中文賽題：搜尋潛水器

希臘的海上巡遊迷你潛艇公司（MCMS）製造能夠將人類運送到海洋最深處的潛水器。潛水器被移動到位置並從(cong) 主船上解開束縛後部署。MCMS現在希望使用他們(men) 的潛水器帶領遊客探索愛奧尼亞(ya) 海底的沉船冒險。然而，在他們(men) 能夠做到這一點之前，他們(men) 需要通過開發安全程序以防潛水器與(yu) 主船失去通信和可能的機械故障包括潛水器推進力喪(sang) 失的情況，來贏得監管機構的批準。特別是，他們(men) 希望你開發一個(ge) 模型來預測潛水器隨時間的位置。不同於(yu) 陸地上或海麵上的典型搜索與(yu) 救援，有缺陷的潛水器可能會(hui) 發現自己位於(yu) 海底或在水下的某個(ge) 中性浮力點。它的位置可能進一步受到海流、海水中不同密度和/或海底地理的影響。你的任務是：

• 定位- 開發一個(ge) 或多個(ge) 模型預測潛水器隨時間的位置。

  • 這些預測有哪些不確定性？
  • 潛水器在事故發生前可以定期向主船發送哪些信息以減少這些不確定性？潛水器需要哪些設備才能做到這一點？

• 準備- 如果需要，你會(hui) 推薦公司在主船上攜帶哪些額外的搜索設備？你可以考慮不同類型的設備，但也必須考慮這些設備的可用性、維護、準備和使用成本。如果需要，救援船需要帶來什麽(me) 額外的設備以協助？

• 搜索- 開發一個(ge) 模型，使用你的定位模型的信息來推薦初始部署點和搜索模式，以盡量減少定位丟(diu) 失潛水器的時間。確定隨時間和累積搜索結果找到潛水器的概率。

• 推斷- 你的模型如何擴展以考慮其他旅遊目的地，如加勒比海？當多個(ge) 潛水器在同一大致區域移動時，你的模型將如何改變？

準備一份不超過25頁的報告，提供你的計劃細節。包括一份兩(liang) 頁的備忘錄，寄給希臘政府，以幫助贏得批準。

你的PDF解決(jue) 方案不得超過總共25頁，應包括：

• 一頁摘要。
• 目錄。
• 你的完整解決方案。
• 參考文獻列表。
• AI使用報告（如果使用不計入25頁限製內）。

注意：對於(yu) 完整的MCM提交沒有具體(ti) 要求的最小頁數。你可以使用最多25頁來展示你的解決(jue) 方案工作和任何你想包含的額外信息（例如：繪圖、圖表、計算、表格）。接受部分解決(jue) 方案。我們(men) 允許謹慎使用AI，如ChatGPT，盡管不必須通過AI創建解決(jue) 方案。如果你選擇使用生成式AI，你必須遵循COMAP的AI使用政策。這將導致你必須在PDF解決(jue) 方案文件的末尾添加一個(ge) 額外的AI使用報告，且不計入你的解決(jue) 方案的總頁數25頁。

術語表

潛水器：潛水器是一種需要由更大的水上交通工具或平台運輸和支持的水下車輛。這區分了潛水器和潛艇，後者是自給自足的，能夠在海上進行長時間獨立操作。

中性浮力發生在物體(ti) 的平均密度等於(yu) 它所浸沒的流體(ti) 的密度時，導致浮力與(yu) 否則會(hui) 使物體(ti) 沉沒（如果物體(ti) 的密度大於(yu) 它所浸沒的流體(ti) 的密度）或上升（如果它較輕）的重力平衡。具有中性浮力的物體(ti) 既不會(hui) 下沉也不會(hui) 上升。

Problem B: Searching for Submersibles

Maritime Cruises Mini-Submarines (MCMS), a company based in Greece, builds submersibles capable of carrying humans to the deepest parts of the ocean. A submersible is moved to the location and deployed untethered from a host ship. MCMS now wishes to use their submersible to take tourists on adventures exploring the bottom of the Ionian Sea for sunken shipwrecks. Before they can do this, however, they need to win approval from regulators by developing safety procedures in case of a loss of communication to the host ship and possible mechanical defects including a loss of propulsion of the submersible. In particular, they would like you to develop a model to predict the location of the submersible over time. Unlike in a typical search and rescue on land or on the surface of a sea, the defective submersible could potentially find itself positioned on the sea floor or at some point ofneutral buoyancyunderwater. Its position could further be affected by currents, differing densities in the sea, and/or the geography of the sea floor. Your tasks are to:

• Locate- Develop a model(s) that predicts the location of the submersible over time.

  • What are the uncertainties associated with these predictions?
  • What information can the submersible periodically send to the host ship to decrease these uncertainties prior to an incident? What kinds of equipment would the submersible need to do so?

• Prepare- What, if any, additional search equipment would you recommend the company carry on the host ship to deploy if necessary? You may consider different types of equipment but must also consider costs associated with availability, maintenance, readiness, and usage of this equipment. What additional equipment might a rescue vessel need to bring in to assist if necessary?

• Search- Develop a model that will use information from your location model(s) to recommend initial points of deployment and search patterns for the equipment so as to minimize the time to location of a lost submersible. Determine the probability of finding the submersible as a function of time and accumulated search results.

• Extrapolate- How might your model be expanded to account for other tourist destinations such as the Caribbean Sea? How will your model change to account for multiple submersibles moving in the same general vicinity? Prepare a report of no more than 25 pages providing the details of your plan. Include a two-page memo of your report addressed to the Greek government to help win approval.

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

• One-page Summary Sheet.
• Table of Contents.
• Your complete solution.
• 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 MCM 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.

Glossary

Submersible: A submersible is an underwater vehicle which needs to be transported and supported by a larger watercraft or platform. This distinguishes submersibles from submarines, which are self-supporting and capable of prolonged independent operation at sea.

Neutral buoyancyoccurs when an object's average density is equal to the density of the fluid in which it is immersed, resulting in the buoyant force balancing the force of gravity that would otherwise cause the object to sink (if the body's density is greater than the density of the fluid in which it is immersed) or rise (if it is less). An object that has neutral buoyancy will neither sink nor rise.