2024年MCM/ICM美国大学生数学建模竞赛B题中英版发布

2024年美赛竞赛时间为2024年2月2日早晨6点-2月6日早晨9点，2024年MCM/ICM美国大学生数学建模竞赛B题中英版一起来看看。

中文赛题：搜寻潜水器

希腊的海上巡游迷你潜艇公司（MCMS）制造能够将人类运送到海洋最深处的潜水器。潜水器被移动到位置并从主船上解开束缚后部署。MCMS现在希望使用他们的潜水器带领游客探索爱奥尼亚海底的沉船冒险。然而，在他们能够做到这一点之前，他们需要通过开发安全程序以防潜水器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，来赢得监管机构的批准。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。不同于陆地上或海面上的典型搜索与救援，有缺陷的潜水器可能会发现自己位于海底或在水下的某个中性浮力点。它的位置可能进一步受到海流、海水中不同密度和/或海底地理的影响。

你的任务是：

·定位 - 开发一个或多个模型预测潜水器随时间的位置。

这些预测有哪些不确定性？

潜水器在事故发生前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，你会推荐公司在主船上携带哪些额外的搜索设备？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。如果需要，救援船需要带来什么额外的设备以协助？

·搜索 - 开发一个模型，使用你的定位模型的信息来推荐初始部署点和搜索模式，以尽量减少定位丢失潜水器的时间。确定随时间和累积搜索结果找到潜水器的概率。

·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大致区域移动时，你的模型将如何改变？

准备一份不超过25页的报告，提供你的计划细节。包括一份两页的备忘录，寄给希腊政府，以帮助赢得批准。

你的PDF解决方案不得超过总共25页，应包括：

一页摘要。

目录。

你的完整解决方案。

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小页数。你可以使用最多25页来展示你的解决方案工作和任何你想包含的额外信息（例如：绘图、图表、计算、表格）。接受部分解决方案。我们允许谨慎使用AI，如ChatGPT，尽管不必须通过AI创建解决方案。如果你选择使用生成式AI，你必须遵循COMAP的AI使用政策。这将导致你必须在PDF解决方案文件的末尾添加一个额外的AI使用报告，且不计入你的解决方案的总页数25页。

术语表

潜水器：潜水器是一种需要由更大的水上交通工具或平台运输和支持的水下车辆。这区分了潜水器和潜艇，后者是自给自足的，能够在海上进行长时间独立操作。

中性浮力发生在物体的平均密度等于它所浸没的流体的密度时，导致浮力与否则会使物体沉没（如果物体的密度大于它所浸没的流体的密度）或上升（如果它较轻）的重力平衡。具有中性浮力的物体既不会下沉也不会上升。

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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.