout Traffic Flow as a Dynamic Fluid System

Abstract

With increasing usage of roundabouts as traffic control mechanisms, it is important to

develop a criteria for the design of efficient roundabouts. We designate vehicle throughput

as the primary measure of a roundabouts efficiency and delay experienced by vehicles

as a secondary measure. We then apply a fluid flow analogy to model traffic density

and throughput within an arbitrary roundabout system with any number of incoming

traffic streams. Due to the distinct differences that separate one-lane roundabouts from

two-lane or multi-lane roundabouts, our model considers each lane case separately and

determine which case is optimal for a given flow of traffic. The model describes the

roundabout system as a non-linear first order partial differential equation relating speed,

traffic density, and traffic flow, all of which are subject to physical constraints. The

model is able to evaluate how many lanes are needed, and whether traffic light controls are

necessary for a given roundabout system. As real-world case study, we apply our model to

a roundabout intersection in Alachua Country Florida, and suggest optimal parameters

to maximize traffic flow through the roundabout. Applying our model to a variety of

roundabout scenarios led us to the following conclusions: traffic lights should not be used

at roundabouts; increasing the radius of the roundabout will increase the throughput,

but the effect is only significant at high rates (above 0.25 vehicles per second per entering

road) of traffic; increasing the number of lanes will always increase throughput, but the

benefit only becomes significant when traffic is heavy; increasing radius increases delay for

vehicles entering at speeds above 20m/s, while decreases delay for those entering below

20m/s.

Team#4330

Traffic circles

Abstract

The use of a tra±c circle is a relatively common means of controlling

tra±c in an intersection. Smaller Tra±c circles can be especially e?ective

in routing and controlling lower levels of vehicular °ow, since tra±c °ows

in only one direction within the circle. In larger intersections, however,

the situation becomes more complicated and congestion can easily form

at higher tra±c levels. In order to ˉnd a means of e±ciently directing

tra±c within a tra±c circle, we have written a continuous car-following

tra±c circle simulation program based on the tra±c dynamics described

by Bando et al. By dividing the tra±c circle into di?erent sections, we

are able to apply SAGE's digraph analysis tools to ˉnd the most e±cient

traversal paths at speciˉc speeds. This allows us to realistically model

car interactions within the tra±c circle in order to determine the most

e±cient methods for controlling the tra±c circle. After analyzing the

results from our analysis, we determined that the best control methods

for small tra±c circles is for cars already within the tra±c circle to yield

to cars entering the tra±c circle. This is signiˉcantly more e±cient than

either having either no control or having incoming vehicles yield to cars

already in the tra±c circle. In the single lane tra±c circle, we ˉnd that

the minimum average travel time is approximately 11.26 seconds for cars

entering at a rate of 1/3 veh/s into a single lane tra±c circle of 40m radius.

Stephen Demjanenko, Jesse Livezey, Joshua Edgerton

A Straightforward Solution to a Roundabout Problem

Abstract

The rst generation of trac circles was prone to trac jams and were poorly

optimized. Over time, trac circles became more eective at controlling trac.

Continuing in this tradition of improvement, we analyze what types of trac

control systems are capable of keeping trac moving at its most ecient rate.

Our model is scalable from small roundabouts to large urban trac circles. Since

all trac circles are unique, it is important to have a general and customizable

model so that we can treat each case separately.

Trac circles are conservative; cars are neither created nor destroyed. As

a result, we chose to model the trac circle by solving the trac ow partial

dierential equation using the Lax-Friedrich nite dierence method. This gives

a macroscopic picture of trac ow in and around the circle. We chose to

simulate stop signs, yield signs and several variants of trac lights with which

a trac engineer can control the ow of cars into and around the trac circle.

In order to optimize the trac control system we dene our metric to be

the number of cars that pass through the circle per unit time. We then ran

several basic cases in order to compare the systems of trac control. Using this

information, several methods of trac control were formulated based on com-

bining the best systems of control, as determined from our initial tests. These

methods then ran on several more complex geometries in order to determine

which method was the best for each geometry. Generalizing the results of these

test cases, we formulate basic guidelines which a civil engineer can use to de-

termine an appropriate method of trac control depending on the properties

of the trac circle. We nd that in general, roads which carry a low volume

of cars should yield while those carrying a high volume of cars should be ow

limited.

Team#4095

CIRCULAR LOGIC

1. Introduction

Trac circles or roundabouts have been in use in Europe and other places around

the world where they often replaced old public squares with the introduction of

the car. Given the irregular nature of some of these old public spaces and and the

streets that entered them, allowing motorist in the circle to proceed unimpeded

while requiring motorist entering the circle to yield may have seemed like the

only way to deal with the situation. Over time, roundabouts have been built as

exible, ecient intersections, often where the incoming road approach irregularly,

or where their trac volume are asymmetrical [6]. More recently, in the United

States. trac circles have been more commonly utilized as ecient replacement

for controlled intersections, indeed their use has been growing exponentially there