历年MCM考题

HUASHI3483

1996 美国大学生数模竞赛题
Problem A

The world's oceans contain an ambient noise field. Seismic disturbances,
surface shipping, and marine mammals are sources that, in different frequency ranges, contribute to this field. We wish to consider how this ambient noise might be used to detect large moving objects, e.g., submarines located below the ocean surface. Assuming that a submarine makes no intrinsic noise, developa method for detecting the presence of a moving submarine, its size, and its direction of travel, using only information obtained by measuring changes to the ambient noise field. Begin with noise at one fixed freqency and amplitude.

Problem B

When determining the winner of a competition like the Mathematical Contest in Modeling, there are generally a large number of papers to judge. Let's say there are P=100 papers. A group of J judges is collected to accomplish the judging. Funding for the contest constains both the number of judges that can be obtained and amount of time that they can judge. For eample if P=100, then J=8 is typical.

Ideally, each judge would read paper and rank-order them, but there are too
many papers for this. Instead, there will be a number of screening rounds in which each judge will read some number of papers and give them scores. Then some selection scheme is used to reduce the number of papers under
consideration: If the papers are rank-ordered, then the bottom 30% that each judge rank-orders could be rejected. Alternatively, if the judges do not rank-order, but instead give them numerical score (say, from 1 to 100),
then all papers below some cut-off level could be rejected.

The new pool of papers is then passed back to the judges, and the process is repeated. A concern is then the total number of papers that judge reads must be substantially less than P. The process is stopped when there are only W papers left. There are the winners. Typically for P=100, W=3.

Your task is to determine a selection scheme, using a combination of
rank-ordering, numerical scoring, and other methods, by which the final W
papers will include only papers from among the "best" 2W papers. (By "best",we assume that there is an absolute rank-ordering to which all judges would agree.) For example, the top three papers. Among all such methods, the one that required each judge to read the least number of papers is desired.

Note the possibility of systematic bias in a numerical scoring scheme. For
example, for a specific collection of papers, one judge could average 70
points, while another could average 80 points. How would you scale your scheme to accommodate for changes in the contest parameters (P, J, and W)?

HUASHI3483

1997 年美国大学生数模竞赛题
Problem A: The Velociraptor Problem

The Velociraptor, Velociraptor mongoliensis, was a predatory dinosaur that lived during the late Cretaceous period, approximately 75 million years ago. Paleontologists think that it was a very tenacious hunter, and may have hunted in pairs or larger packs. Unfortunately, there is no way to observe its hunting behavior in the wild as can be done with modern mammalian predators. A group of paleontologists has approached your team and asked for help in modeling the hunting behavior of the velociraptor.
They hope to compare your results with field data reported by biologists studying the behaviors of lions, tigers, and similar predatory animals.

The average adult velociraptor was 3 meters long with a hip height of 0.5
meters and an approximate mass of 45 Kg. It is estimated that the animal
could run extremely fast, at speeds of 60 km/hr., for about 15 seconds.
After the initial burst of speed, the animal needed to stop and recover
from a buildup of lactic acid in its muscles.

Suppose that Velociraptor prey on Thescelosaurus neglectus, a herbivorous
biped approximately the same size as the Velociraptor. A biomechanical
analysis of a fossilized thescelosaurus indicates that if could run at a
speed of about 50km.hr. for long periods of time.

Part1
Assuming the velociraptor is a solitary hunter, design a mathematical model
that describes a hunting strategy for a single velociraptor stalking and
chasing a single thescelosaurus as well as the evasive strategy of the
prey. Assume that the thecelosaurus can always detect the velociraptor when
in comes within 15 meters, but may detect the predator at even greater
ranges (up to 50 meters) depending upon the habitat and weather conditions.
Additionally, due to its physical structure and strength, the velociraptor
has a limited turning radius when running at full speed. This radius is
estimated to be three times the animal's hip height. On the other hand, the
thescelosaurus is extremely agile and has a turning radius of 0.5 meters.

Part 2
Assuming more realistically that the velociraptor hunted in pairs, design a
new model that describes a hunting strategy for two velociraptors stalking
and chasing a single thescelosaurus as well as the evasive strategy of the
prey. Use the other assumptions and limitations given in Part 1.


Problem B: Mix Well For Fruitful Discussions

Small group meetings for the discussion of important issues, particularly
long-rang planning, are gaining popularity. It is believed that large
groups discourage productive discussion and that a dominant personality
will usually control and direct the discussion. Thus, in corporate board
meetings the board will meet in small groups to discuss issues before
meeting as a whole. These smaller groups still run risk of control by a
dominant personality. In an attempt to reduce this danger it is common to
schedule several sessions with a different mix of people in each group.

A meeting of an Tostal Corporation will be attended by 29 Board Members of
which nine are in-horse members(i.e., corporate employees). The meeting is
to be an all-day affair with three sessions scheduled for the morning and
four for the afternoon. Each session will take 45 minutes, beginning on the
hour from 9:00 A.M. to 4:00 P.M., with lunch scheduled at noon. Each
morning session will consist of six discussion group with each discussion
group led by one of the corporation's six senior officers. None of these of
officers are board members. Thus each senior officer will lead three
different discussion groups. The sessions will consist of only four
discussion groups.

The president of the corporation wants a list of board-member assignments
to discussion group for each of seven sessions. The assignments should
achieve as much of a mix of members as much as possible. The ideal
assignment would have each board member with each other board member in a
discussion group the same number of times while minimizing common
membership of groups for the different sessions.

The assignments should also satisfy the following criteria:
1.For the morning sessions, no board member should be in the same senior
officer's discussion group twice.
2.No discussion group should contain a disproportionate number of in-house
members.

Give a list of assignments for members 1-9 and 10-29 and officers 1-6.
Indicate how well the criteria in the precious paragraphs are met. Since it
is possible that some board members will cancel at the last minute or that
some not scheduled will show up, an algorithm that the secretary could use
to adjust the assignments with an user to make assignments for future
meetings involving different levels of participation for each type of
attendee.

HUASHI3483

1998 年美国大学生数模竞赛题
Problem A:
Introduction:
Industrial and medical diagnostic machines known as Magnetic Resonance Imagers (MRI) scan a three-dimensional object such as a brain, and deliver their results in the
form of a three-dimensional array of pixels. Each pixel consists of one number indicating a color or a shade of gray that encodes a measure of water concentration in a
small region of the scanned object at the location of the pixel. For instance, 0 can picture high water concentration in black (ventricles, blood vessels), 128 can picture a
medium water concentration in gray (brain nuclei and gray matter), and 255 can picture a low water density in white (lipid-rich white matter consisting of myelinated
axons). Such MRI scanners also include facilities to picture on a screen any horizontal or vertical slice through the three-dimensional array (slices are parallel to any of the
three Cartesian coordinate axes). Algorithms for picturing slices through oblique planes, however, are proprietary. Current algorithms are limited in terms of the angles and
parameter options available; are implemented only on heavily used dedicated workstations; lack input capabilities for marking points in the picture before slicing; and tend
to blur and "feather out" sharp boundaries between the original pixels.
A more faithful, flexible algorithm implemented on a personal computer would be useful (1) for planning minimally invasive treatments, (2) for calibrating the MRI
machines, (3) for investigating structures oriented obliquely in space, such as postmortem tissue sections in animal research, (4) for enabling cross-sections at any angle
through a brain atlas consisting of black-and-white line drawings. To design such an algorithm, one can access the values and locations of the pixels, but not the initial data
gathered by the scanner.

Problem:
Design and test an algorithm that produces sections of three-dimensional arrays by planes in any orientation in space, preserving the original gray-scale values as closely as possible.

Data Sets:
The typical data set consists of a three-dimensional array A of numbers A(i,j,k) which indicates the density A(i,j,k) of the object at the location (x,y,z)_{ijk} . Typically,
A(i,j,k) can range from 0 through 255. In most applications; the data set is quite large. Teams should design data sets to test and demonstrate their algorithms. The data sets
should reflect conditions likely to be of diagnostic interest. Teams should also characterize data sets that limit the effectiveness of their algorithms.

Summary:

The algorithm must produce a picture of the slice of the three-dimensional array by a plane in space. The plane can have any orientation and any location in space. (The
plane can miss some or all data points.) The result of the algorithm should be a model of the density of the scanned object over the selected plane.

Problem B:
Background:
Some college administrators are concerned about the grading at A Better Class (ABC) college. On average, the faculty at ABC have been giving out high grades (the
average grade now given out is an A-), and it is impossible to distinguish between the good and mediocre students. The terms of a very generous scholarship only allow
the top 10% of the students to be funded, so a class ranking is required.

The dean had the thought of comparing each student to the other students in each class, and using this information to build up a ranking. For example, if a student obtains
an A in a class in which all students obtain an A, then this student is only "average" in this class. On the other hand, if a student obtains the only A in a class, then that
student is clearly "above average". Combining information from several classes might allow students to be placed in deciles (top 10%, next 10%, etc.) across the college.

Problem:
Assuming that the grades given out are (A+, A, A-, B+, ... ) can the dean's idea be made to work? Assuming that the grades given out are only (A, B, C, ... ) can the dean's
idea be made to work? Can any other schemes produce a desired ranking? A concern is that the grade in a single class could change many student's deciles. Is this possible?

Data Sets:
Teams should design data sets to test and demonstrate their algorithms. Teams should characterize data sets that limit the effectiveness of their algorithms.

HUASHI3483

1999年美国大学生数学建模竞赛题目（第十五届）
　　第十五届美国大学生数学建模通讯赛（MCM-99）于1999年2月5－8日进行。竞赛组委会主任F.R.Giordano在给系主任和过去MCM参加者的信中指出MCM是一项面向中学和大学学生旨在团队合作的情形下激励和改善学生问题解决及写作的能力的竞赛。MCM-99的一个新的特点是增加第三题（题C）：一个与数学、化学、环境科学和环境工程有关的跨学科的实际问题。参赛队可以从网上得到一个实际的污染问题的数据。参赛题C的队要单独报名，一个学校可以有两个队报名参赛C题。因此，一个学校至多可以报名6个队参加MCM-99。

　　MCM-99问题A 强烈的碰撞
　　美国国家航空和航天局（NASA）从过去某个时间以来一直在考虑一颗大的小行星撞击地球会产生的后果。
　　作为这种努力的组成部分，要求你们队来考虑这种撞击的后果，加入小行星撞击到了南极洲的话。人们关心的是撞到南极洲比撞到地球的其它地方可能会有很不同的后果。
　　假设小行星的直径大约为1000米，还假设它正好在南极与南极洲大陆相撞。
　　要求你们对这样一颗小行星的撞击提供评估。特别是，NASA希望有一个关于这种撞击下可能的人类人员伤亡的数量和所在地区的估计，对南半球海洋的食物生产的破坏的估计，以及由于南极洲极地冰岩的大量融化造成的可能的沿海岸地区的洪水的估计。
　　
　　MCM-99问题B 非法的集会
　　在许多公众设施的用于公众集会的房间里都有指示牌，指明在本室的人员超过指定数目，那将是非法的，这个指定的数目可能室根据一有紧急情况时能从房间出口撤离的速度来确定的。类似地，在电梯和其他设施中常有“最大容量”之类地张贴告示。
　　试研制一个数学模型：什么数目可以作为“合法地容量”张贴在指示牌上。作为你们的求解的一部分，你们要讨论与火警或其他紧急情况不通的决定房间（或空间）中的人数为“非法”的准则。还有，你们构造的模型要考虑在诸如（带有桌、椅的）自助餐厅那样带有可移动家具的房间、体育馆、游泳池，以及有成排作为和走道的报告厅之间的差别。你们可能希望对比在各种不同的环境－电梯、报告厅、游泳池、自助餐厅和体育馆－下可能得出的结论的相似之处或不通之处。收集诸如摇滚音乐会和足球比赛那些能提出特特定条件的数据。把你们的模型应用与你们学院（或邻镇）的一个或多个公众设施。试把你们的结果和这些设施所指示的容量（如果有张贴的话）进行比较。如果用了后，你们的模型看来会引起提高容量的当事人的兴趣的话，试给当地的报纸写一篇捍卫你们分析的文章。
　　MCM-99问题C 大地污染
　　背景 若干实践中重要但理论上困难的数学问题与污染的评估有关。这种问题之一就是根据只是在被怀疑为已污染地区的周围而不必直接在该地区中测得的很少的测量数据来导出不易进入的地下的渗漏污染物的位置和数量、以及污染源的精确估计。
　　例子 数据可通过http://www.comap.com/mcm.prodata.xis 找到。
　　该数据集（一种电子表格文件an Excel file），它能卸载到大多数电子数据表（spreadsheets）展示了从1990到1997在10个监测井处地下水中污染物的数量数据。单位是微克（mg/l）。8个测井的位置和高度是已知的并在下表给出。头两个数是在一张地图的直角格点上井的位置的坐标。第三个数是井中水面高出平均海平面的高度（以英尺计）。
井号 x－坐标（英尺计） y－坐标（英尺计） 高度 （英尺计）
MW-1 4187.5 6375.0 1482.23
MW-3 9062.5 4375.0 1387.92
MW-7 7625.0 5812.5 1400.19
MW-9 9125.0 4000.0 1384.53
MW-11 9062.5 5187.5 1394.26
MW-12 9062.5 4562.5 1388.94
MW-13 9062.5 5000.0 1394.25
MW-14 4750.0 2562.5 1412.00

　　数据集中另两个井（MW-27和MW-33）的位置和高度不同。在该数据集中你还会看到数字后面的字母T(Top),M(Middle)或B(Bottom)，它们分别表示测量是在井的含水层的顶部、中部和底部进行的。因此，MW-7B和MW-7M是来自同一个井，但分别是底部和中部的测量。此外，其它的测量数据表明水有流向该区域中的MW-9号井的趋势。
　　问题一 试建立一个数学模型来决定在由该数据集来表示的区域和时间里是否由任何新的污染物产生。若有，试识别新的污染物并估计它们的污染源的位置和时间。
　　问题二 在收集任何数据之前，会提出下列问题：是否拟议中的数据类型和模型能给出关于污染物所在的位置和数量的我们想要的估计。液态的化学物质会从埋置在均匀的土壤的储藏中的一个储藏罐中渗漏。因为若要在许多大罐的下面去探测的费用会过分昂贵而且危险，所以只能在储藏设备的边缘地区附近或在看来更合适的地区的表面进行测量。试决定只是在整个储藏罐的边界的外面或表面进行什么样类型的测量以及测量数目可以用于一个数学模型以决定上楼是否发生，何时发生，何处（从哪个罐）发生，以及渗漏多少液体。

HUASHI3483

2002年美国大学生数学建模竞赛题目
2002 Mathematical Contest in Modeling （MCM）Problems
原文下载网址：http://www.comap.com/undergraduate/contests/


如问题A
作者：Tjalling Ypma
标题：风和喷水池
在一个楼群环绕的宽阔的露天广场上，装饰喷泉把水喷向高空。刮风的日子，风把水花从喷泉吹向过路行人。喷泉射出的水流受到一个与风速计（用于测量风的速度和方向）相连的机械装置控制，前者安装在一幢邻近楼房的顶上。这个控制的实际目标，是要为行人在赏心悦目的景象和淋水浸湿之间提供可以接受的平衡：风刮得越猛，水量和喷射高度就越低，从而较少的水花落在水池范围以外。

你的任务是设计一个算法，随着风力条件的变化，运用风速计给出的数据来调整由喷泉射出的水流。（王强译）


Problem A
Authors: Tjalling Ypma
Title: Wind and Waterspray

An ornamental fountain in a large open plaza surrounded by buildings squirts water high into the air. On gusty days, the wind blows spray from the fountain onto passersby. The water-flow from the fountain is controlled by a mechanism linked to an anemometer (which measures wind speed and direction) located on top of an adjacent building. The objective of this control is to provide passersby with an acceptable balance between an attractive spectacle and a soaking: The harder the wind blows, the lower the water volume and height to which the water is squirted, hence the less spray falls outside the pool area.
Your task is to devise an algorithm which uses data provided by the anemometer to adjust the water-flow from the fountain as the wind conditions change.



问题B
作者：Bill Fox 和 Rich West
标题：航空公司超员订票

你备好行装准备去旅行，访问New York城的一位挚友。在检票处登记之后，航空公司职员告诉说，你的航班已经超员订票。乘客们应当马上登记以便确定他们是否还有一个座位。

航空公司一向清楚，预订一个特定航班的乘客们只有一定的百分比将实际乘坐那个航班。因而，大多数航空公司超员订票?也就是，他们办理超过飞机定员的订票手续。而有时，需要乘坐一个航班的乘客是飞机容纳不下的，导致一位或多位乘客被挤出而不能乘坐他们预订的航班。

航空公司安排延误乘客的方式各有不同。有些得不到任何补偿，有些改订到其他航线的稍后航班，而有些给予某种现金或者机票折扣。

根据当前情况，考虑超员订票问题：

航空公司安排较少的从A地到B地航班
机场及其外围加强安全性
乘客的恐惧
航空公司的收入迄今损失达数千万美元

建立数学模型，用来检验各种超员订票方案对于航空公司收入的影响，以求找到一个最优订票策略，就是说，航空公司对一个特定的航班订票应当超员的人数，使得公司的收入达到最高。确保你的模型反映上述问题，而且考虑处理“延误”乘客的其他办法。此外，书写一份简短的备忘录给航空公司的CEO（首席执行官），概述你的发现和分析。（王强译）


Problem B
Authors: Bill Fox and Rich West
Title: Airline Overbooking

You're all packed and ready to go on a trip to visit your best friend in New York City. After you check in at the ticket counter, the airline clerk announces that your flight has been overbooked. Passengers need to check in immediately to determine if they still have a seat.

Historically, airlines know that only a certain percentage of passengers who have made reservations on a particular flight will actually take that flight. Consequently, most airlines overbook-that is, they take more reservations than the capacity of the aircraft. Occasionally, more passengers will want to take a flight than the capacity of the plane leading to one or more passengers being bumped and thus unable to take the flight for which they had reservations.

Airlines deal with bumped passengers in various ways. Some are given nothing, some are booked on later flights on other airlines, and some are given some kind of cash or airline ticket incentive.

Consider the overbooking issue in light of the current situation:

Less flights by airlines from point A to point B
Heightened security at and around airports
Passengers' fear
Loss of billions of dollars in revenue by airlines to date

Build a mathematical model that examines the effects that different overbooking schemes have on the revenue received by an airline company in order to find an optimal overbooking strategy, i.e., the number of people by which an airline should overbook a particular flight so that the company's revenue is maximized. Insure that your model reflects the issues above, and consider alternatives for handling "bumped" passengers. Additionally, write a short memorandum to the airline's CEO summarizing your findings and analysis.

[此贴子已经被作者于2003-12-21 14:05:04编辑过]

HUASHI3483

2002年美国大学生交叉学科建模竞赛题目
2002 Interdisciplinary Contest in Modeling （ICM） Problem
原文下载网址：http://www.comap.com/undergraduate/contests/icm/

如果我们过分扫荡自己的土地，将会失去各种各样的蜥蜴。



佛罗里达灌木蜥蜴是一种灰色或灰褐色小蜥蜴，遍布于佛罗里达中部和大西洋沿岸地区的沙质高地上。Florida濒危动植物委员会把这种灌木蜥蜴归类为濒危的生物。

在网址http://www.comap.com/undergraduate/contests/icm/2002problem/scrublizard.pdf你将会找到一份有关这种佛罗里达灌木蜥蜴的实情说明。
佛罗里达灌木蜥蜴的长期存活，有赖于保留适当的空间搭配和灌木丛生地带的规模。
任务1：讨论在佛罗里达州促使灌木蜥蜴丧失适当栖息地的各种因素。为了保留这些栖息地，你会提出哪些建议？并且论述实现你的建议的各种障碍。
任务2：利用表1中提供的数据估计数值Fa（成年蜥蜴平均产卵量）；Sj（处在出生和第一个繁殖季节之间的幼年蜥蜴存活率）；Sa（成年蜥蜴平均存活率）。

表1
摘要数据是关于一群灌木蜥蜴的，它们先被捕捉尔后连续跟踪四年。幼小蜥蜴（0岁）在出生当年夏季不产卵。所有其他雌蜥蜴的平均孵卵量与身体尺寸成比例，正如线性函数y=0.21*(SVL)-7.5所表示的，其中y是孵卵量，而SVL是鼻子到肛门以mm为单位的长度。

年度 年龄 存活总数 雌蜥蜴存活数平均雌蜥蜴身长 (mm)
1?? 0 ??972 ??495?? 30.3
2?? 1?? 180?? 92 ???45.8
3 ??2?? 20???11 ???55.8
4?? 3?? 2 ???2??? 56.0

任务3：人们推测，参数Fa ，Sj 和Sa与一片灌木地带的露天沙质区的规模和总量有关联。利用提供在表2中的数据构造若干函数来针对不同地带估计Fa ，Sj 和Sa 。此外，构造函数对给定地带评估其承载灌木蜥蜴的能力C。

表2
关于8个灌木地带的摘要数据，包括灌木蜥蜴的生命变化速率。对于每个地带，雌蜥蜴的年产卵量（Fa），幼小蜥蜴存活率（Sj），以及成年蜥蜴存活率（Sa），连同地带规模和露天沙质栖息地的总量列在一起。


灌木地带 地带规模 (公顷) 沙质栖息地 (公顷) Fa Sj Sa 密度(蜥蜴数/公顷)



a         11.31                4.80           5.6   0.12    0.06     58
b         35.54               11.31           6.6   0.16    0.10     60
c        141.76               51.55           9.5   0.17    0.13     75
d         14.65               7 .55           4.8   0.15    0.09     55
e         63.24               20.12           9.7   0.17    0.11     80
f        132.35               54.14           9.9   0.18    0.14     82
g          8.46                1.67           5.5   0.11    0.05     40
h        278.26               184.32         11.0   0.19    0.15    115
任务4：已有许多动物研究表明，在一个栖息地带中，食物，空间，掩蔽地，抑或繁殖配偶可能受限制的，这就导致动物个体在各个地带之间迁徙。有关灌木蜥蜴的迁徙原因缺少明确的证据。不过，确有百分之十的幼年蜥蜴在各个地带之间游走，而这种迁徙会影响一个地带中群体规模。成年蜥蜴显然不迁徙。利用下面直方图中给出的数据估计在任何两个地带i和j之间经迁徙而存活的蜥蜴的概率。

表3 直方图
幼年蜥蜴的迁徙数据，是经由个体标记，释放，再捕获直到6个月后获取的。对于再捕捉的测量工作是在距离释放地点方圆750m内进行。




任务5：对于表3中给出的地表形貌，建立模型估计灌木蜥蜴的整个群体规模。而且，确定哪些地带适于灌木蜥蜴栖息，哪些地带会不支持一个有生存力的群体。
对于一个展布在Avon Park Air Force Range上面的具有29个地带的地表形貌，下面的表格列出了各个地带规模和露天沙质栖息地。参看：
http://www.comap.com/undergraduate/contests/icm/2002problem/map.jpg
给出的一张地表形貌的地图。

灌木带标识 地带规模(公顷) 沙质栖息地(公顷)



1           13.66            5.38
2           32.74           11.91
3            1.39            0.23
4            2.28            0.76
5            7.03            3.62
6            14.47           4.38
7            2.52            1.99
8            5.87            2.49
9            22.27           8.44
10            19.25          7.58
11            11.31          4.80
12            74.35         19.15
13            21.57          7.52
14            15.50          2.82
15            35.54          11.31
16            2.93            1.15
17            47.21          10.73
18            1.67            0.13
19            9.80            2.23
20            39.31           7.15
21            2.23            0.78
22            3.73            1.02
23            8.46            1.67
24            3.89            1.89
25            1.33            1.11
26            0.85            0.79
27            8.75            5.30
28            9.77            6.22
29            13.45           4.69
任务6：空中摄影业已确定，在佛罗里达灌木区域内，植被密度一年增长6%左右。请针对一个可控燃烧政策提出建议。

HUASHI3483

MCM2000
问题A  空间交通管制

为加强安全并减少空中交通指挥员的工作量，联邦航空局(FAA)考虑对空中交通管制系统添加软件，以便自动探测飞行器飞行路线可能的冲突，并提醒指挥员。为完成此项工作，FAA的分析员提出了下列问题。

要求A: 对于给定的两架空中飞行的飞机，空中交通指挥员应在什么时候把该目标视为太靠近，并予以干预。

要求B: 空间扇形是指某个空中交通指挥员所控制的三维空间部分。给定任意一个空间扇形，我们怎样从空中交通工作量的方位来估量它是否复杂？当几个飞行器同时通过该扇形时,在下面情形所确定的复杂性会达到什么程度：（1）在任一时刻？（2）在任意给定的时间范围内？（3）在一天的特别时间内？在此期间可能出现的冲突总数是怎样影响着复杂性来的？

提出所添加的软件工具对于自动预告冲突并提醒指挥员，这是否会减少或增加此种复杂性？

在作出你的报告方案的同时，写出概述（不多于二页）使FAA分析员能提交给FAA当局Jane Garvey ，并对你的结论进行答辩。



问题B  无线电信道分配

我们寻找无线电信道配置模型.在一个大的平面区域上设置一个传送站的均衡網絡,以避免干扰.一个基本的方法是将此区域分成正六边形的格子(蜂窝状),如图1.传送站安置在每个正六边形的中心点.

容许频率波谱的一个区间作为各传送站的频率.将这一区间规则地分割成一些空间信道,用整数1,2,3,…来表示.每一个传送站将被配置一正整数信道.同一信道可以在许多局部地区使用,前提是相邻近的传送站不相互干扰. 根据某些限制设定的信道需要一定的频率波谱,我们的目标是极小化频率波谱的这个区间宽度.這可以用跨度这一概念.跨度是某一个局部区域上使用的最大信道在一切滿足限制的配置中的最小值.在一个获得一定跨度的配置中不要求小于跨度的每一信道都被使用.

令s为一个正六边形的一侧的长度.我们集中考虑存在两种干扰水平的一种情况.

要求A: 频率配置有几个限制,第一,相互靠近的两个传送站不能配给同一信道.第二,由于波谱的传播,相互距离在2s內的传送站必须不配给相同或相邻的信道,它们至少差2.在這些限制下,关于跨度能说些什么.

要求B: 假定前述图1中的格子在各方向延伸到任意远,回答要求A.

要求C: 在下述假定下,重复要求A和B.更一般地假定相互靠近的传送站的信道至少差一个给定的整数k,同时那些隔开一点的保持至少差1.关于跨度和关于设计配置的有效策略作为k的一个函数能说点什么.

要求D: 考虑问题的一般化,比如各种干扰水平,或不规则的传送站布局.其他什么因素在考虑中是重要的.

要求E: 写一篇短文(不超过两页)给地方报纸,阐述你的发现.

HUASHI3483

ICM2000
问题：大象题

大象群落的兴衰 归根到底，如果象群对于栖息地造成不尽人意的影响，就要考虑对它们的驱除，即使是运用淘汰法则。国家地理杂志（地球年鉴）1999年12月

在位于南非的一个巨大的国家公园里，栖息着近乎11000只象。管理策略要求一个健康的环境以便维持11000只象的稳定群落。公园的管理员们逐年统计象的总数。在过去的20年间，整个群落经受驱除得以保持其总数尽量接近11000只。这个过程涉及枪杀（对于大部分）和每年转移近乎600到800只象到异地。

近年来，公众抗议枪杀这些象。此外，即使每年转移少量的象也是不可能了。然而，一种避孕注射法开发成功，它可以在两年期间内阻止一只成熟的母象受孕。

下面是一些关于这个公园内象的信息：

很少发生象本身移入移出该公园的事。

性别比非常接近1:1，而且采取控制措施力求维持均衡。

新生幼象的性别比也是1:1左右。双胞胎的机会接近于1.35%。

母象在10岁和12岁之间第一次怀孕，平均每3.5年产下一个崽儿，直到60岁左右为止。怀孕期约为22个月。

避孕注射使一只母象每个月发情（但不怀孕）。象通常在3.5年内仅求偶一次，所以，上述按月周期能够引起附加的反应。

一只母象可以每年注射而没有任何有害的影响。一只成熟的母象在上次注射后两年内将不能怀孕。新生幼象中的70%到80%活到一岁，其后，存活率非常高（超过95%）并且在各年龄段一致，直到60岁左右；假定象死于70岁之前是恰当的。在这个公园内没有狩猎，偷猎也是微乎其微。

公园管理部门有一个粗略的数据文件，其中列出近两年内由这个地区运出的象的大致年龄和性别。这组数据可在网站http://www.comap.com/icm/icm2000data.xls上找到。可惜的是，没有关于在这个公园内被射杀和留下来的象的可用数据。你的全部任务是发展和利用模型来研究避孕注射会如何用于控制象的数量。特别是：

任务1：发展和利用一个模型来推测年龄在2岁到60岁之间象的合理存活率。并且推测这个大象群落的当前年龄结构。

任务2：估计每年有多少只母象需要避孕注射以保持这个群落固定在11000只象左右。说明被处理数据的不确定性如何影响你的估计。试加评论这个群落年龄结构的任何改变以及会如何影响旅游者。（你或许要前瞻30-60年左右。）

任务3：假如每年转移50至300只象是可行的，这会怎样减少承受避孕注射的象只数量？试加评定避孕注射和转移之间的折衷办法。

任务4：若干反对避孕注射的人提出疑问，如果发生一场大量象只的突然灭绝（由于疾病或不受控制的偷猎），即使立即停止避孕注射，这个群落重新壮大的能力也会受到严重阻碍。对这个顾虑进行研究并作出回应。

任务5：这个公园的管理部门不相信建模。他们特别表示，由于缺少完整的数据，任何通过模型来引导他们作出决定的尝试都构成一种愚弄。除了你的技术报告之外，请附上一份字斟句酌写给公园管理部门的报告（最多三页），对于他们的疑虑作出回应并且给予劝告。还要提出一些办法来增加公园管理部门对于你的模型和结论的信赖程度。

任务6：如果你的模型有效，南非的其他大象公园会乐于采用它。请为各种规模的公园（300至25000只象）准备一项避孕注射计划，同时带有略微不同的存活率和转运可能性。

HUASHI3483

MCM1985
Problem A       Animal Populations


Choose a fish or mammal for which appropriate data are available to model it accurately. Model the animal's natural interactions with its environment by expressing population levels of different groups in terms of the significant parameters of the environment. Then adjust the model to account for harvesting in a form consistent with the actual method by which the animal is harvested. Include any outside constraints imposed by food or space limitations that are supported by the data. Consider the value of the various quantities involved, the number harvested, and the population size itself, in order to devise a numerical quantity that represents the overall value of the harvest. Find a harvesting policy in terms of population size and time that optimizes the value of the harvest over a long period of time. Check that the policy optimizes that value over a realistic range of environmental conditions.



Problem B       Strategic Reserve Management

Cobalt, which is not produced in the US, is essential to a number of industries. (Defense accounted for 17% of the cobalt production in 1979.) Most cobalt comes from central Africa, a politically unstable region. The Strategic and Critical Materials Stockpiling Act of 1946 requires a cobalt reserve that will carry the US through a three-year war. The government built up a stockpile in the 1950s, sold most of it off in the early 1970s, and then decided to build it up again in the late 1970s, with a stockpile goal of 85.4 million pounds. About half of this stockpile had been acquired by 1982.

Build a mathematical model for managing a stockpile of the strategic metal cobalt. You will need to consider such questions as: How big should the stockpile be? At what rate should it be acquired? What is a reasonable price to pay for the metal? You will also want to consider such questions as: At what point should the stockpile be drawn down? At what rate should it be drawn down? At what price is it reasonable to sell the metal? How should it be allocated?

HUASHI3483

MCM1986
Problem A       Hydrographic Data



The table below gives the depth Z of water in feet for surface points with rectangular coordinates X, Y in yards [table of 14 data points omitted]. The depth measurements were taken at low tide. Your ship has a draft of five feet. What region should you avoid within the rectangle (75,200) x (-50, 150)?



Problem B       Emergency-Facilities Location

The township of Rio Rancho has hitherto not had its own emergency facilities. It has secured funds to erect two emergency facilities in 1986, each of which will combine ambulance, fire, and police services. Figure 1 indicates the demand [figure omitted], or number of emergencies per square block, for 1985. The "L" region in the north is an obstacle, while the rectangle in the south is a part with shallow pond. It takes an emergency vehicle an average of 15 seconds to go one block in the N-S direction and 20 seconds in the E-W direction. Your task is to locate the two facilities so as to minimize the total response time.

Assume that the demand is concentrated at the center of the block and that the facilities will be located on corners.

Assume that the demand is uniformly distributed on the streets bordering each block and that the facilities may be located anywhere on the streets.