2012数学全国卷HIMCM 2014美国中学生数学建模竞赛试题

2020年11月16日发(作者：柯劭态)
2014 HiMCM Problems
HIMCM 2014美国中学生数学建模竞赛试题

Problem A: Unloading Commuter Trains
Trains arrive often at a central Station, the nexus for many commuter trains from suburbs
of larger cities on a “commuter” line. Most trains are long (perhaps 10 or more cars long).
The distance a passenger has to walk to exit the train area is quite long. Each train car has
only two exits, one near each end so that the cars can carry as many people as possible.
Each train car has a center aisle and there are two seats on one side and three seats on the
other for each row of exit a typical station of interest, passengers must exit the
car, and then make their way to a stairway to get to the next level to exit the station.
Usually these trains are crowded so there is a “fan” of passengers from the train trying to
get up the stairway. The stairway could accommodate two columns of people exiting to the
top of the commuter train platforms have two tracks adjacent to the platform.
In the worst case, if two fully occupied trains arrived at the same time, it might take a long
time for all the passengers to get up to the main level of the a mathematical
model to estimate the amount of time for a passenger to reach the street level of the
station to exit the complex. Assume there are n cars to a train, each car has length d. The
length of the platform is p, and the number of stairs in each staircase is q. Use your model
to specifically optimize (minimize) the time traveled to reach street level to exit a station
for the following:
问题一:通勤列车的负载问题
在中央车站 ,经常有许多的联系从大城市到郊区的通勤列车“通勤”线到达。大多数火车很
长(也许10个或更多的 汽车长)。乘客走到出口的距离也很长，有整个火车区域。每个火车
车厢只有两个出口,一个靠近终端, 因此可以携带尽可能多的人。每个火车车厢有一个中心
过道和过道两边的座椅，一边每排有两个座椅，另 一边每排有三个座椅。走出这样一个典型
车站,乘客必须先出火车车厢,然后走入楼梯再到下一个级别的 出站口。通常情况下这些列车
都非常拥挤,有大量的火车上的乘客试图挤向楼梯，而楼梯可以容纳两列人 退出。大多数通
勤列车站台有两个相邻的轨道平台。在最坏的情况下,如果两个满载的列车同时到达,所 有的
乘客可能需要很长时间才能到达主站台。建立一个数学模型来估计旅客退出这种复杂的状况
到达出站口路上的时间。假设一列火车有n个汽车那么长,每个汽车的长度为d。站台的长
度是p，每个 楼梯间的楼梯数量是q。使用您的模型具体来优化(减少)前往主站台的时间，
有如下要求:

Requirement 1. One fully occupied train's passengers to exit the train, and ascend the
stairs to reach the street access level of the station.
要求1. 一个满载乘客的火车，所有乘客都要出火车。所有乘客都要出楼梯抵达出主站台的
路上。

Requirement 2. Two fully occupied trains' passengers (all passengers exit onto a
common platform) to exit the trains, and ascend the stairs to reach the street access level
2014 HiMCM Problems
of the station.
要求2. 两个满载列车的乘客都要出车厢(所有乘客出到一个公用站台), 所有乘客都要出楼
梯抵达出主站台的路上。

Requirement 3. If you could redesign the location of the stairways along the platform,
where should these stairways be placed to minimize the time for one or two trains'
passengers to exit the station?
要求3. 如果你能重新设计楼梯沿着站台的位置，那么这些楼梯应放置在哪,以缩短一列或两
列火车的乘客出站所用的时间?

Requirement 4. How does the time to street level vary with the number s of stairways
that one builds?
要求4. 乘客到达出主站台的路上所用的时间跟构建楼梯的台阶数有怎样的关系?

Requirement 5. How does the time vary if the stairways can accommodate k people, k
an integer greater than one?
要求5. 如果楼梯可以容纳K个人，那么时间会如何变化？（k是大于1的整数）

In addition to the HiMCM format, prepare a short non- technical article to the director of
transportation explaining why they should adopt your model to improve exiting a station.
除了要遵循HiMCM规范，准备一个简短的非技术物品向运输主管解释为什么他们应该采取
你的模型 来提高出站效率。
2014 HiMCM Problems

Problem B: The Next Plague?
In 2014, the world saw the infectious Ebola virus spreading in western Africa. Throughout
human history, epidemics have come and gone with some infecting andor killing
thousands and lasting for years and others taking less of a human toll. Some believe these
events are just nature’s way of controlling the growth of a species while others think they
could be a conspiracy or deliberate act to cause harm. This problem will most likely come
down to how to expend (or not expend) scarce resources (doctors, containment facilities,
money, research, serums, etc…) to deal with a crisis.
问题二:下一场瘟疫?
在2014年，世界看到了感染埃博拉病毒在西非蔓 延。纵观人类历史，流行病来了又走，有
些感染带到来然后杀死成千上万的人并且持续数百年，另外一些 流行病导致少量的人员伤亡
的。一些人认为，这些事件只是大自然控制物种的生长方式，而其他人则认为 ，这可能是一
个阴谋，或者是故意行为造成伤害。这个问题很可能会归咎到如何花费（或不花费）稀缺资
源（医生，防护设施，资金，科研，精华素等），以应对危机。

Section A:
A routine humanitarian mission on an island in Indonesia reported a small village where
almost half of its 300 inhabitants are showing similar symptoms. In the past week, 15 of
the “infected” have died. This village is known to trade with nearby villages and other
islands. Your modeling team works for a major center of disease control in the capital of
your country (or if you prefer, for the International World Health Organization).
A部分：
一个常见的人道主义报道：在印度 尼西亚的一个岛上的一个小村庄，那里的300名居民中
几乎有一半都出现了类似的症状。在过去的一周 ，15个“传染”着已经死亡。这个村与附近
的村庄和其他岛屿进行交易而出名。您的建模团队工作的疾 病控制的一个主要中心是在贵国
的首都（或者，如果你喜欢，可以说是国际世界卫生组织）。

Requirement 1: develop a mathematical model that performs the following functions as
well as howwhen to best allocates these scarce resources and…
- Determines and classifies the type and severity of the spread of the disease
- Determines if an epidemic is contained or not
- Triggers appropriate measures( when to treat, when to transport victims, when to restrict
movement, when to let a disease run its course, etc…) to contain a disease.
Note: While you may want to start with the well-known “SIR” family or models for parts of
its problem, consider others,
modifications to the SIR, multiple models, or creating your own.
要求1：建立一个数学模型执行以下功能，包括如何何时最佳分配这些稀缺资源...
- 确定和分类的疾病传播的类型和严重程度
- 或者，确定疫情是否受控（爆发）
- 引发适 当的措施（什么时候治疗，什么时候运送受害者，什么时候限制转移，什么时候让
疾病听其自然，等等） 去控制某种疾病。
2014 HiMCM Problems
注意：此时你可能要开始使用 著名的“SIR”模型或该模型的部分，或者考虑别的修正后的SIR
模型，多个模型，或者创建自己的 模型。

Requirement 2: Based on the information given, your model, and the assumptions your
team has made, what initial recommendations does your team have for your country
center for disease control? (Given 3-5 recommendations with justifications)
Additional Situational Information: A multi- national research team just returned to your
country capital after spending
7 days gathering information in the infected village.
要求2：根据所给出的信息、你的模型以及你的团队所做的假设，你的团队需要为你的国家
疾病预防控 制中心给出哪些初步建议？（3-5给出建议，理由）
附加态势信息：多国研究小组花7天聚集在村里被感染后的信息返回给你的国家首都。

Requirement 3: You can ask them up to 3 questions to improve your model. What would
your ask and why?
Additional Situational Information: The mulit-national research team concluded that the
disease:
-Appears to spread through contact with bodily fluids of an infected person
-The elderly and children are more likely to die if infected
-A nearby island is starting to show similar signs of infection
-One of the researchers that returned to your capital appears infected
要求3：你可以要求问他们3个问题，以改进你的模型。你需要问什么问题和原因？
附加情境信息：多国研究小组得出结论一致认为，本病：
-出现传播是通过与受感染者的体液接触
-如果感染，老人和儿童更容易感染
-附近的岛上开始出现类似感染的迹象
-一个研究者返回到贵国首都出现感染

Requirement 4:How does the additional information above changemodify your model?
要求4：如何根据以上的附加信息改版修改模型？

Requirement 5:Write a one-page synopsis of your findings for your local non-technical
news outlet.
要求5：写下你的发现，为当地的非技术广播电台(或电视台)提供新闻稿。