2017 MCM Problems ABC Notes by magic2728

建模的小伙伴们，知道你们都在挑灯夜战，为了梦想做最后的努力啦，先为你们的努力喝彩！

magic说到做到，会在比赛时候给大家一波神秘助攻！下面的内容是我对MCM ABC三道题的解析，希望能祝各位拿大奖一臂之力！

解析的格式是题意信息标注 + 模型要点分析，看看大家题意理解有没有跑偏，重点是否突出。上过我相关课程的同学可对照“审题三部曲”里关于三类信息标注的暗号系统查看，没上过的也没关系，可以直接跳到每个题目末尾看“函数建模法”的解析哦！

Problem A: Managing The Zambezi River (imp1)

The Kariba Dam on the Zambezi River is one of the larger dams in Africa.(bk1) Its construction was controversial, and a 2015 report by the Institute of Risk Management of South Africa included a warning that the dam is in dire need of maintenance.(pr1) A number of options are available to the Zambezi River Authority (ZRA) that might address the situation.(bk2s) Three options in particular are of interest to ZRA:

(Option 1) Repairing the existing Kariba Dam, (Option 2) Rebuilding the existing Kariba Dam, or (Option 3) Removing the Kariba Dam and replacing it with a series of ten to twenty smaller dams along the Zambezi River.

There are two main requirements for this problem:

Requirement 1 ZRA management requires a brief assessment of the three options listed, with sufficient detail to provide an overview of potential costs and benefits associated with each option.(imp1, spm1) This requirement should not exceed two pages in length, and must be provided in addition to your main report.(rsc1, mss1)

Requirement 2 Provide a detailed analysis of Option (3) - removing the Kariba Dam and replacing it with a series of ten to twenty smaller dams along the Zambezi river. This new system of dams should have the same overall water management capabilities as the existing Kariba Dam while providing the same or greater levels of protection and water management options for Lake Kariba that are in place with the existing dam.(rsc2) Your analysis must support a recommendation as to the number and placement of the new dams along the Zambezi River.(spm2, imp2)

In your report for Requirement 2, you should include a strategy for modulating the water flow through your new multiple dam system that provides a reasonable balance between safety and costs.(imp3) In addition to addressing known or predicted normal water cycles, your strategy should provide guidance to the ZRA managers that explains and justifies the actions that should be taken to properly handle emergency water flow situations (i.e. flooding and/or prolonged low water conditions).(rsc3) Your strategy should provide specific guidance for extreme water flows ranging from maximum expected discharges to minimum expected discharges.(imp4) Finally, your recommended strategy should include information addressing any restrictions regarding the locations and lengths of time that different areas of the Zambezi River should be exposed to the most detrimental effects of the extreme conditions.(rsc4)

Your MCM submission should consist of three elements: a standard 1 page MCM Summary Sheet, a 1-2 page brief assessment report (Requirement 1), and your main MCM solution (Requirement 2) not to exceed 20 pages for a maximum submission of 23 pages.(rsc4) Note: Any appendices or reference pages you include will not count towards the 23 page limit.

又到了每年美赛给大家写评注的时间了，今年工作实在太忙，发送稍晚，小伙伴们久等了！

速度言归正传！

此题的背景是大坝修建防水的规划问题，模型类型有评价函数建立和用优化方法对修建方案求解的考察。

题目中两个requirement即需要建立的两个模型和回答的两个问题，文末也提到了，应该以这两个问题的解决作为整体线索来架构整个文章，requirement之前的部分为话题介绍，带过即可。

Requirement1：assessment明显暗示了需要一个评价模型，并且提到了需要考察的评价点指标为cost and benefits，于是按照评价问题的四部曲“找指标，定关系，参数估计，评价评价”执行建模过程即可，最后一定要把solution，即对三个option的评价结果予以正面回答才算完成此部分。由于本部分篇幅仅有1~2页，所以对量化评价的要求不是很高，不好定量的部分可以通过描述带过去。

Requirement2：此部分是在1中评估option3为最优方案的基础上的方案实施方法，参数取值的进一步细化的优化问题，是一个典型的用优化思想来做决策的模型，优化要素如下：

X：number and placement of the new dams

(restrictions for range: the same overall water management capabilities)

Y：same or greater levels of protection and water management options, reasonable balance between safety and costs

F: strategy for modulating the water

Parameter：dam size and cost, variation of water environment and etc.

这里的优化目标暗示为花费和保护安全能力的平衡，翻译成数学模型的语言就是，二者归一化，特征化以后的加权求和作为总的综合优化目标（优化目标第一层转化），能够用的作为方案的决策变量也很简单，n个大坝的数量和位置，当然每个大坝的能力，造价，水流环境变化等音素可以作为参数假设为定值以免去处理，简化问题！（如果有能力异化每个大坝的构建，花费是减弱假设提高模型模拟能力，调整水流环境变化为灵敏度分析！）题目最后的难点在水流的物理建模模拟，以测算怎样的大坝组合可以达到多少年一遇的洪水，这是流体大坝的经典物理建模。（连接决策变量与子优化目标，得到最终整体优化表达式）最后接上前面的综合评价模型计算该状况下优化目标的值，小小优化一下，即可得到最终结论。

最后提醒一下，有了modeling，一定要有完整的答案作为solution哦，并且注意题干最后一段对排版的限制，此题就说到这里，祝大家好运！

Problem B: Merge After Toll(imp1)

Multi-lane divided limited-access toll highways use “ramp tolls” and “barrier tolls” to collect tolls from motorists.(bk1) A ramp toll is a collection mechanism at an entrance or exit ramp to the highway and these do not concern us here. A barrier toll is a row of tollbooths placed across the highway, perpendicular to the direction of traffic flow.(bk1’) There are usually (always) more tollbooths than there are incoming lanes of traffic (see former 2005 MCM Problem B). So when exiting the tollbooths in a barrier toll, vehicles must “fan in” from the larger number of tollbooth egress lanes to the smaller number of regular travel lanes.(bk1’’) A toll plaza is the area of the highway needed to facilitate the barrier toll, consisting of the fan-out area before the barrier toll, the toll barrier itself, and the fan-in area after the toll barrier.(bk1’’’) For example, a three-lane highway (one direction) may use 8 tollbooths in a barrier toll. After paying toll, the vehicles continue on their journey on a highway having the same number of lanes as had entered the toll plaza (three, in this example).(rsc1)

Consider a toll highway having L lanes of travel in each direction and a barrier toll containing B tollbooths (B > L) in each direction. Determine the shape, size, and merging pattern of the area following the toll barrier in which vehicles fan in from B tollbooth egress lanes down to L lanes of traffic.(spm1) Important considerations to incorporate in your model include accident prevention, throughput (number of vehicles per hour passing the point where the end of the plaza joins the L outgoing traffic lanes), and cost (land and road construction are expensive).(imp1) In particular, this problem does not ask for merely a performance analysis of any particular toll plaza design that may already be implemented. The point is to determine if there are better solutions (shape, size, and merging pattern) than any in common use.(spm1’)

Determine the performance of your solution in light and heavy traffic.(rsc1) How does your solution change as more autonomous (self-driving) vehicles are added to the traffic mix?(spm1’’) How is your solution affected by the proportions of conventional (human-staffed) tollbooths, exact-change (automated) tollbooths, and electronic toll collection booths (such as electronic toll collection via a transponder in the vehicle)?(spm1’’’)

Your MCM submission should consist of a 1 page Summary Sheet, a 1-2 page letter to the New Jersey Turnpike Authority(mss1), and your solution (not to exceed 20 pages) for a maximum of 23 pages. Note: The appendix and references do not count toward the 23 page limit.

这道题披着2005mcm收费站问题的外衣，考察的是2014mcm车道安排问题里智能交通管理思想的延伸。整个题目只有一个大模型和背景，却考察全面，不断深入，是美赛延续下来的经典系列。

文中spm1处determine暗示出本题的主题模型类型为优化问题，背景显然是交通类的规划，值得注意的是，2005年题目中的决策变量在这里变成了给定的未知参数，不再需要优化，而只要做灵敏度分析而已。

X：shape, size, and merging pattern，这里提到的几个决策变量都是需要进一步参数化后再量化求解的，注意方案设计和实际问题的吻合合理，符合常识的决策范围空间的设定，尤其是merging pattern这一点需要方案式的设计；

Y：accident prevention, throughput and cost，这里比2005新增了事故的决策目标，增加了问题的复杂度和难度，不过思路本质一致，无非是多目标归一化特征化后加权求和作为终极优化目标求解；

F：关于不同地形状，大小，通行方案，需要花费多少，车流会如何运行，最后会导致怎样的事故率和通行效率，车流模拟仍然可以采用元胞自动机，流体力学，排队论等模型去建模。注意题目特意强调了车流量的变化，你的模型在各种车流量状态下的效用水平的期望综合得分才是最终优化目标。最后，F还应该根据条件不同予以不同设计：a. 如果是单个车辆程序化的自动驾驶车辆，b. 如果收费站有不同的模式和效率，c. 如果整个系统信息共通，可以一同管理调度，那么我们的建模在a上可以增加对车辆管理的规范性约束，c可以联动所有车一同调度和控制，b也是merging pattern的一部分，这里可以看作模型的拓展延伸，当在比如元胞自动机的模型框架下，我们应当对应建立a，c不一样的汽车行文随即控制模型和考虑b作为新增的决策变量来优化最终的结果。文章要和提问的方式一样，从简单到复杂，体现递进式的逻辑结构。

Parameter：B，L，记得做灵敏度分析，而不是优化这两个参数，他们的性质是给定的未知参数。

不知道大家的思路有没有根据以上分析变得清晰明确？最后的letter mission不能忘记，建模好运！

Problem C: “Cooperate and navigate”

Traffic capacity is limited in many regions of the United States due to the number of lanes of roads.(pr1) For example, in the Greater Seattle area drivers experience long delays during peak traffic hours because the volume of traffic exceeds the designed capacity of the road networks.(pr1’) This is particularly pronounced on Interstates 5, 90, and 405, as well as State Route 520, the roads of particular interest for this problem.(pr1’’)

Self-driving, cooperating cars have been proposed as a solution to increase capacity of highways without increasing number of lanes or roads.(bk1) The behavior of these cars interacting with the existing traffic flow and each other is not well understood at this point.(pr2)

The Governor of the state of Washington has asked for analysis of the effects of allowing self-driving, cooperating cars on the roads listed above in Thurston, Pierce, King, and Snohomish counties. (See the provided map and Excel spreadsheet).(spm1) In particular, how do the effects change as the percentage of self-driving cars increases from 10% to 50% to 90%?(spm1’) Do equilibria exist? Is there a tipping point where performance changes markedly? Under what conditions, if any, should lanes be dedicated to these cars? Does your analysis of your model suggest any other policy changes?(imp1s)

Your answer should include a model of the effects on traffic flow of the number of lanes, peak and/or average traffic volume, and percentage of vehicles using self-driving, cooperating systems.(spm1’’) Your model should address cooperation between self-driving cars as well as the interaction between self-driving and non-self-driving vehicles.(imp2) Your model should then be applied to the data for the roads of interest, provided in the attached Excel spreadsheet.(rsc1)

Your MCM submission should consist of a 1 page Summary Sheet, a 1-2 page letter(mss1) to the Governor’s office, and your solution (not to exceed 20 pages) for a maximum of 23 pages. Note: The appendix and references do not count toward the 23 page limit.

Some useful background information:(rsc2s)

· On average, 8% of the daily traffic volume occurs during peak travel hours.

· The nominal speed limit for all these roads is 60 miles per hour.

· Mileposts are numbered from south to north, and west to east.

· Lane widths are the standard 12 feet.

· Highway 90 is classified as a state route until it intersects Interstate 5.

· In case of any conflict between the data provided in this problem and any other source, use the data provided in this problem.

Definitions:(rsc3s)

milepost: A marker on the road that measures distance in miles from either the start of the route or a state boundary.

average daily traffic: The average number of cars per day driving on the road.

interstate: A limited access highway, part of a national system.

state route: A state highway that may or may not be limited access.

route ID: The number of the highway.

increasing direction: Northbound for N-S roads, Eastbound for E-W roads.

decreasing direction: Southbound for N-S roads, Westbound for E-W roads.

一个道路设计交通问题还不够？那就再来一个自动驾驶强化版的交通规划问题，这种一个纯物理，两个社会规划类问题的MCM改版后的模式相信大家越来越熟悉了，体现了对现代社会类问题的重视和经典方法在新场景问题下的灵活运用，这也是未来能够学而有所用的方向指引哦~

开篇提到了交通拥堵的problem同时自己给了解决方案自动驾驶，自问自答以后重点就不在对model可决策变量的优化，而在过程的函数表达与仿真了。在允许一定量的自动驾驶汽车进入交通系统以后道路运载能力变化是以参数之一对系统作灵敏度分析的要点，对这一过程的仿真模拟仍然可以采用经典的元胞自动机方法或流体模型，由于没有服务过程，故排队论建模的抽象策略不再可取。唯一有一点点优化意味的可能仅在于对cooperation方案的小优化（用定方案带入也不算错，但是优化了可能是亮点），而number of lanes, peak and/or average traffic volume, and percentage of vehicles using self-driving, cooperating systems都是模型考虑参数化输入，都是要灵敏度分析的测试点。由此模拟而得到结果道路运载力的结果仿真值，而重点不在于优化，不过构建函数的过程和优化问题并无两样，是一个规划类的问题建模。

还是梳理一下模型的各个要素供大家参考：

X：percentage of vehicles using self-driving，cooperating systems

Parameter：number of lanes, peak and/or average traffic volume

Y：capacity

F：对引入一定比例自动驾驶汽车以后车辆运行方式的模拟，得到整个道路在各种X，parameter影响下的Y值，元胞自动机和流体力学模型都可以使用。

最后提醒下imp1s里的几个关于参数灵敏度分析后对结果的横向比较的等价优化过程结果方向的暗示以及对使用自动驾驶车比例的决策建议，还有rsc2，3s里的变相条件要全然满足不要自由发挥，最后记得提交letter结果作为solution的组成部分。