爱华网
PROBLEM A:The Leaves of a Tree
"How much do the leaves on a tree weigh?" How might one estimatethe actual weight of the leaves (or for that matter any other partsof the tree)? How might one classify leaves? Build a mathematicalmodel to describe and classify leaves. Consider and answer thefollowing:
• Why do leaves have the various shapes that theyhave?
• Do the shapes “minimize” overlapping individualshadows that are cast, so as to maximize exposure? Does thedistribution of leaves within the “volume” of the tree and itsbranches effect the shape?
• Speaking of profiles, is leaf shape (generalcharacteristics) related to tree profile/branching structure?
• How would you estimate the leaf mass of a tree?Is there a correlation between the leaf mass and the sizecharacteristics of the tree (height, mass, volume defined by theprofile)?
In addition to your one page summary sheet prepare a one pageletter to an editor of a scientific journal outlining your keyfindings.
PROBLEMB:Camping along the Big LongRiver
Visitors to the Big Long River (225 miles) can enjoy scenic viewsand exciting white water rapids. The river is inaccessible tohikers, so the only way to enjoy it is to take a river trip thatrequires several days of camping. River trips all start at FirstLaunch and exit the river at Final Exit, 225 miles downstream.Passengers take either oar- powered rubber rafts, which travel onaverage 4 mph or motorized boats, which travel on average 8 mph.The trips range from 6 to 18 nights of camping on the river, startto finish.. The government agency responsible for managing thisriver wants every trip to enjoy a wilderness experience, withminimal contact with other groups of boats on the river.Currently,Xtripstravel down the Big Long River each year during a six month period(the rest of the year it is too cold for river trips). ThereareYcamp sites on theBig Long River, distributed fairly uniformly throughout the rivercorridor. Given the rise in popularity of river rafting, the parkmanagers have been asked to allow more trips to travel down theriver. They want to determine how they might schedule an optimalmix of trips, of varying duration (measured in nights on the river)and propulsion (motor or oar) that will utilize the campsites inthe best way possible. In other words, how many more boat tripscould be added to the Big Long River’s rafting season? The rivermanagers have hired you to advise them on ways in which to developthe best schedule and on ways in which to determine the carryingcapacity of the river, remembering that no two sets of campers canoccupy the same site at the same time. In addition to your one pagesummary sheet, prepare a one page memo to the managers of the riverdescribing your key findings.
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接到题之后发现B题比较好想。我的经验是：A题描述空泛，大量的工作需要自己完成，发散性强；B题较实际，针对性强，即使思路不是很清楚，上手也比较快。于是在城际上一直在想，感觉有两种方法，一种是用MonteCarlo模拟，另外一种是图像法。（图略）
第二天，我们讨论半天才发现有4天时间，于是紧张的神经顿时松懈下来，开始看日和。期间不时地讨论一下，始终没有达成共识，但通过查阅资料，原型已经搞清楚，是一个关于Grand Canyon Rafting的一个旅游项目，在美国的同学应该很清楚，可在其他国家就未必了。我想COMAP组委会的人可能觉得美国本土的同学获奖太少，大部分奖项都被其他国家拿走了，心里不爽，于是出了一道偏向于本土同学的题。
今年有两个很有意思的情况：
1.据不完全统计，去年国内做C题的比例相当少；而今年只是在南开，我认识的人里面就有不到一半是做C题的。可能是看A和B都不好做吧。
2.比赛刚开始的时候，做A题的很多，根据是数学中国和校苑数模的帖子数量，当然这种推理不能够完全成立，但至少说明国内讨论A题的同学要比B题的同学多一些；而第三天开始，B题帖子的数量就超过A题了。可能是B题后期更值得讨论，也有可能是觉得A题不好做转到B题了吧。
这种现象其实是值得关注的。据往年的统计来看，数学建模正在悄然升温，而且就我学习的专业而言，经济金融类的大趋势是数理方面。在申请国外的学校过程中，也比较重视有关数学的课程和课外活动。如今数学的地位在各个领域不断地提升，我们是否应当顺应潮流投身发展数学的事业当中？这个问题值得思考，本人也在权衡。当下，统计专业就业很好，但理论数学和应用数学却受到了限制，国内大部分同学申请统计专业，极少申请理论数学。
第三天下午我们去找了老师，讨论了一下，有些启发，但具体的方法没有。不过又一次明确了数学建模的目的：解决实际问题，不要因为是数学建模而觉得必须用到某种数学方法。用我的理解，数学建模是一种解决实际问题的方式，具体方法是：集中高智商人群，利用各种工具（主要是数学），最终得到问题的最优或次优解。建模并不像做数学题那样有明确的定理和思路，发散性更强，并且虽要求严谨，但更侧重解决问题。
晚上又讨论了一下，觉得仿真不太靠谱，即便是做了很强的假设，180天的仿真也是个很难完成的任务。
第四天，不能再耽搁了，起床之后立刻开始着手干活。三人各种分工，我负责用latex写论文，其他两人负责画图，写思路，以及具体方法的实现。据我们搞数学的说，这次是用到了组合数学，我不是很了解。而且编程方面，一是这个问题确实不容易写成代码，二是没有现成的代码可以参考，三是时间不多了，所以我们只是象征性的写了一些放在附录。
从早上八点多熬到了转天14号的八点多，终于写完了，提交邮箱solutions@comap.com。之后就是把学校里所有队伍的论文寄送出去，附上地址：
MCM/ICM Coordinator
COMAP, Inc.
175 Middlesex Turnpike.,Suite3B
Bedford,MA01730
USA
貌似用邮政太贵，前年有用联邦快递的，好像很不错，下次试一下。
具体方法就不再赘述了，结果出来之后可以参考OutStanding 论文。
js，xt都是很好的伙伴，感谢他们。
