2020 MCM Problem A Move North Notes

本系列文章是2020MCM美赛试题解析，距离比赛已经一段时间了，补发出来，仅供参考。

2020 MCM Problem A Move North(bk1)

Global ocean temperatures affect the quality of habitats for certainocean-dwelling species.(pr2) When temperaturechanges are too great for their continued thriving, these species move to seekother habitats better suited to their present and future living andreproductive success.(pr2’) One example ofthis is seen in the lobster population of Maine, USA that is slowly migratingnorth to Canada where the lower ocean temperatures provide a more suitablehabitat.(pr2’’) This geographic population shift can significantly disrupt the livelihoodof companies who depend on the stability of ocean-dwelling species.(pr2’’’)

Your team has been hired as consultants by a Scottish North Atlanticfishery management consortium. The consortium wants to gain a betterunderstanding of issues related to the potential migration ofScottish herring and mackerel from their current habitats near Scotlandif and when global ocean temperatures increase.(imp1) These two fishspecies represent a significant economic contribution to the Scottish fishingindustry.(imp1’) Changes in population locations of herring and mackerel could make iteconomically impractical for smaller Scotland-based fishing companies, who usefishing vessels without on-board refrigeration, to harvest and deliver freshfish to markets in Scotland fishing ports,(imp1’’)

Requirements

1.      Build amathematical model to identify the most likely locations for these twofish species over the next 50 years, assuming that water temperatures are goingto change enough to cause the populations to move.(spm1)

2.      Based upon howrapidly the ocean water temperature change occurs, use your model to predictbest case, worst case, and most likely elapsed time(s) until thesepopulations will be too far away for small fishing companies to harvest if thesmall fishing companies continue to operate out of their current locations.(spm2)

3.      In light of yourpredictive analysis, should these small fishing companies make changes to theiroperations?(spm3)

·        a. If yes, use your model to identify andassess practical and economically attractive strategies for small fishingcompanies. Your strategies should consider, but not be limited to, realisticoptions that include(imp2s)

§ Relocating some or all of afishing company’s assets from a current location in a Scottish port to closerto where both fish populations are moving;

§ Using some proportion of small fishingvessels capable of operating without land-based support for a period of timewhile still ensuring the freshness and high quality of the catch.

§ Other options that your teammay identify and model.

·        b. If your team rejects the need for anychanges, justify reasons for your rejection based on your modeling results asthey relate to the assumptions your team has made.

4.      Use your model toaddress how your proposal is affected if some proportion of the fishery movesinto the territorial waters (sea) of another country.(spm3’)

5.      In addition toyour technical report, prepare a one- to two-page article for Hook Line andSinker magazine to help fishermen understand the seriousness of the problem andhow your proposed solution(s) will improve their future business prospects.(mss1)

Your submission should consist of:

·        One-page Summary Sheet

·        Table of Contents

·        One- to Two-page Article

·        Your solution of no more than 20 pages,for a maximum of 24 pages with your summary, table of contents, and article.

Note: Reference List and any appendices do not count toward the page limit andshould appear after your completed solution. You should not make use ofunauthorized images and materials whose use is restricted by copyright laws.Ensure you cite the sources for your ideas and the materials used in yourreport.

Glossary

·        Fishery: The collectionof fish of a given species and the area that they inhabit.

·        Habitat:  The typeof environment in which an organism or group normally lives or occurs.

·        Small Fishing Company:  A companyengaged in commercial fishing with limited or very limited financial resourcesto invest in new equipment/vessels.

·        Territorial Waters (sea): “as defined bythe 1982 United Nations Convention on the Law of the Sea, is a belt of coastalwaters extending at most 12 nautical miles (22.2 km; 13.8 mi) from the baseline(usually the mean low-water mark) of a coastal state. The territorial sea isregarded as the sovereign territory of the state, although foreign ships(military and civilian) are allowed innocent passage through it, or transitpassage for straits; this sovereignty also extends to the airspace over andseabed below.” [Territorial Waters. (n. In Wikipedia. Retrieved January 28,2020, from https://en.wikipedia.org/wiki/Territorial_waters.]

 

Analysis

 

看完本题，感觉美赛已经出不出什么新意了，题目的结构，背后的数学模型逻辑几乎年年一样，真是觉得背后应该是题库了。

 

很久以前美赛出过海平面上升的问题（2008 MCM A Take a Bath），曾经那篇很著名的O奖论文无非是用三个递进的模型从最简单到最复杂解决了这个估计和预测。

 

12年了，组委会又搬出了全球变暖问题，只不过上次是冰川融化，这次是气温升高不利于动物繁衍，那这又有什么区别呢？

 

第一段提出背景，标准的GRE段落展开，起因：温度上升；结果：寻找栖息地；举例说明；最后是破坏公司生计。

 

第二段有一个关键暗示：Scottish herring and mackerel是会随着温度变化而迁徙的，我们需要搞清楚这个机理。这也是后面解决问题模型的核心。

 

看问题：

1. 模型1：自变量：温度T足够的升高；因变量，鱼类迁徙位置P。显然这里只需要模拟出未来温度变化以及在不同地域分布的规律T（p，t），然后鱼类的迁徙就取所有时刻温度处在其适宜范围内的区域就可以了，注意，鱼类适宜温度T0，是个常数。是各个地方的温度（用一个连续曲面方程或者离散二阶张量表达）随时间变化，鱼类取各个时间内温度适宜的位置坐标：

 

即求所有t时候，T（p，t） = T0的关于位置p的解，当然也可能t是个范围，求p的范围。

 

2. 模型2：这里所谓最好，最坏和最可能，就是温度变化参数a取最小，最大和期望或众数的时候的模型灵敏度分析的结果。显然，公司的运营收益B是因变量，自变量是温度变化导致的鱼类活动区域（来自于模型1），把这个函数细化建模以后为B(p)，则结果为：

求p0，使得B（p）= 0，取T（p0，t，a）= T0，在·a取以上三个值的时候，t的解，即为所求。

 

3. 模型3：改变经营方式，无非涉及到需要添加的固定成本和最后增加的收益率的权衡。在某个a参数下，根据模型2已经知道原方案亏本的时间，那么，在修改经营方式以后，我们可以以一定时间为界计算整体收益和风险，取期望等来计算改变方案在某a下比较E（B（p））和E（B‘（p））的影响，由此来根据不同情况，如果a大可能有需要，a小不需要，得到类似的结论即可。其中B’是修改方案以后的方案的最优解，也是本问的模型，期望是对不同时间得到的不同p求的。

 

4. 以上模型3没有考虑这些渔业会迁徙，加入这个参数对计算B（p）的影响即可，比如当收益率到达某个临界值就会恶性循环，然后加剧问题，来凸显我们的方案重要性即可。

 

一共就这3个模型，T（p，t），求给定值T0，所有t时候的方程的解；B（p）建模，求B（p） = 0的P的解，带入T（p0， t， a） = T0，求t；建模B‘（p），计算E（B’(p) – B(p)）,来计算方案可行与否；最后，在B‘（p）中引入渔业迁徙因素升级重算。

 

注意全程对a的取值进行分布建模以后对所有模型的灵敏度分析，十分重要。

 

分析到这，同学们加油！

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