MCM 2015 Summary Sheet for Team 35565
Team Control Number
For office use only T1 ________________ T2 ________________ T3 ________________ T4 ________________
35565
Problem Chosen
For office use only
F1 ________________ F2 ________________ F3 ________________ F4 ________________
B
Summary
The lost MH370 urges us to build a universal search plan to assist searchers to locate the lost plane effi-ciently and optimize the arrangement of search plans.
For the location of the search area, we divided it into two stages, respectively, to locate the splash point and the wreckage?s sunk point. In the first stage, we consider the types of crashed aircraft, its motion and different position out of contact. We also consider the Earth?s rotation, and other factors. Taking all these into account, we establish a model to locate the splash point. Then we apply this model to MH370. we can get the splash point in the open water is 6.813°N 103.49°E and the falling time is 52.4s. In the second stage, considering resistances of the wreckage in different shapes and its distribution affected by ocean currents, we establish a wreckage sunk point model to calculate the horizontal displacement and the angle deviation affected by the ocean currents. The result is 1517m and 0.11°respectively. Next, we extract a satellite map of submarine topography and use MATLAB to depict seabed topography map, determining the settlement of the wreckage by using dichotomy algorithm under different terrains. Finally, we build a Bayesian model and calculate the weight of corresponding area, sending aircrafts to obtain new evidence and refresh suspected wreckage area.
For the assignment of the search planes, we divide it into two stages, respectively, to determine the num-ber of the aircraft and the assignment scheme of the search aircraft. In the first stage, we consider the search ability of each plane and other factors. And then we establish global optimization model. Next we use Dinkelbach algorithm to select the best n search aircrafts from all search aircrafts. In the second stage, we divide the assignment into two cases whether there are search aircrafts in the target area. If there is no search aircraft, we take the search area as an arbitrary polygon and establish the subdivision model. Considering the searching ability of each plane, we divide n small polygons into 2n sub-polygons by using NonconvexDivide algorithm, which assigns specific anchor points to these 2n sub-polygons re-spectively. If there exist search aircrafts, we divide the search area into several polygons with the search aircrafts being at the boundary of the small polygons. To improve search efficiency, we introduce” max-imize the minimum angle strategy” to maximize right-angle subdivision so that we can reduce the turning times of search aircraft. When we changed the speed of the crashed plane about 36m/s, the latitude of the splash point changes about 1°.When a wreck landing at 5.888m out from the initial zone, it will divorce from suspected searching area, which means our models are fairly robust to the changes in parameters.
Our model is able to efficiently deal with existing data and modify some parameters basing the practical situation. The model has better versatility and stability. The weakness of our model is neglect of human factors, the search time and other uncontrollable factors that could lead to deviation compared to practical data. Therefore, we make some in-depth discussions about the model, modifying assumptions establish-ment above, to optimize our model.
Searching For a Lost Plane
Control#35565
February 10, 2014
Team # 35565 Page 3 of 57
Contents
1 Introduction.............................................................................................................................................5 1.1 Restatement of the Problem ……………………………………….................................….5 1.2 Literature Review……………………………………………………………………..…...…...6 2 Assumptions and Justifications...................................................................................................7 3 Notations………………………………………………………………………………………………….7 4 Model Overview……………...............................................................................................................105 Modeling For Locating the Lost Plane..................................................................................10 5.1 Modeling For Locating the Splash Point………………………………….11 5.1.1 Types of Planes………………………………………………...…11 5.1.2 Preparation of the Model—Earth Rotation ………………...….....12 5.1.3 Modeling……………………………………………………….....13
5.1.4 Solution of The Model…………………………………………....14
5.2 Modeling For Locating Wreckage………………………………………...15
5.2.1 Assumptions of the Model………………………………………..16
5.2.2 Preparation of the Model………………………………………….16 5.2.3 Modeling…………………………………………………………..21 5.2.4 Solution of the Model……………………………………………..25
5.3 Verification of the Model……………………………………..……………26 5.3.1 Verification of the Splash Point……………………………….…..26 5.3.2 Verification of the binary search algorithm………………………..27 6 Modeling For Optimization of Search Plan……………………………...……29 6.1 The Global Optimization Model…………………………………………...29 6.1.1 Preparation of the Model…………………………………………..29
6.1.2 Modeling……………………………………………………….….31
6.1.3 Solution of the Model……………………………………..……….31 6.2 The Area Partition Algorithm……………………….…………….…………..……...……33 6.2.1 Preparation of the Model……………………….………………….33
6.2.2 Modeling………………………………………………………..…34
6.2.3 Solution of the Model……………………………………….…….35 6.2.4 Improvement of the Model……………………………..…………36 7 Sensitivity Analysis……………………………………………………………..38 8 Further Discussions……………………………………………………………..39 9 Strengths and Weaknesses………………………………………………..…….41 9.1 Strengths ……………………………………………………..…….………41 9.2 Weaknesses…………………………………………………………….…..42 10 Non-technical Paper……………………………………………………………..42
Team # 35565 Page 4 of 57
Team # 35565 Page 5 of 57
1 Introduction
An airplane (informally plane) is a powered, fixed-wing aircraft that is propelled for-ward by thrust from a jet engine or propeller. Its main feature is fast and safe. Typi-cally, air travel is approximately 10 times safer than travel by car, rail or bus. Howev-er, when using the deaths per journey statistic, air travel is significantly more danger-ous than car, rail, or bus travel. In an aircraft crash, almost no one could survive [1]. Furthermore, the wreckage of the lost plane is difficult to find due to the crash site may be in the open ocean or other rough terrain.
Thus, it will be exhilarating if we can design a model that can find the lost plane quickly. In this paper, we establish several models to find the lost plane in seawater and develop an op-timal scheme to assign search planes to model to locate the wreckage of the lost plane.
1.1 Restatement of the Problem
We are required to build a mathematical model to find the lost plane crashed in open water. We decompose the problem into three sub-problems:
? Work out the position and distributions of the plane?s wreckage ? Arrange a mathematical scheme to schedule searching planes
In the first step, we seek to build a model with the inputs of altitude and other factors to locate the splash point on the sea-level. Most importantly, the model should reflect the process of the given plane. Then we can change the inputs to do some simulations. Also we can change the mechanism to apply other plane crash to our model. Finally, we can obtain the outputs of our model.
In the second step, we seek to extend our model to simulate distribution of the plane wreckage and position the final point of the lost plane in the sea. We will consider more realistic factors such as ocean currents, characteristics of plane.
We will design some rules to dispatch search planes to confirm the wreckage and de-cide which rule is the best.
Then we attempt to adjust our model and apply it to lost planes like MH370. We also consider some further discussion of our model.
美赛数模论文英文版
