2009國際建模競賽題

美國（國際）大學生數學建模競賽，是一項國際級的權威競賽項目，其中，2009年國際建模競賽題被當作經典樣題，和該期獲獎論文一起，為後來者參考學習。

基本介紹

• 中文名：2009國際建模競賽題
• 時間：2009年2月5日至2月9日
• 賽事：美國（國際）大學生數學建模競賽
• 簡稱：MCM/ICM

競賽介紹

時間

2009年美國（國際）大學生數學建模競賽於2009年2月5日至2月9日舉行。

內容

美國大學生數學建模競賽（MCM/ICM），是一項國際級的競賽項目，更是現今各類數學建模競賽之鼻祖。MCM/ICM 是 Mathematical Contest in Modeling 和 Interdisciplinary Contest in Modeling 的縮寫，即“數學建模競賽”和“交叉學科建模競賽”。 　MCM 始於 1985 年，ICM 始於 2000 年，由競賽的主持者是美國數學及其套用聯合會COMAP，並得到美國運籌及工業和套用數學協會、美國工業與套用數學學會、美國數學協會等多個組織的贊助。美國賽著重強調研究問題、解決方案的原創性、團隊合作、交流以及結果的合理性。 　美國國際大學生數學建模競賽從1985年開始舉辦，英文全稱“Mathematical Contest in Modeling”，縮寫為“MCM”。MCM的每個參賽隊由3名隊員和1名指導教師組成，比賽為期四天，每次只有兩個考題，每隊只需任選一題。在四天的參賽時間內參賽者可以使用包括計算機、軟體包、教科書、雜誌和手冊等資源。比賽時要求就選定的賽題每個隊在連續四天的時間裡寫出論文，它包括：問題的適當闡述；合理的假設；模型的分析、建立、求解、驗證；結果的分析；模型優缺點討論等。數學建模競賽宗旨是鼓勵大學師生對範圍並不固定的各種實際問題予以闡明、分析並提出解法，通過這樣一種方式鼓勵師生積極參與並強調實現完整的模型構造的過程。以競賽的方式培養學生套用數學進行分析、推理、證明和計算的能力；用數學語言表達實際問題及用普通人能理解的語言表達數學結果的能力；套用計算機及相應數學軟體的能力；獨立查找文獻，自學的能力，組織、協調、管理的能力；創造力、想像力、聯想力和洞察力。他還可以培養學生不怕吃苦、敢於戰勝困難的堅強意志，培養自律、團結的優秀品質，培養正確的數學觀。

賽題插圖

我國參賽情況

我國一些著名大學從1989年起開始參加這項賽事，並經常在該項賽事中取得令人矚目的成績，大大提升了學校的國際知名度，如上海交通大學、浙江大學等。該競賽每年一屆，一般在二月份的一個周末舉行，歷時四天。每個參賽學校最多報四支隊，其中每個系最多報兩支隊。由於比賽時間正值中國農曆新年，為我國大學師生參賽帶來了一定困難。

競賽詳情

競賽題目

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題 設計一個交通環島

在許多城市和社區都建立有交通環島，既有多條行車道的大型環島（例如巴黎的凱旋門和曼谷的勝利紀念碑路口），又有一至兩條行車道的小型環島。有些環島在進入口設有“停車”標誌或者讓行標誌，其目的是給已駛入環島的車輛提供行車優先權；而在一些環島的進入口的逆向一側設立的讓行標誌是為了向即將駛入環島的車輛提供行車優先權；還有一些環島會在入口處設立交通燈（紅燈會禁止車輛右轉）；也可能會有其他的設計方案。

這一設計的目的在於利用一個模型來決定如何最優地控制環島內部，周圍以及外部的交通流。該設計的目的在於可利用模型做出最佳的方案選擇以及分析影響選擇的眾多因素。解決方案中需要包括一個不超過兩頁紙，雙倍行距列印的技術摘要，它可以指導交通工程師利用你們模型對任何特殊的環島進行適當的流量控制。該模型可以總結出在何種情況之下運用哪一種交通控制法為最優。當考慮使用紅綠燈的時候，給出一個綠燈的時長的控制方法（根據每日具體時間以及其他因素進行協調）。找一些特殊案例，展示你的模型的實用性。

B題

如題～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.

這個問題涉及到手機革命的能源問題。手機使用率迅速增加，許多人使用手機並放棄了固定電話。這方面的用電能使用會帶來什麼後果？每個手機都配備了電池和充電器。

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

目前認為美國是一個人口約為3億人的國家，從現有數據估計家庭號為h，每個家庭有M個成員，以前是使用座機電話的。現在，假設所有的座機被手機取代，也就是說每個家庭成員都有手機。在當前美國這種用電模型即過渡又是穩定的，分析應該考慮到對行動電話充電的需要，同時行動電話不能像固定電話那樣持續使用也是一個現實問題（比如說行動電話可能會丟失或者損壞）

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.

考慮到第二個“偽美國”--一個約3億人口、跟當前美國具有相同的經濟地位的國家，然而，這個新興國家既沒有固定電話也沒有行動電話，從這個國家的能源角度看，用什麼最佳方式為這個國家提供電話服務，當然，手機有很多固定電話所不具有的用處和社會影響。這個討論是關於單獨使用固定電話或者單獨使用行動電話，或者混合使用兩種電話帶來的廣泛或潛在的影響。

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.

手機需要定期充電。但是許多人在不考慮手機是否要充電的情況下，總是將充電器一直插在電器插槽上。在前兩個假設成立的基礎上，建立能源浪費的數學模型。另外，假定美國以石油作為電力來源，計算、解釋浪費的石油量。

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.

在估計不同電器設備（電視、DVR、電腦外圍設備等）所使用能源數量時，電器特性：沒有充電設備 要求用精確的數據建立模型 模型是關於當前美國每天所浪費的能源 以原油（桶/天）計量

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.

現在考慮人口及經濟成長在未來的50年內的情況。如何使這個假設中的美國發展壯大。 對於每一個10年的今後50年內進行能源的需求預測，前提是在你前三次的分析基礎上而進行的提供的電話服務。另外還有一個假設是：電力來自石油。解釋你預測到的石油桶數目。

獲獎論文選登

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

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