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2015年美赛B题论文

时间:2016-03-11


Where is the MH 370?
Abstract
Where is the crashed MH 370? This is an issue of global concern. In this article, the search work for the crashed aircraft is divided into three stages:determining the fall area, select the search location, arrange rescue equipment.To solve problems, we have set up three mathematical models. According to physics equations,we have established a differential equations model that can describe the crashed procedure of the aircraft.By combined maritime related cases,we have calculated the theoretical appeared area of the aircraft. Because of the large area of theory, it will be split into many small regions of equal area. With the limited search capability,we need to find a small piece where the aircraft is most likely to exist in.Then we use the conditional probability to establish a maritime search model and have got the actual search area and search paths. Each time a search is completed.We use a Bayesian probability formula to update the appearing probability of the aircraft in each small area if the crashed aircraft is not found.Besides,we resolve the model to acquire the actual search area and search paths. From an economic point of view, we have created a scheduling model of the search appliances with the existed search equipment. Then we made reasonable arrangements for personnel and equipment based on the results of the model.

Keywords:Differential Equations

Conditional Probability

Bayesian Methods Nonlinear Programming

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CONTENTS 1. Introduction….………………………………………………………………2 2. Assumptions………………………………………………………………….2 3. Explanation of notations………………………………………………….. 3 4. Model One:the Aircraft Crashed Model
4.1 Analysis of Model……………………………………………………………4 4.2 Model Building………………………………………………………………4 4.3 Solutions to the Model……………………………………………………….5 4.3 Testing the Model…………………………………………………………….6

5. Model Two:the Maritime Search Model
5.1 Analysis of Model…………………………………………………………….6 5.2 Bayesian Methods…………………………………………………………….7 5.3 Model Building……………………………………………………………….8

6. Model Three:the Search DevicesScheduling Model
6.1 Analysis of Model………………………………………………………….…8 6.2 Building the Model………………………………………………………….. .8 6.3 Model Solving……………………………………………………………… ...9

7. Conclusions………………………………………………………….………. ..9 8. Strength and Weakness
8.1 Model One……………………………………………………………………10 8.1 Model Two…………………………………………………………………... 10 8.1 Model Three………………………………………………………………..…11

9. References……………………………………………………………………..11 10.Paper Concerning Future Search Plans………………………………..….12 11. Appendix………………………………………………………………….. …14

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1. Introduction
Although science and technology are advanced rapidly in recent years,the crash incidents still occur now and then .Take Malaysia Airlines MH370 forexample, its crash have already attracted hundreds of millions of people's attention. In the case that it cannot send out any signal, the rescuers have to determine the best search strategy as soon as possible. In addition, due to the diversification of the search appliance, we have given the best scheduling schemes of the search appliance. The problems we have settled are listed as follow: ? ? ? How to determine fall point of the aircraft in the open sea? If we can search onlyparticulararea of seaevery time, how to determine the possible search region? When some important parameters of search equipment are known,how to get the best scheduling solution of the search devices? In order to deal with those problems above,we found some practical and efficient methods. ? At the beginning,we established a physical model to describe theprocedure of the aircraft falling from the sky to the sea and gotthe possible crashed region of theplane. ? ? Moreover,we built a search model of Bayesian probability updating and obtained more realistic search strategies. Last but not least,we found optimal scheduling scheme by establishing scheduling model of search equipment based on minimal costs.

2. Assumptions
? ? ? ? ? ? There is no land in the search sea. The ocean currents in the search sea are very complex. When the aircraft falls down, the airplane did not explode. When the aircraft falls down, the plane fuselage remains level. When the aircraft falls down, its acceleration of gravity remainsunchanged. There are only two search devices:planes and ships.They can be scheduled together.

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3. Explanation of notations
Table 1 Symbol G F f M the gravity of the aircraft rising force of the aircraft from the air resistanceof the aircraft from the air quality of the aircraft horizontal acceleration of the aircraft when falling down vertical acceleration of the aircraft when falling down the density of atmosphere coefficient of resistance coefficient of rising force extension area of aircraftwing bottom surface areaof aircraft speedin the horizontal direction speedin the vertical direction advancing speed of searching equipment maximum stay time in task searching area the width of sweeping the sea the area of maritime searchregion the number of the aircraft the number of the ship the speed of the aircraft the speed of the ship the cost of an aircraft per hour the cost of a ship per hour the scanning width of an aircraft the scanning width of a ship the maximum time to complete one search task the actual time of useto complete one search task moving distance of searching equipment in everysmall square Notation Meaning

ax

ay ?
Cw

Cu
S1
S2

vx
vy

v T w

S

a
b

v1

v2
c1

c2 w1
w2

T0

t
zi

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4. Model One:
4.1 Analysis ofModel
The aircraft will fall down after the engine lost power.At this time, the forces of the aircraft are shown in Figure 1. There are the gravity G, rising force F of the aircraft from the air and resistance f of the aircraft from the air. F

f

G Figure 1the Forces of the Aircraft

The acceleration of the aircraft is resolved into horizontal acceleration and vertical acceleration.Thenweestablished dynamic equations in the plane coordinate system.The dynamic equations are: f ? Max
G ? F ? Ma y

Moreover,accelerations are defined as:

d 2x d2y ax ? 2 , a y ? 2 dt dt
By referring to material,we knew about the formulas below.
1 2 ? Cw S1 vx 2 1 2 F ? ? Cu S 2 v y 2 f ?

4.2 Model Building
Consequently,the model can be summarized as the differential equations below.
2 ?1 d 2x ? dx ? ? C S ? M ? ? ? w 1? dt ? dt 2 ?2 ? 2 2 ? 1 ? C S ? dy ? ? Mg ? M d y ? ? u 2? dt 2 ? dt ? ?2

The initial conditionsare described as follow:

? dx dy ? 240, ?0 ? dt dt t ?0 ? t ?0 ? x ? 0 ? ? 0, y ? 0 ? ? 10000 ?

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4.3 Solutions to the Model
By looking for information,we acquired relevant information of Malaysia Airlines MH370 as shown in Table 2. Table 2 Related Parameters of MH370 M 200000kg

?
Cw
Cu

0.849kg / m3

0.08 1.2
130 m2
200 m2

S1
S2

Use MATLAB to solve the equations.It takes 81.9071 seconds for MH370to crash into the sea.When it crashed into the sea,itsspeedin the horizontal direction is 167.3729 minutes per second and speed in the vertical direction is 138.6997 minutes per second.Besides,its AbscissaX is 16328 meters while ordinateYis 0.0062 meters. Additionally,we obtained curve of solutions for the equations byMATLAB.The crashed track of MH370 is shown in Figure 2.
10000 Plane crashed track 9000 8000
Vertical distance/meter

7000 6000 5000 4000 3000 2000 1000 0

0

0.2

0.4

0.6

0.8 1 1.2 1.4 Horizontal distance/meter

1.6

1.8 x 10

2
4

Figure 2the Crashed Track of MH370

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4.4 Testing the Model
In the Aircraft Crashed Model, we cannot calculate the exact crashed time of the plane due to a computer error. But the error is within a certain range(0.62%), and therefore results of the model are with higher confidence.

5. Model Two:the Maritime Search Model
5.1 Analysis of Model
In Model One, we determined the theoretical placement of the aircraft.However, it may still be some distance aheadafter the plane lost contact with the flight.So it is possible to translate the theoretical impact point forward along the original direction of flight of the aircraft. Regard the round having a circle of theoretical placement and a radius of twenty kilometers as the search area.Then round collections whose circles are in a straight line are set as the possible search area, shown in Figure 3.

Figure 3

Searching Area

In order to facilitate the solution to the problem, we make this area approximately a rectangular region, as shown in Figure 4.

Figure 4

Rectangle Area of Searching

This area is divided into small squares with the number of N. What’s more, we suppose the event Bi ?i ? 1,2,?, N ? stands for the incident that the aircraft is in the small square iand the event A represents the incident that the plane crashed. Therefore, the probability of the plane crashed just into a small squarei.

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P ? A ? Bi ? ? P ? B ? P ? A Bi ?
Material that we have found shows that:

P ? A Bi ? ? 1 ? e
P ? Bi ? ?

? zi w

Here, we assumed that the probability of search at first time.

1 , i ? 1, 2,?, N N

5.2 Bayesian Methods
Bayesian analysis, a method of statistical inference (named forEnglish mathematician Thomas Bayes) that allowsone to combine prior information about a population parameter with evidence from information contained in a sample to guide the statistical inference process. A prior probability distribution for a parameter of interest is specified first. The evidence is then obtained and combined through an application of Bayesian theorem to provide a posteriorprobability distribution for the parameter. The posteriordistribution provides the basis for statistical inferences concerning the parameter.

The Bayes Formula is represented as follow:
P( Ai ? B) P( B | Ai ) P( Ai ) ? n (i ? 1, 2,? , n) P( B) ? P( B | Ai ) P( Ai )
i ?1

P( Ai | B) ?

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5.3 Model Building ? Optimization Model
As a result, we have come to the optimization model search.
? zi ? ? max f ? ? P ? A ? Bi ? ? ? P ? Bi ? ?1 ? e w ? i ? x(1) i ? x(1) ? ? x( n ) x( n )

? x( n ) ? ? zi ? vT st. ? i ? x(1) ? x ,? , x ?{1, 2,?, N } (n) ? (1)
? Information Updating
The first t + 1 time search, we have to update the probability of the incident according to existing information.The corresponding formulas are represented as follow. As for the small square area having been searched in the first t time search:

Pt ?1 ( Bi ) ?

e
j ? x(1)

? zi w

?e

x( n )

?zj w

Pt ( Bi )
j?{ x(1) ,?x( n ) }

Pt ? B j ? ?

?

Pt ( B j )

As for the small square area having not been searched in the first t time search:

Pt ?1 ( Bi ) ?

Pt ( Bi )
j ? x(1)

?e

x( n )

?zj w

Pt ? B j ? ?

j?{ x(1) ,?x( n ) }

?

Pt ( B j )

6. Model Three: the Search DevicesScheduling Model
6.1 Analysis of Model
We regarded minimum costsas the goal of the model. From this, we can create scheduling model of the search appliances.

6.2 Building the Model
Based on the goal of minimum costs, we established scheduling model of the search equipment.

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min C = x c1 t ? y c2 t ?v1 w1 t x ? v2 w2 t y ? S ?t ? 24 T 0 ? ? x ?y ? 3 ? ?0 ? x ? a ?0 ? y ? b ?
6.3 Model Solving
Through relevant information, we set some parameters as follow. There are ten aircrafts with the cost of one hundred dollars per hour, a search speed of seven hundred kilometers per hour and the sweep width of one kilometer. There are thirty vessels with the cost of thirty dollars per hour, a search speed of one hundred kilometers per hour and the sweep width of one kilometer. The total search area is 8100 square kilometers and search tasks must be completed within fifteen days. With the LINGOsoftware,we calculated the optimal scheduling scheme: nine aircrafts, three ships. Thus it took 1.23 hours to accomplish search tasks and the smallest search cost is 1,215 dollars

7. Conclusions
? the Aircraft Crashed Model
By referring to material, the horizontal velocity and vertical velocity of the airplane cannot disintegrate the plane when it crashed. The assumption that the airplane did not explode has been proved reasonable. But in reality, when the aircraft's engine failed, the pilot would lower theaircraft nose.The aircraft glide a distance as well.Thus falling direction of the aircraft was not level.As a result, there exists a certain bias between the calculation results of the model and the actual situation.

? the Maritime Search Model
After determining maritime search area, due to the complex situation at sea, when we first searched for the location information of the aircraft, we made no more accurate inference.Therefore, we thought that the probability that the aircraft appeared in any point of search area is the same. So before the first search, all the waters are

Team #48627Page10of14 theoretically equal area most likely to find the location of the aircraft. If the aircraft is not found after the first end of the search, then we used Bayesian approach to update the probability of finding the aircraft in each region. We re-solved the maritime search model.As expected,we found the aircraft position and the sea search path of the maximum probability.

? the Search DevicesScheduling Model
According to the actual situation, the best scheduling solution is using nine aircrafts and three ships.This scheme can ensure the completion of the search task with minimal costs. But how much actual work time spent searching is a more important factor, it was not taken into consideration in the program given above.

8. Strength and Weakness
8.1Model One
Strength
? ? ? The model is reasonable by model testing. Solutions to the differential dynamic equations we established are easy to implement. We have found the theoretical crashed placement of Malaysia Airlines MH370.

Weakness
? ? Only the method of calculating crashed site is given. There are no discussions about the possible region that the aircraft may fell into.

8.2Model Two
Strength
? ? Basedon Bayesian methods, we have proposed the practical detection probability model. Discuss the crash probability at various points in the searching area.

Weakness
? We possibly found additional information about the new discovery of aircraft debris and the location of black box signal. Additionalinformation had an effect on Bayesian Information updating. Furthermore, it was not

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8.3Model Three
Strength
? ? Regard minimalcosts as the goal. Discuss the dispatch of search and rescue equipment

Weakness
? The model may lead to a waste of time owing to the blind pursuit of lowcosts.

9. References
[1]The Aircraft Lost Contact Search Program Based on Differential Equations and NonlinearProgramming[EB/OL]. http://wenku.baidu.com/link?url=pZh0bMYn_L52FQbTUhqdnqb6pz6pztQ8AqogpF_ E6XVQoOyrotdHIUR1soKPU2FlI5kXdzjana6oIA7Wpn7TG2KVFESRN5J9NrRz9 YG8CPS. 2015-11-2 [2]Zhou Changyin, Zhao Yutang, Sun Yaxing.Updating Crashed Plane Detection Probability Model Based on Bayesian Information [J] mathematical modeling and its applications, 2015,4 (2): 71-78 [3] Wu Zhihua.Using Mathematical Methods to Find theWreckage[J]. Science Humanities. [4]The Problem of Finding the Black Box Model Based on Maritime Search and GlobalOptimization[EB/OL]. http://wenku.baidu.com/link?url=KrdxNu5Dwuv7iltDrKzx1OQxK1u89X5TqfgUT_F zeORa4jACo_FQAdVu7oIqsIfXO903eHOIYp3RkMXRjx4nR9Pm6X1R4VhXrDt6g TttIWe. 2015-9-6

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10. Paper Concerning Future SearchPlans
On March 8th of 2014,Malaysia Airlines MH370 burst out crashing at twenty-two past oneof Malaysia Local Time. It lost contact with Air Traffic Control during a transition of airspace between Malaysia and Vietnam whilst en-route to Beijing.There were 227 passengers,2 flight crews and 10 cabin crews on board. Today, standing here, we must first extend our deepest apologies to families of the victims, we will try to find the truth about the crash with the fastest speed at all costs to give an account of the victims. For the future of search, we developed a rigorous program, which is divided into three stages: find the general area of the aircraft crashed in the sea, search the most likely location of aircraft in this area, and find equipment and personnel participating in the search arrangements search for work in a timely manner. Next I will describe the three stages in detail:

Team #48627Page13of14 First of all, through studying historical data and information returned before the crash, we identified the aircraft may fall on a rectangular sea. Due to the large area of this sea, we try to find a small sea where the plane is available with the most possibility, in order to ensure the timeliness of search efforts, we mainly use aircraft to search with the aid of ship and work immediately after the best scheduling solution decided, we will continue to repeat the process until we find the crashed plane eventually. We once again express our sympathy to all those who have been affected by the terrible accident. It has been a hard time for all who have tried their best in the search for MH370. We have never wavered in our commitment to continue our efforts to find MH370 and bring closure for everyone, most of all for the families of the passengers and crew of MH370.

11. Appendix

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Solving the plane crashed Model:
function [k,vx,vy,xx,yy]=zhuiluo(t0) for t=t0:0.0001:90 if(20000-(-7000/849*283^(1/2)*t+5000000/2547*log(1/2*exp(2547/500+21/2500*28 3^(1/2)*t)+1/2*exp(2547/500)))<=0) k=t; break; end end t=0:0.0001:k; x=500000000/11037*log(33111/6250000.*t+1); y=20000-(-7000/849*283^(1/2).*t+5000000/2547*log(1/2*exp(2547/500+21/2500*2 83^(1/2).*t)+1/2*exp(2547/500))); plot(x,y); xx=500000000/11037*log(33111/6250000*k+1); yy=20000-(-7000/849*283^(1/2)*k+5000000/2547*log(1/2*exp(2547/500+21/2500* 283^(1/2)*k)+1/2*exp(2547/500))); vx=240/(33111/6250000*k+1); vy=7000/849*283^(1/2)-7000/849*283^(1/2)*exp(2547/500+21/2500*283^(1/2)*k)/( 1/2*exp(2547/500+21/2500*283^(1/2)*k)+1/2*exp(2547/500)); axis([0,20000,0,10000]); grid on; xlabel('Horizontal distance/meter'); ylabel('Vertical distance/meter'); legend('Plane crashed track');

Find the best possible position of aircraft:
model: sets: num/1..81/:x; endsets max=@sum(num(i):1/810-1/810*e^(-0.01*x(i))); @for(num(i):@sum(num(j):x(j))=300); @for(num:@GIN(x)); end

The optimal scheduling program of search device:
model: sets: num/1/:x,y,t; endsets min=@sum(num:100*x*t+30*y*t); @for(num:700*t*x+100*y*t>=8100); @for(num:t<=24*15); @for(num:y>=x/3); @for(num:@GIN(x);@GIN(y);); @for(num:@BND(0,x,10);@BND(0,y,30)); end


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