2016 MCM notes by magic2728

2016-2-12 20:59| 发布者: magic2728| 查看: 616| 评论: 4|原作者: magic2728

摘要: Analysis of 2016 MCM Problems2016 MCM Problem A A Hot Bath(bk1)A person fills a bathtub with hot water from asingle faucet and settles into the bathtub to cleanse and relax.(bk1') Unfortunately, the b ...

Analysis of 2016 MCM Problems2016 MCM Problem A A Hot Bath(bk1)

A person fills a bathtub with hot water from asingle faucet and settles into the bathtub to cleanse and relax.(bk1') Unfortunately, the bathtub is not a spa-styletub with a secondary heating system and circulating jets, but rather a simple watercontainment vessel.(pr1) After a while, the bath gets noticeably cooler, so the person adds a constanttrickle of hot water from the faucet to reheat the bathing water.(pr1'， imp1) The bathtub is designed in such a way that whenthe tub reaches its capacity, excess water escapes through an overflow drain.(pr1'')

Develop a model of the temperature of the bathtub water in space and time to determine the best strategy the person in the bathtub can adopt tokeep the temperature even throughout thebathtub and as close aspossible to the initial temperature without wasting too much water.(spm1)

Use your model to determine the extent to whichyour strategy dependsupon the shape and

volume of the tub, the shape/volume/temperature of the person in the bathtub, and the motions made by the person in the bathtub. If the personused a bubble bath additive while initially filling the bathtub to assist in cleansing, how wouldthis affect your model’s results?(spm1')

In addition to the required one-page summary foryour MCM submission, your report must

include a one-page non-technical explanation forusers of the bathtub that describes your strategy while explaining why it is so difficult to get anevenly maintained temperature throughout the bath water.(mss1)

Analysis：

本题是物理背景的优化问题，十分常见。

第一段提到洗热水澡的背景和水容易凉的问题以及加热水解决却常常浪费水的问题，第二段描述了优化模型的主体：

decision variable：strategy，量化后即为细流的水速控制，水温和水源位置以及其他可能合理地操作（比如同时排水，但不能加热，pr1'暗含的暗示，否则就不是bathhub了）；

objective：even temperature， close to initial， waste，即温度和空间的函数变化范围小，平均温度和原始差距小，并浪费水量小，定义各自产生对人的效用影响函数（浪费水的效用就是对应水费的原始效用），即可得到三个子目标汇总的得到的总目标；

objective function：热传导模型和水流模型足够描述在一定入水速，水温和水源位置为自变量，周围环境温度，人和水缸的属性作为参数时，温度变化的规律和水的流动过程，进而得到实时温度-（s，t）的函数，进而得到所有的子目标和最终目标。

（t = f1（strategy），ob1，2，3 = f234（t），ob = f5（ob1,2,3），共五个函数需要定义，f1是物理过程建模，连续的微分方程，离散的元胞自动机均可，f234是三个指定指标用原始测量值描述的评价模型，f5是以效用（可以有其他自己定义的含义）作为总目标的三个目标因子评价模型）

optimization：给定参数的一组参考值后，对三个维度的自变量穷举搜索，MonteCarlo 或者启发式搜素，得到一定程度的最优解。

第三段提到几个提醒的做灵敏度分析和鲁棒性分析的点，照做即可，还可以针对其他可能变动的参数进一步作该分析。

最后一段为mission，提醒需要把均匀性和保持性都做到的困难总结出来。

优化模型的四个要点中，此题重点和空白的地方在目标函数的确立，其中有多个层次需要分层次清晰考虑，而决策变量和决策目标均有明确的要求，根据从信息标注得到的模型类型和背景类型，在该模型类型下夹逼出我们需要完成的任务（目标函数确立和最优解搜索），审题就全部完成了。

祝大家建模顺利！

2016 MCM Problem B Space Junk（bk1）

The amount of small debris in orbit around earthhas been a growing concern.（bk1'） It is estimated that more than 500,000 pieces of space debris, alsocalled orbital debris, are currently being tracked as potential hazards to space craft.（pr1） The issue itself became more widely discussed in thenews media when the Russian satellite Kosmos-2251 and theUSA satellite Iridium-33 collided on 10 February, 2009.(pr1')

A number of methods to remove the debris havebeen proposed.(bk2) These methods include small, space-based water jets and high energy lasersused to target specific pieces of debris and large satellites designed to sweep up the debris, amongothers.(bk2) The debris rangesin size and mass from paint flakes to abandoned satellites.(bk3) The debris’ high velocity orbits make capturedifficult.(pr2)

Develop a time-dependent model to determine the best alternative or combination of alternatives that a private firm could adopt as a commercialopportunity to address the space debris problem.(spm1)

Your model should include quantitative and/orqualitative estimates of costs, risks, benefits, as well as other important factors.(imp1 of spm1) Your model should be able to assess independentalternatives as well as combinations of alternatives and be able toexplore a variety of important “What if?” scenarios.(rsc1 of spm1)

Using your model, determine whether an economically attractive opportunityexists or no such opportunity is possible.(spm1') If a viable commercial opportunity exists as analternative solution, provide a comparison of the different options forremoving debris, and include a specific recommendation as to how the debris should beremoved. If no such opportunity is possible, then provide innovative alternatives for avoidingcollisions.

In addition to the required one-page summary foryour MCM submission, your report must

include a two-page Executive Summary thatdescribes the options considered and major

modeling results, and provides a recommendationfor a particular action, combination of actions, or no action, as appropriate from your work.(mss1) The Executive Summary should be written for high level policy makers and news media analysts whodo not have a technical background.(imp2)

Analysis：

本题为物理社会混合领域背景，以及含有优化，评价，预测全部模型类型的综合性问题，好在虽然考察面广，但是每一个点都不难，各个击破即可。

第一段提到太空残骸出现的背景和问题，第二段提到两种可行策略和残骸状态（这里背景要求严格，要像rcs一样遵守）以及仍然存在的困难，第三段提出需要我们决策去除残骸的方案（以及第四段给出相关要求和暗示），无一例外地要转化成给定决策目标后的优化问题来解决，我们继续用条件问题夹逼法，找出模型类型需要的要素和已知之间的差距，找到我们建模需要完成的gap：

decision variable：选择单一或者组合的方案，由于可行方案已经给出，这里方案的量化十分简单，选择或者权重组合而已，rsc1要求两种情况分别考虑（给公司的报酬数量可作为参数）；

objective：并不是直觉上的残骸的清理效果，而是商业公司能够盈利最多，即文中commerialopportunity的含义；

objective function：profit = revenue – cost，所有的因素都要归结到收入或者损失上，最后得到的即为利润，此乃评价模型的归一加权思想（在上一题对人的效用即用到）而这里costs, risks,是方案产生的物理效果估计，是物理类的机理为主导的预测模型，benefits则为社会类预测，更多靠数据作为依据，其他因素还需要在此框架下补充。（imp1的理解）

optimization：低维搜索应该就够了，不复杂，重点也不在这。

第五段要求使用模型回答问题，做出决策，在最优方案下保证能够有足够的把握使得商业市场接纳，通过调整报酬参数使得方案总是可行的，如果你有别的策略也行，言之成理即可，这是在用优化结果去得到一个限制条件不等式，进而解出参数的范围即为所求。

此题为背景模型大融合，每种模型类型以及典型背景类型都有涉及，优化部分的特色搜索最优解并不难，而用评价和预测方案去构建目标函数是此题最大得特征。

最后的mission完成就大功告成啦！

2016 MCM Problem C The Goodgrant Challenge（bk1）

The Goodgrant Foundation is a charitableorganization that wants to help improve educational performance of undergraduates attending collegesand universities in the United States.（bk1'） To do this, the foundation intends to donate a total of $100,000,000 (US100 million) to an appropriate group of schools per year, for five years, starting July 2016. In doing so, they do not wantto duplicate the investments and focus of other large grantorganizations such as the Gates Foundation and Lumina Foundation.(rsc1)

Your team has been asked by the GoodgrantFoundation to develop a model to determine an optimal investment strategy that identifies the schools, the investment amount per school, the return on that investment, and the time duration that the organization’s money should beprovided to have the highest likelihood of producing a strong positiveeffect on student performance.(spm1) This strategy should contain a 1 to N optimized and prioritizedcandidate list of schools you are recommending for investment based on each candidate school’sdemonstrated potential for effective use of private funding, and an estimated return on investment(ROI) defined in a manner appropriate for a charitable organization such as the Goodgrant Foundation.(imp1)

To assist your effort, the attached data file (ProblemCDATA.zip) contains informationextracted from the U.S. National Center on Education Statistics (www.nces.ed.gov/ipeds), which maintains an extensive database of survey information onnearly all post-secondary colleges and universities in the United States, and the College Scorecard data set(https://collegescorecard.ed.gov)which contains various institutional performance data.(rsc1) Your model and subsequent strategy must be basedon some meaningful and defendable subset of these two datasets.(rsc1')

In addition to the required one-page summary foryour MCM submission, your report must include a letter to the Chief Financial Officer (CFO) ofthe Goodgrant Foundation, Mr. Alpha Chiang, that describes the optimal investment strategy, yourmodeling approach and major results, and a brief discussion of your proposed concept of areturn-on-investment (ROI) that the Goodgrant Foundation should adopt for assessing the 2016 donation(s)and future philanthropic educational investments within the United States. This letter should beno more than two pages in length.(mss1)

Analysis：

社会背景的优化问题，最后变成决策。条件众多，一一满足，还给定了数据作函数参数训练的依据，我们在有限的自由度下完成优化问题的审题：

decision variable：strategy，受总预算的约束，而school,investment amount, return, time皆为给定参数;

objective: student performance，是个体表现的期望

objective function: 用社会类预测模型，给定形式后，用所给数据训练/估计出参数，即为所求；

optimization：按照函数搜索即可。