# HIMCM 2014美国中学生数学建模竞赛试题

2014 HiMCM Problems

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 seats.To 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 stairs.Most 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 station.Build 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 and/or 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 how/when 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

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