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大学生,数学建模,美国大学生数学建模竞赛,MCM/ICM,2024年美赛特等奖O奖论文
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Problem Chosen
B
2024
MCM/ICM
Summary Sheet
Team Control Number
2407038
Unlocking the Abyss: A Dynamical Model for Deep-Sea Adventure
Safety and Rescue Strategies
In the ever-growing realm of adventure tourism, the exploration of sunken shipwrecks by riding
submersibles has emerged as an exciting experience and popular activity, which allows enthusiasts to
witness the silent beauty of maritime history beneath the waves. However, due to the complexity of the
ocean current and weak communication underwater, the event that a submersible loses contact with the
host ship under the sea continues, and positioning the missing submarine and carrying out rescue will
be difficult. Predicting the trajectory of the submarine and developing the best search strategy are
problems faced by both marine administrators and mathematicians. In order to handle this challenge, we
divide the problem into three key points and establish three models separately.
Model I: Based on the knowledge of the underwater environment, the current effect, seawater density,
and seafloor topography are three main points influencing submersible movement. We first utilize the
Autoregressive Integrated Moving Average (ARIMA) Time Series Model and ridge regression to fit
the three-dimensional continuous current and seawater density distribution. According to the dynamics
function of the submersible considering complex factors, we build a model to solve these equations
and predict the trajectory of the missing submersible. The prediction results are shown in Figure 6.
Furthermore, we analyze the uncertainty of the current. After randomly input time series, it has an average
uncertainty of 2.39%.
Model II: During the preparation stage, necessary search equipment for rescue must be deployed on the
host ship and rescue vessel. There are different indicators of various types of search equipment including:
equipment cost, maintenance cost, availability, usage, and readiness, which are firstly averagely processed
and quantified. Our model uses the objective functions of minimum total cost, availability, and preparation
time. Then, a Muti-Objective Optimization is established based on a Genetic Algorithm to obtain the
Pareto set of solutions for the host ship and rescue vessel, respectively. Then, we choose three optimal
results as different equipment allocations for the host ship and rescue vessel. The final results are shown
in Table 8.
Model III: To decide the search strategy, we rasterize the region into grids and estimate the dynamic
probability distribution based on our prediction results and the Poisson distribution function. Inspired
by Bayes’s theorem, we adjust the probability distribution according to the information obtained from the
previous search. Ultimately, we divide the search into many time intervals and accumulate the probability
of finding the submersible. The results of the accumulated probability are shown in Figure 10.
Additionally, we test the scalability and sensitivity of this model. Applying the model to another
tourist destination, the Caribbean Sea, the trajectory corresponds to the current trend in that sea area,
which means it has a fine scalability. Then, we change the time interval value in our model to test its
sensitivity. After calculation, the final probability change rate is less than 5.8% if the time interval change
rate is less than 10%. The results show that our model is not sensitive to changes in time intervals. Finally,
we formulate a memo to the Greek government to obtain official approval.
Keywords: Submersible Trajectory; Dynamical Analysis; Grid Analysis; Bayes Theorem; Sensitivity
Analysis
Team #2407038 Page 1
Contents
1 Introduction 2
1.1 Problem Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Restatement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.4 Our Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Assumptions 4
3 Notations 4
4 Model Preparation 5
4.1 Data Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
4.2 Data Collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
4.3 Data Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
4.3.1 Data Extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
4.3.2 Conversion from longitude and latitude to distance . . . . . . . . . . . . . . . . . 5
5 Trajectory Prediction Model Based on Dynamics Analysis 6
5.1 Advanced Time Series Analysis: Delving Deep into Ocean Current Speed Data . . . . . . 6
5.2 Continuous Current Speed Distribution Regression Model . . . . . . . . . . . . . . . . . 8
5.3 Dynamical Model based on Newtonian mechanics . . . . . . . . . . . . . . . . . . . . . . 9
5.4 Discussion on Key Equipment and Technologies . . . . . . . . . . . . . . . . . . . . . . . 10
6 Comprehensive Equipment Allocation Optimization Model 11
6.1 Preliminary Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
6.2 Model Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
6.2.1 Parameter Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
6.2.2 Data Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
6.2.3 Objective Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
6.3 Genetic Algorithm for Multi-objective Optimization . . . . . . . . . . . . . . . . . . . . . 14
6.4 Results and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
7 Grid Probability Search Technique based on Poisson Distribution 15
7.1 Regional Rasterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
7.2 Poisson Probability Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
7.3 Probability Calculation Based on Bayesian Theory . . . . . . . . . . . . . . . . . . . . . 17
7.4 Results and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
8 Expansion of the Model 19
8.1 Model Applied in Different Areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
8.2 Rescue of Multiple Submersibles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
9 Sensitivity Analysis 21
Team #2407038 Page 2
1 Introduction
1.1 Problem Background
There have been 38 accidents involving submarines and submersibles since 2000, and one of them
was the “Titan implosion” last year [1]. Once an incident in deep water happens, an urgent international
search and rescue operation is about to begin. Greece-based company Maritime Cruises Mini-Submarines
(MCMS) manufactures submersibles for deep-sea explorations and wishes to provide tourists with the
chance to adventure the depths of the Ionian Sea and explore sunken shipwrecks. However, it is vital to
have safety procedures to cope with the possible risks underwater. Consequently, we are asked by MCMS
to design a four-step procedure for them.
1.2 Restatement of the Problem
To ensure security on board, a four-step safety procedure is modeled, including Locate, Prepare, Search,
and Extrapolate. After thorough background reading, each procedure can be formulated into a sub-problem
as follows:
• LOCATE:
Develop a predictive model for the submersible’s location over time. Identify uncertainties in pre-
dictions. Determine data the submersible can transmit to the host ship to reduce uncertainties. Specify
required equipment on the submersible for data transmission.
• PREPARE:
Recommend additional search equipment for the host ship, considering cost, availability, maintenance,
readiness, and usage factors. Also, outline the additional equipment a rescue vessel might need for
assistance.
• SEARCH:
Create a model integrating location predictions to suggest optimal deployment points and search
patterns for search equipment, minimizing the time to locate a lost submersible. Calculate the probability
of finding the submersible based on time and accumulated search results.
• EXTRAPOLATE:
To adapt the model for other tourist destinations like the Caribbean Sea, integrate region-specific data
such as currents, weather patterns, and underwater topography. For multiple submersibles in the same
vicinity, incorporate individual identifiers and update the prediction algorithm to manage simultaneous
movements.
1.3 Literature Review
The task consists of two main models to be constructed: the prediction model of the submersible path
and the search methodology of the rescue vessel. According to existing studies, predicting the submersible
tracks has two main methods: Neural Network Predictions [2] and Computational Dynamics Simulations
[3]. The neural Network-based Method uses a neural network (LSTM) to capture the disturbance like
effects of currents or geography, which has remarkable accuracy with the proper network. At the same
time, the computational and time costs are unavoidable. Computational Dynamics Simulations start from
essential physics and kinetics models, and the simulation runs simultaneously with happening situations.
Its accuracy relies heavily on the dynamical models and the formal information.
Team #2407038 Page 3
The underwater target search methods mainly include random search, geometry search, and heuristic
search. This task requires searching by probability based on the existing results and time, a heuristic
search method using prior knowledge of the target. Based on our findings, a few studies use mathematical
models to tackle this issue: Limited studies have addressed this issue with mathematical models. Yao et
al. applied expectation-maximization (EM) for static target search, but it is unsuitable for moving targets
[4]. In [5], Juan, Li et al. use the RRT algorithm and neural network to improve the exploration capability
in multiple-target search in a changing environment, but the search efficiency is relatively low.
1.4 Our Work
The task involves establishing a four-step procedure to ensure the safety of deep-sea exploration, which
mainly includes:
1. Based on the currents, seawater density, and seafloor geography of the Ionian Sea, a Trajectory
Prediction Model based on dynamics analysis is established.
2. Multi-Objective Optimization evaluates the rescue search equipment and then decides whether to be
installed on the host ship and rescue vessels.
3. Based on the location obtained from the Trajectory Prediction Model, we rasterize the search area
and construct the probability model from the search results and time.
4. Extrapolation of the model tests the transferability of situations like different oceans and multiple
submersibles lost.
In order to avoid complex descriptions and intuitively reflect our workflow, the flowchart is shown in
Figure 1.
Figure 1: Work Flow
Team #2407038 Page 4
2 Assumptions
Before establishing a mathematical model for the motion trajectory of a defective submersible under
the sea, we make some assumptions to make the model easier to realize.
• Assumption 1: The changes in ocean currents are periodic.
Explanation: The periodic changes in ocean currents result from the combined effects of various
complex factors, including but not limited to wind, Earth’s rotation, and solar radiation.
• Assumption 2: The submersible’s weight and volume do not change after it loses contact with
the host ship. The size of the submersible is supposed to be 670 cm × 280 cm × 250 cm
Explanation: A submersible’s weight and volume will decide its gravity and buoyancy and further
influence the trajectory underwater. To simplify the model, we ignore the changes in these factors for a
defective submersible.
• Assumption 3: When a submersible breaks down and loses contact with the host ship, it will
lose the ability to provide propulsion and change its volume simultaneously.
Explanation: If the submersible still has propulsion, the operation of the driver after contact loss will
be unpredictable, which means the later position of the submersible will be unpredictable. Additionally, if
the submersible can still reach the water’s surface with its propulsion, discussing how to predict the motion
and carry out the rescue underwater is meaningless.
• Assumption 4: When a submersible breaks down and loses contact with the host ship, it is
always positioned on the sea floor or at some point of neutral buoyancy underwater.
Explanation: Since the purpose of a submersible is to allow tourists to visit underwater landscapes and
search for underwater boats, it will be suspended at some position under the sea or just stop on the seabed.
Then, we can suppose that the submersible is still when accidents happen.
• Assumption 5: When a rescue vessel searches for the missing submersible, it can detect
submersible as long as they are in the same latitude and longitude coordinates.
Explanation: Due to the complex environment and terrain, the rescue vessel may miss the submersible
even though they are close enough. The missing probability is low and unpredicted, so we view it as 0,
which means the rescue vessel can find the target under the water.
3 Notations
Table 1: Notations Used in this Paper
Symbol Definition Symbol Definition
ˆ
𝑌
𝑡
Predicted ocean current velocity vector 𝑓
𝐶
(𝑥) Cost objection function
𝑅
𝑡
Residual sequence 𝐺
𝑠
Grid size
𝐽
(
𝜃
)
Ridge regression function 𝑣 Velocity
𝑀𝑆𝐸 (𝜃) Mean square error 𝑃
(
𝜒 = 𝑘
)
The Poisson distribution
𝐸𝐶 Environmental coefficient 𝑝(𝐴) Probability of event A
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