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An Optimization Model and Simulation of a Wrecked
Plane to Seek for in the Sea
Hong Fang and Xinglong Ren*
College of Science, Huazhong Agricultural University, Wuhan, China
*Corresponding author
Abstract—It is of great necessity to explore a method to seek for a
wrecked plane in the sea. In this paper we think of a static
situation and establish a model that can be applied to quickly
search the plane fallen into different oceans. Firstly, we develop a
parabola model to simulate the process of plane falling. Take the
projection of where the plane finally signaled on the sea as the
center, and the distance between the two signaling positions plus
the horizontal distance of the parabolic movement as radius, we
can initially define the most likely falling area. Secondly, we
apply Kruskal minimal spanning tree algorithm to minimize the
total searching route. Then we regard suspicious neighboring
goals as a single zone in different ways and each plane searches
only one zone. Considering time cost and fuel cost, a multi-
objective programming model is established to optimize the
searching scheme. Finally we obtain the minimum cost and time
for the whole searching.
Keywords-district division; multi-objective programming;
Kruskal minimum spanning tree
I. INTRODUCTION
In recent years, it is not uncommon that the increasing
number of airplane crash issues occurs pretty frequently
worldwide. To find the lost plane which fell into the ocean,
rescue the survivors in time, and research what cause airline
disasters, which contributes to reducing the opportunity of
more airplane crash accidents, experts from all over the world
have devoted tremendous amount of time and energy in study
of searching for disappeared airplanes. Rescue service is a
significant action associated with all the public and private
resources, aiming at supervising hazards, corresponding with
outside, determining the location of victims and transferring
them to safer place. The government puts emphasis on rescue
service as an indispensable part of emergency rescue system. It
guarantees the safety of citizens’ lives, and convinces people
of their nation’s good reputation. Furthermore, it’s very critical
to save their fellows. Therefore, provided that the government
establishes a rescue system with high efficiency and takes
appropriate measures promptly, it will make great differences.
In this paper, we simulate the process of plane’s falling and
proposed an efficient approach about how to search for lost
planes on the sea.
II. A PARABOLA MODEL FOR SIMULATING
Commonly, the plane sends a signal to the ground console
every five minutes, including its position, speed, direction etc.
We can know the final position of the plane by the last signal it
has sent. As Figure I shows, Point M is where the plane sends
the last signal, and Point N is where it is supposed to send
another signal five minutes later.
Regardless of air resistance and other factors, and assume
that the plane moves at uniform motion in a straight line
between Point M and Point N. The possible falling point
equally distributes on the line MN. If the plane falls from
Point M, then it will drop to Point C in a parabolic track, and if
the plane falls from Point N, then it will drop to Point D. Point
C and Point D are on the sea.
As we do not now the orientation of the plane at last time,
it may do parabola movement in any direction[1]. So the
possible falling area on the sea is a circle.
FIGURE I. PARABOLA MODEL
FIGURE II. THE MOST LIKELY FALLING AREA ON THE SEA
Point A and Point B partly represents the projection of
Point M and Point N on the sea. Segment AC and Segment BD
International Conference on Applied Mathematics, Simulation and Modelling (AMSM 2016)
© 2016. The authors - Published by Atlantis Press
201
represent the horizon distance of the parabola movement. Set
the length of Segment MN as l, so as Segment AB, and the
length of Segment AC and Segment BD as d.
Take Point A as center, the length of Segment AC and
Segment AD as radius, two circles are drawn, as Figure II
shows. The dashed part between the two circles is the possible
area where the plane may drop into.
Assume that the flight altitude of the plane h is 10000
meters, and the horizontal flight speed v is 800 kilometers per
hour. According to parabolic equation:
2
2
1gth = (1)
tvd *= (2)
Time needed for falling t is 2010 seconds, and the
horizontal distance of the parabolic movement d is
9
2020000 meters.
Assume that the plane flies normally, according to the
following equation:
1
*tvL = (3)
We can get the horizontal flight distance in five minutes l
is 3
200000 meters.
So the farthest distance the plane can arrive, namely, the
radius of the outside circle of the possible falling area R is
calculated as the following:
dLR += (4)
Combining all four equations above, we can obtain the
results:
kmmR 604.76
9
60000020*20000 ≈
+
=
According to data available, the detection range of a
remote surveillance radar is from 300 kilometers to 500
kilometers, which is much greater than R. So if put one that
radar at Point A, we can scan all suspicious goals in that area.
Red dots in Figure 2 represent all suspicious goals.
III. KRUSKAL MINIMUM SPANNING TREE
Aiming at making the rescue plane rapidly search for all
suspicious targets which are detected by radar in the possible
region, we adopt the approach which is based on Kruskal
minimum spanning tree[2].
Minimum spanning tree is a spanning tree in connected
graph with n nodes. It is the minimal connected sub-graph
with all nodes of the original graph, and has the smallest
number of edges. To simulate the process of searching for
targets in the suspicious region detected by radar, we adopt the
dots generated randomly by MATLAB. Those dots simulate
the suspect place where plane might fall. Using Kruskal
algorithm can guarantee that the searching road is the
shortest[3].
-2 0 2 4 6 8 10 12 14 16
x 10
4
0
5
10
15
x 10
4
FIGURE III. SUSPICIOUS GOALS AND MINIMAL SPANING TREE
As it is indicated in Figure III, dots where the plane which
is simulated randomly might fall present an irregular equal
possibility distribution. Take every direction of the flight into
consideration, the red dots represent the location when the last
time the plane emitted the signal and fell down. The blue
circles represent the boundary when the plane have flown for
five minutes and arrived at the next signal emitting location.
The black dots are generated randomly by MATLAB,
representing possible falling locations. The blue segments
constitute the minimum spanning tree as the shortest searching
road.
IV. DIVISION OF THE SEARCHING AREA
According to the minimum spanning tree constituted as
Figure III shows, we divide near points into different regions.
As there are many ways of dividing, we adopt four dividing
schemes, the results are showed in Figure IV.
FIGURE IV. SEARCHING AREA DIVIDING SCHEMES
202
As we can see in Figure IV, there are respectively three,
four, four and five separated regions in every picture. Plus all
blue segments, we can obtain the shortest route of searching.
Then we can find the goal plane with the fastest speed if only
we scan all suspicious targets in shadowed areas[4].
After dividing the regions, assume that airports nearest to
every regions can assign rescue planes so as to scan all these
regions. Airports are represented by red points in the pictures.
As Figure IV shows, there are airports A1, B1, C1 in picture a,
airports A2, B2, C2, D2 in picture b, airports A3, B3, C3, D3 in
picture c, airports A4, B4, C4, D4, E5in picture d.
To minimum the cost and make the lasting time for
searching as short as possible, we should build a model to
compare all these area dividing schemes in order to find out
the optimal method. So a multi-objective programming model
is established as following.
V. A MULTI-OBJECTIVE PROGRAMMING MODEL
Assume that there are i kinds of rescue planes and j
areas. ij
xmeans the number of ith kind of rescue planes
assigned to scan the jth area. ij
ymeans the scanning length of
every ith kind of rescue plane in jth area. ij
zmeans the distance
between the airport where the ith kind of rescue planes take off
and the farthest point in jth area. j
dist means the total route
needing scanning in the jth area. i
vmeans the speed of the ith
kind of rescue planes. i
tmeans the longest time that the ith kind
of plane can last for flying, i
rmeans the amount of fuel the ith
kind of rescue plane consumes in per time-step. p means the
price of per cell fuel. i
smeans the intrinsic cost of employing
the ith kind of rescue plane, including wages of officers,
constructing cost assigned to the specific plane etc.
A. Object Fuction
To minimum the cost and make the lasting time for searc
Aiming to minimize the time needed[5]. Time needed for each
plane is calculated by dividing the distance it flies by its speed.
Plus all distances of every single plane, we then get the total
time needed. Use the equation below:
∑∑
=
ij i
ijij
v
yx
T*
min (5)
Aiming to minimize the total cost for searching, which is
consisting of fuel cost and intrinsic cost. Fuel cost can be
calculated by multiplying the amount of fuel consumed by the
price per cell, intrinsic cost is given by collecting data. As a
result of lack of data, here we consider the intrinsic cost of
every kind of rescue planes is equal, Use the equation below:
∑∑ +=
ij iiji
i
ijij sxpr
v
yx
J***
*
min (6)
B. Constraint conditions
The total distance of all rescue planes assigned to one area
can fly should be greater than distance needing scanning of the
specific area. This condition is expressed as the equation
below:
(
)
,...,2,1* =≥
∑
jdistyx j
iijij (7)
The sum of time for scanning and time for returning should
not exceed the plane can last for flying. This condition is
expressed as the equation below:
(
)
()
,...2,1=≤
+
∑it
v
zy
ji
i
ijij (8)
C. Data and Results
By searching the Internet for information, rescue planes we
may employ are listed below in Table I:
TABLE I. PARAMETERS OF RESCUE PLANES MAY BE EMPLOYED
Type Flight speed Maximum flight
range Fuel consumed
/time-step
Type 1 530km/h 5500km 4650kg/h
Type 2 639km/h 5000km 5100kg/h
Because the price of fuel per ton is 4140 yuan at present,
using Lingo to solve the model above, we can obtain the result
of our model. One is the shortest lasting time: min
t=1.26 h,
and the other is the Minimum rescuing cost:
min
S=27657331yuan.
VI. A SUPPLEMENT TO DYNAMIC SIMULATION
Due to the promoting action of water and wind, a floating
object on the sea is always drifting forward. Normally, as for
crashed planes fallen into the sea, we simply the issue to the
sea surface, and the position of the crashed plane will change
irregularly under the impact of oceanic currents and wind
pressure. If we can accurately determine the initial location
and time of the incident, then its trajectory can be described by
modeling.
In order to find out the position of the crashed plane at any
time, we need to predict its orientation and speed of drifting.
Assume that the ocean velocity direction relative to the
ground is horizontal axis(x), the floater speed relative to the
water environment is vertical axis(y). Regard the floater drift
velocity as a vector, its decomposition along horizontal axis(x)
is curr
V, another decomposition along vertical axis(y) is
leeway
V. Leeway is formulated assuming that the floater on the
surface is affected by oceanic currents and wind pressure,
making it drift deviated from wind direction. Leeway may turn
right clockwise around the wind direction, or turn left
anticlockwise wind direction. Set leeway as
α
, then the ocean
velocity direction relative to the ground is:
203
α
cos*
driftcurr vv = (9)
the floater speed relative to the water environment is:
α
sin*
driftleeway vv = (10)
Monte-Carlo method is on the basis of statistics, and is
well applied in complicated systems. According to equations
above, we adopt Monte-Carlo method to simulate the change
track of plane.
Use MATLAB to generate initial positions of crashed
planes, the result is revealed in Figure V:
FIGURE V. INITIAL POSITIONS GENERATED BY MATLAB
The figure above simulates the possible positions at
original time. As time goes by, the positions of all points turn
in all directions. Take 12 hours as a time interval, respectively
imitate the positions in 12, 24 and 36 hours based on drift
model. The results is shown in Figure VI:
FIGURE VI. POSITIONS IN FURTHER TIMES
We divide the whole zone into square grids with side
length of 50000 meters. According to Figure 6, the possibility
of the plane’s falling into each grid changes as time changes.
The possibility is defined as the number of points in each
square grid.
Regarding each square grid as a node, we start the
searching with nodes with maximum probability. Combining
the principle of proximity, if nodes in one probability level
searched thoroughly, then the searching plane should find a
nearest node with probability in a lower level to restart
searching.
Relying on Matlab, we can find out the optimal route for
searching as is shown in Figure VII.
FIGURE VII. OPTIMAL SEARCHING ROUTE
As calculated, the total route the searching planes need to
fly is approximately 1000 kilometers. If needed, several planes
should be assigned to search partly.
VII. CONCLUSION
In this paper, we analyze the method to search for a
wrecked plane on the sea in static situation. At first, we seek
for a possible area the plane may fall into to limit the
searching area, which can reduce searching range and decrease
cost in a degree. Secondly, based on Kruskal minimum
spanning tree algorithm, we optimize the whole searching
route.Of course there are some short comings, such as, In the
dropping process of the plane, we ignore air resistance and
other factors, but the actual situation may be much more
complicated and as few effective data are available in our
model establishing model, we just simulate the process on
computer, which is not very convincing. And as a supplement,
a dynamic simulation is also done, and a feasible search
scheme is given.
What’s more, we can do some further discussion. Firstly,
in parabola model, in order to make sure the possible falling
area of the plane more exactly, factors such as air resistance
should be involved. Secondly, in district dividing model, more
schemes can be listed, which may optimize the searching route
for a further step. And at last, we can consider more kinds of
planes for searching.
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