2024年HIMCM美國中學數學建模競賽C題

中文賽題：音樂巡演建模

注意：隻有所有成員年齡都小於(yu) 14歲半的參賽隊才能選擇問題 C。

泰勒·斯威夫特（Taylor Swift）將在 12 月結束創紀錄的 Eras 巡演。整個(ge) 巡演包括 152 場演出，每場約 3.5 小時。在美國，約有 72,000 名觀眾(zhong) 出席每場演出，單場收入達 1300 萬(wan) 美元。Eras巡演成為(wei) 首個(ge) 收入超過 10 億(yi) 美元的巡演。然而，許多粉絲(si) 發現買(mai) 票並非易事，票價(jia) 高達數千美元且售罄迅速。此外，Taylor Swift 對其團隊成員非常慷慨，給予樂(le) 隊成員、舞者、燈光和音響技術員、餐飲人員、卡車司機和其他工作人員豐(feng) 厚的獎金。但並非所有音樂(le) 人的巡演都如 Eras 巡演般極端。

你是 MidMCM 音樂(le) 經紀公司的巡演經理。一位音樂(le) 人（可以是 Taylor Swift 這樣的個(ge) 人，或像 iKON 這樣的團體(ti) ）向你尋求幫助，計劃其下一次巡演。你的團隊需要分析這位音樂(le) 人的過往巡演，以建立一個(ge) 模型，幫助規劃下一次巡演。

任務

1. 開始：熟悉音樂人及其巡演曆史。
   • 選擇一位有過巡演的音樂人。
     • 他們為什麽要巡演？
     • 巡演持續多長時間？
     • 一場巡演包含多少場演出？
     • 他們巡演的地點有哪些？
   • 研究音樂人一次過去的巡演，確定巡演計劃中應考慮的因素：
     • 運行單場演出需要考慮哪些因素？
     • 計劃多場巡演需要考慮哪些因素？
     • 成功的巡演有哪些關鍵因素
2. 建立模型：開發一個模型，幫助規劃音樂人的巡演。(提醒：您可以對您認為需要但找不到的任何信息進行假設 ）。
   • 你可以考慮（但不限於）以下方麵：
     • 演出場次
     • 場地選擇
     • 票價
   • 製定一個“最佳”巡演計劃，使其達成音樂人的目標，例如：
     • 最大化觀眾人數
     • 最大化場地覆蓋率
     • 最大化利潤
     • 其他目標？
3. 應用模型：分析過去並規劃未來。
   • 使用你的模型分析所研究的過去巡演。
     • 識別巡演的優缺點。
     • 巡演整體是成功還是失敗？
   • 使用模型幫助規劃音樂人的未來巡演。
     • 模型可以在哪些方麵幫助規劃未來巡演？
     • 模型建議音樂人未來巡演該如何安排？
4. 分享模型：寫一封1-2頁的信給音樂人，描述模型幫助規劃的可能新巡演，解釋該巡演如何最能滿足其目標。
5. 反思：你的模型是否能幫助規劃其他音樂人的巡演？適用於所有音樂人嗎？

你的PDF解決(jue) 方案總頁數不超過25頁，包括：

• 一頁的總結。
• 目錄。
• 完整解決方案。
• 一至兩頁的信件。
• 參考文獻列表。
• AI使用報告（如果使用，不計入25頁限製）。

注意：在 MidMCM 提交中，並沒有要求最低頁數限製。你的解決(jue) 方案及其他附加信息（例如：圖畫、圖表、計算、表格）總共可以使用最多25頁。接受部分解決(jue) 方案。我們(men) 允許在解決(jue) 問題時謹慎使用生成式 AI 工具（如 ChatGPT），但並不是必須使用。如果選擇使用生成式 AI，你必須遵循 COMAP 的 AI 使用政策。這將需要在提交文件末尾添加一份 AI 使用報告，但該報告不計入解決(jue) 方案的 25 頁限製。

HiMCM/MidMCM最新：在線提交流程

本文旨在幫助和指導參與(yu) HiMCM/MidMCM 的學生和指導老師。COMAP 在文中提供了關(guan) 於(yu) 新的在線提交流程的信息，需使用新的在線提交頁麵（提交鏈接）。完成提交時，你需要準備好團隊控製編號、指導老師ID和所選問題編號。

術語表

• 假設：用於替代未知或不確定信息的假設或推測。

MidMCM 指南

COMAP提供了一篇法官評論文章和 2022 年 MidMCM 競賽的優(you) 秀論文。這些資源為(wei) 指導老師和學生提供了指導。以下是關(guan) 於(yu) MidMCM 提交結構的一般建議：

所有解決(jue) 方案必須以PDF格式提交，內(nei) 容整合在一個(ge) PDF文件中。MidMCM團隊可以手寫(xie) 數學公式、圖表、表格、草圖等，並將其照片包含在PDF中。隨著學生進入高中和HiMCM，我們(men) 期望提交的文件為(wei) 電子版。MidMCM允許指導老師在技術上協助學生將各部分整合為(wei) 一個(ge) PDF文件。

與(yu) HiMCM類似，提交文件的頁數上限為(wei) 25頁，但不要求達到25頁，較短的提交也是可以接受的。提交文件的所有部分（包括文本、圖表、表格、圖片等）必須包含在一個(ge) 不超過25頁的PDF文件中。接受部分解決(jue) 方案。若使用了生成式AI（如ChatGPT），則需在解決(jue) 方案後附上AI使用報告，但該報告不計入25頁限製。

完整的解決(jue) 方案提交一般包含以下部分：

• 執行摘要：在完成所有工作後編寫的一頁摘要，為文件的第一頁，概述工作內容並包含實際結果。
• 目錄：列出文件中的主要項目，展示文件的結構。
• 問題引言和重述：介紹問題，用自己的語言重述問題和要求。
• 假設及其依據：列出為簡化和解決問題而做出的假設，並解釋做出這些假設的原因。
• 變量定義：定義在模型和方程中使用的所有變量。
• 模型和解決方案的展示：確保滿足所有要求，詳細描述解決問題的過程，展示並解釋所有工作。使用適當的表現形式（如方程、表格、圖表、圖片等）幫助讀者理解解決過程。
• 分析：評估模型和解決方案的優勢（優點）和局限性（缺點）。
• 結論段落：在文件末尾寫一個總結段落，概述結果和/或提出未來工作的建議。
• 參考文獻：列出所有用於解決問題的來源（例如：網頁、報紙或雜誌文章等）。
• AI使用報告：若使用了生成式AI（如ChatGPT），需按照指導附上AI使用報告。該報告附在提交文件最後部分，不計入25頁的限製。

Problem: The Modeling Musical Tour

Note: Only teams with all members younger than 14 ½ years old may choose Problem C.

Taylor Swift will be wrapping up her record-breaking Eras Tour in December. In total, the tour schedule included 152 shows that each run around 3.5 hours long. For the US shows, approximately 72,000 fans attend each concert bringing in1billion. However, getting a ticket to a concert has been no easy feat for many fans. Tickets can be thousands of dollars and sell out quickly. Taylor Swift has also made headlines for being incredibly generous with the members of her team – giving significant bonuses to band members, dancers, lighting and sound technicians, caterers, truck drivers, and other crew members. However, not all musical performers’ tours are as extreme as The Eras Tour.

You are a tour manager at MidMCM Music Agency. A musical performer of your choice (it could be an individual performer like Taylor Swift, a group like iKON, etc.) has approached you to help them plan their next tour. Your team needs to analyze the musical performer’s past tours to develop a model to help plan their next tour.

Requirements

1. Get Started. Familiarize yourself with a musical performer and their touring history.
   • Choose a musical performer that has had a concert tour in the past.
     • Why did they go on tour?
     • How long is a tour?
     • How many shows are in a tour?
     • Where did they go on tour?
   • Investigate one of the musical performer’s past tours to identify the factors that should go into planning a tour. Some things to consider:
     • What factors need to be considered to run a single show on a tour?
     • What factors need to be considered when planning multiple shows for a tour?
     • What makes a successful music tour?
2. Create your model. Develop a model to help plan a tour for the musical performer. (Reminder: You can make Assumptions about any information you think you need and can not find.)
   • Some things you may consider include, but are not limited to:
     • Number of Concerts
     • Venues
     • Ticket Prices
   • What will be the “best” plan for the performer to meet their goals for the tour?
     • Maximizing fan attendance?
     • Maximize venues?
     • Maximizing profits?
     • Something else?
3. Apply your model. Analyze the past and plan for the future.
   • Use your model to analyze the past tour you investigated.
     • Identify the strengths and weaknesses of their tour.
     • Was the tour an overall success or failure?
   • Use your model to help plan a future tour for the musical performer.
     • What aspects of a future tour can your model help with?
     • What does your model suggest the performer should do on a future tour?
4. Share your model. Write a one- to two- page letter to the musical performer that describes a possible new tour that your model has helped plan. Explain why this tour will best meet the goals of going on tour for the musical performer.
5. Reflect. How might you be able to use your model to help plan a tour for another musical performer? Will it work for every performer?

Your PDF solution of no more than 25 total pages should include:

• One-page Summary Sheet.
• Table of Contents.
• Your complete solution.
• One- to two-page IOC letter
• References list.
• AI Use Report (If used does not count toward the 25-page limit.)

Note: There is no specific required minimum page length for a complete HiMCM submission. You may use up to 25 total pages for all your solution work and any additional information you want to include (for example: drawings, diagrams, calculations, tables). Partial solutions are accepted. We permit the careful use of AI such as ChatGPT, although it is not necessary to create a solution to this problem. If you choose to utilize a generative AI, you must follow the COMAP AI use policy. This will result in an additional AI use report that you must add to the end of your PDF solution file and does not count toward the 25 total page limit for your solution.

NEW HiMCM/MidMCM: Online Submission Process The purpose of this article is to assist and guide students and advisors participating in HiMCM/MidMCM. In the article, COMAP, provides information about the new online submission process using the new online submission page https://forms.comap.org/242386224483964. You will need your team's control number, advisor id number and your problem choice to complete your submission.

Glossary

• Assumptions: hypotheses or educated guesses that take the place of an unknown or uncertain piece of information.

Guidance for MidMCM

COMAP has a Judges’ Commentary article along with the Outstanding papers from the 2022 MidMCM contest available. The commentary article provides guidance to both advisors and students. We also provide the following general guidance about MidMCM submission organization.

Solutions must be in PDF format and submitted in one PDF document. This does not preclude MidMCM teams from doing mathematics, graphs, tables, sketches, etc. by hand and including pictures of their work in the single PDF document submission. As students move to high school and the HiMCM, we expect that submissions will be typed. For the MidMCM, advisors may technically assist students in putting their solution components into one PDF format file for submission.

As with HiMCM, there is a 25-page limit for the submission document. This does not mean your solution must be 25 pages. A shorter submission is certainly acceptable. All portions of your submission (text, graphs, tables, charts, pictures, etc.) must be within one PDF document that is 25 pages or less. We accept partial solutions. Then, if you have utilized any artificial intelligence programs, such as ChatGPT, you will append an AI Use Report after your solution – that report does not count towards the 25-page limit on the solution portion of your PDF.

In general, a complete solution submission is organized as follows:

• Executive Summary – Write this summary after you have done all your work. This one-page summary is Page #1 of your solution document. It provides an overview of your work and includes actual results.
• Table of Contents – List the major items in your solution document to show the organization of your paper.
• Introduction and Restatement of the Problem – Introduce the problem. Restate the problem and requirements in your own words.
• Assumptions with Justifications – State any assumptions you made to simplify and solve the problem and state why you made those assumptions.
• Variable Definitions – Define any variables you use in your model and equations.
• Presentation of Model and Solution – Ensure you address all requirements and describe what you are doing in solving the problem. Show and explain all your work. Use representations that help you tell the reader how you solved the problem (for example: equations, tables, graphs, pictures, etc.).
• Analysis of Your Work – Address any strengths (good points) and limitations (weaknesses) of your model and solution.
• Concluding Paragraph – End your solution paper with a final concluding paragraph that summarizes your results and/or makes recommendations for future work.
• Reference List – List any sources that you used to solve the problem (for example, website pages, newspaper or magazine articles, etc.).
• AI Use Report – This only applies if you used generative AI systems such as ChatGPT. Follow the guidance provided here. These additional pages do not count toward the 25-page limit for the rest of your submission.