2024年美国大学生数学建模竞赛B题中英版

中文赛题：搜寻潜水器

希腊的海上巡游迷你潜艇公司（MCMS）制造能够载人深入海洋最深处的潜水器。潜水器被移动到位置并从主船上无缆绳部署。MCMS现在希望利用他们的潜水器带领游客探险爱奥尼亚海底寻找沉船残骸。然而，在他们能够做到这一点之前，他们需要通过制定安全程序来赢得监管机构的批准，以应对与主船失去通信和可能的机械故障，包括潜水器推进力丧失的情况。特别是，他们希望你开发一个模型来预测潜水器随时间的位置。与陆地或海面上的典型搜救不同，故障潜水器可能会发现自己位于海底或水下某个中性浮力点。其位置还可能受到海流、海水不同密度和/或海底地理的影响。你的任务是：

• 定位 - 开发一个模型（或多个模型）预测潜水器随时间的位置。

  • 这些预测的不确定性有哪些？

  • 潜水器可以定期向主船发送哪些信息以减少这些不确定性？潜水器需要哪些设备才能做到这一点？

• 准备 - 如果需要，你会建议公司在主船上携带哪些额外的搜寻设备？你可以考虑不同类型的设备，但也必须考虑这些设备的可用性、维护、准备和使用成本。救援船只可能需要携带哪些额外设备以协助搜救？

• 搜寻 - 开发一个模型，利用你的定位模型的信息来推荐部署点和搜索模式，以最小化定位丢失潜水器的时间。

  确定随时间和累积搜索结果而变化的找到潜水器的概率。

• 推断 - 你的模型如何扩展以考虑其他旅游目的地，如加勒比海？当多个潜水器在同一区域活动时，你的模型将如何变化？准备一份不超过25页的报告，详细说明你的计划。包括一份两页的备忘录，地址给希腊政府，以帮助赢得批准。

您的PDF解决方案不得超过25页，应包括：

• 一页摘要表。

• 目录。

• 您的完整解决方案。

• 参考文献列表。

• AI使用报告（如果使用，不计入25页限制内）。

注意：对于完整的MCM提交，没有具体要求的最低页数。您可以使用多达25页来展示您的解决方案工作和任何您想包含的额外信息（例如：图画、图表、计算、表格）。我们接受部分解决方案。我们允许谨慎使用AI，如ChatGPT，虽然不是解决这个问题的必需条件。如果您选择使用生成性AI，必须遵守COMAP AI使用政策。这将导致您必须在PDF解决方案文件末尾添加额外的AI使用报告，且不计入解决方案的25页总限制内。

术语表

潜水器：潜水器是一种水下车辆，需要由更大的水上船只或平台运输和支持。这与潜艇不同，潜艇是自给自足的，能够在海上进行长时间的独立操作。

中性浮力发生在物体的平均密度等于其所处流体的密度时，导致浮力平衡重力，否则物体会沉没（如果物体的密度大于其所处流体的密度）或上升（如果密度较低）。具有中性浮力的物体既不会沉没也不会上升。

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 of neutral buoyancy underwater. 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 buoyancy occurs 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.