2018 MCM Problem B C Notes

2018 MCM Problem B: How Many Languages?(pr1)

Background(bk1s): There are currently about 6,900 languages spoken on Earth. About half the world’s population claim one of the following ten languages (in order of most speakers) as a native language: Mandarin (incl. Standard Chinese), Spanish, English, Hindi, Arabic, Bengali, Portuguese, Russian, Punjabi, and Japanese. However, much of the world’s population also speaks a second language. When considering total numbers of speakers of a particular language (native speakers plus second or third, etc. language speakers), the languages and their order change from the native language list provided. The total number of speakers of a language may increase or decrease over time because of a variety of influences(imp1) to include, but not limited to, the language(s) used and/or promoted by the government in a country, the language(s) used in schools, social pressures, migration and assimilation of cultural groups, and immigration and emigration with countries that speak other languages. Moreover, in our globalized, interconnected world there are additional factors that allow languages that are geographically distant to interact.(imp2) These factors include international business relations, increased global tourism, the use of electronic communication and social media, and the use of technology to assist in quick and easy language translation.

Retrieved from https://en.wikipedia.org/wiki/List_of_languages_by_total_number_of_speakers on January 17, 2018.

Problem: A large multinational service company, with offices in New York City in the United States and Shanghai in China, is continuing to expand to become truly international. This company is investigating opening additional international offices and desires to have the employees of each office speak both in English and one or more additional languages. The Chief Operating Officer of the company has hired your team to investigate trends of global languages and location options for new offices(pr2).

Part I:

A. Consider the influences and factors described in the background paragraph above, as well as other factors your group may identify. Based on projected trends, and some or all of these influences and factors, model the distribution of various language speakers over time. (spm1)

B. Use your model to predict what will happen to the numbers of native speakers and total language speakers in the next 50 years.(spm2) Do you predict that any of the languages in the current top-ten lists (either native speakers or total speakers) will be replaced by another language? Explain. (mss1)

本题是一道预测类问题，以语言的使用人数为背景。还记得预测类问题寻找变量的两个类型和得到表达式的两个思路吗？都用上以后本题就迎刃而解了。

Y: distribution of language speakers, 每个人对语言选择是个多选而且有权重的值，因为第二，三语言的存在，所以f(p, l)为人p对语言l的掌握程度（有掌握/不掌握区分以及掌握程度差别），对每个人排序后即为他的第一二三语言，而语言的分布则为掌握人数以及掌握者排序值的分布，都可以从f(p, l)计算得来。而每个人对语言的f(p, l)可以根据x: factors 计算出来。

X: government in a country, the language(s) used in schools, social pressures, migration and assimilation of cultural groups, and immigration and emigration; interconnected world; business relations, increased global tourism, the use of electronic communication and social media, and the use of technology; 以及其他能够考虑到的因素，把这些自变量纳入考虑的前提是，能够获取相应的数据，否则不要自找麻烦。

F: 考虑以上变量X对Y增量或存量的影响，建立回归关系并估计系数得到答案。这是关联预测，也可以独立预测用Y历史数据直接推断；表达式方面，基于统计的回归关系可行，而从机理上考虑每一个变量可能的作用方式再汇总也很好。

Spm1只要求对静态的Y，X关系进行估计，而spm2则是Y(t) = f(t, Y(t - k), X(t))的预测推演，注意这两者的回答重点的区别。

最后，mss1可以由spm2的结果予以回答。只要细致小心，自圆其说，本题难度并不大，想出彩还是要费一番功夫。

万变不离其宗，这是一道和2012 mcm A题数树叶问题和2010 mcm B题犯罪地点预测问题一脉相承的题目，改变了研究的对象主体，但是其数学模型本质却是一致的，这便是学习能够在脑海里留下的价值。

2018 MCM Problem C: Energy Production (bk1)

Background(bk1s): Energy production and usage are a major portion of any economy. In the United States, many aspects of energy policy are decentralized to the state level. Additionally, the varying geographies and industries(imp1) of different states affect energy usage and production. In 1970, 12 western states in the U.S. formed the Western Interstate Energy Compact (WIEC), whose mission focused on fostering cooperation between these states for the development and management of nuclear energy technologies. An interstate compact is a contractual arrangement made between two or more states in which these states agree on a specific policy issue and either adopt a set of standards or cooperate with one another on a particular regional or national matter.

Problem: Along the U.S. border with Mexico, there are four states – California (CA), Arizona (AZ), New Mexico (NM), and Texas (TX) – that wish to form a realistic new energy compact focused on increased usage of cleaner, renewable energy sources. Your team has been asked by the four governors of these states to perform data analysis and modeling to inform their development of a set of goals for their interstate energy compact.

The attached data file “ProblemCData.xlsx” provides in the first worksheet (“seseds”) 50 years of data in 605 variables on each of these four states’ energy production and consumption, along with some demographic and economic information. The 605 variable names used in this dataset are defined in the second worksheet (“msncodes”).

Part I:

A. Using the data provided, create an energy profile for each of the four states. (spm1)

B. Develop a model to characterize how the energy profile of each of the four states has evolved from 1960 – 2009.(spm2) Analyze and interpret the results of your model to address the four states’ usage of cleaner, renewable energy sources in a way that is easily understood by the governors and helps them to understand the similarities and difference between the four states. Include in your discussion possible influential factors of the similarities and differences (e.g. geography, industry, population, and climate).

C. Determine which of the four states appeared to have the “best” profile for use of cleaner, renewable energy in 2009. Explain your criteria and choice. (spm3)

D. Based on the historical evolution of energy use in these states, and your understanding of the differences between the state profiles you established, predict the energy profile of each state, as you have defined it, for 2025 and 2050 in the absence of any policy changes by each governor’s office.(spm4)

Part II:

A. Based on your comparison between the four states, your criteria for “best” profile, and your predictions, determine renewable energy usage targets for 2025 and 2050 and state them as goals for this new four-state energy compact. (spm1234’)

B. Identify and discuss at least three actions the four states might take to meet their energy compact goals. (mss1)

Part III: (mss2)

Prepare a one-page memo to the group of Governors summarizing the state profiles as of 2009, your predictions with regard to energy usage absent any policy changes, and your recommended goals for the energy compact to adopt.

Your submission should consist of:

· One-page Summary Sheet,

· One-page memo,

· Your solution of no more than 20 pages, for a maximum of 22 pages with your summary and memo.

· Note: Reference list and any appendices do not count toward the 22-page limit and should appear after your completed solution.

Attachments:

ProblemCData.xlsx Includes two worksheets seseds and msncodes.

References:

State Energy Data System (SEDS) Complete Dataset through 2009 (All 50 states) https://catalog.data.gov/dataset/state-energy-data-system-seds-complete-dataset-through-2009#sec-dates

这道题和近两年改版成3+3模式的MCM-ICM方式的出题逻辑比较接近，都是集预测，评价，优化三类函数考察于一体的完整任务，整体工作量大但细细拆分以后，每个具体任务，并不那么难。

Spm1的energy profile即为描述统计任务，统计的是4个州作为总体分别605个能源相关变量随时间变化的分布情况以及变化状态，并汇总加以比较，最好还能提取出特征供更多参考。

Spm2则特意强调了energy profile随时间变化的度量，可见spm1可以只对单州单国家作描述而其他可在spm2中完成，预测本身最好用独立预测，因为没有其他变量可以参考，而 geography, industry, population, and climate等这些因素可以用来关联预测以解释不同的预测结果，以此得到4个州的不同因素是如何导致数据中反映的变化趋势的。

Spm3则是需要建立评价模型汇总这个energy profile而得到总评价指标，要给出机理解释，并用这个指标优化选择出最好的州。

Spm4需要把spm2中的预测表达式推演到未来，在政策等外界因素不变的情况下综合利用独立和关联预测的方案进行，这个结果也是接下来执行层优化方案的baseline估计；

Part 2中，有了energy profile，prediction，evaluation，便可以计算出以一个基本的预算方案下，evaluation结果的分布，我们取分布的一定统计量作为发展目标即为所求。至于mss1的actions，言之成理即可，能够对应上影响evaluation的各个影响因素当然最佳。

Part3即为把内容按照memo格式递交政府，即使没有此问，这个任务也是建模的基本步骤：回答原问题，也是需要完成的，把之前计算得到的结论用人话再转述一遍即可。

这道题比B题好做在有参考数据，可以把精力锁定在建模而不是找数据上，但是考察点比B题更多更全面，需要扎实的建模基础，包括写作功底才能完整地完成整个任务。但是数据最大的问题都解决了，不靠完美的模型取胜，还靠什么呢？

本次2018 MCM试题分析到这里就结束了，欢迎继续关注一年一度的建模盛会，我在这里等你。