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For office use only
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T2 ________________
T3 ________________
T4 ________________
34025
Problem Chosen
B
For office use only
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2015 Mathematical Contest in Modeling (MCM) Summary Sheet
(Attach a copy of this page to your solution paper.)
For the search problem of a lost plane, on the foundation of search theory we build
a generic mathematical search model which aims at various types of search planes and
crash planes. Furthermore, we optimize the model on the target of achieve the
maximum the probability of success (POS) and reduce the total cost in a time unit.
Firstly we employ Multi-mode hybrid methods to obtain the initial probability of
containment (POC) which weighted the uniform distribution and track distribution.
Secondly we employ Analysis Hierarchy Process (AHP) to consider the synthesized
detecting ability of search planes, so as to obtain the probability of detection (POD).
Then we obtain the distribution scheme of the quantity of planes according to the
curve of the settlement probability, we divide our search strategy into two categories:
drift strategy and fallout strategy. We obtain the distribution schemes of the types of
search planes by establishing a 0-1 programming.
Aimed at the two search strategies we establish dynamic POC model on the
foundation of Bayesian. Owing to the influence of ocean environment and electronics,
we establish their own programming with multiple objectives and multiple stages and
optimize them by Parallel Selected Genetic Algorithm(PSGA). Especially, in fallout
strategy, we solve the problem of detecting depth of sonar adopting Neural Network.
In drift strategy, we create a new model named ‘Nuggets-Gold miners’, thus
enhancing the POC obviously. In addition, we simulate our generic strategy with a
particular case: search an Airbus-321 plane crashed in the Indian Ocean. Ultimately,
we calculate out that the optimal POS for a year’s searching is equal to 62.751% and
the total cost is about $3,455,000,000.
In further result analysis, we find that the increasing of the quantity of planes can’t
effectively enhance POS. Moreover, with time going by, POS increase slowly. We
make error analysis which resulted by probable error and the migration of navigation.
Aimed at the deficiencies of search strategy, we develop the model in the end of essay.
Searching for a lost plane
# Team 34025
Team #34025
Page 2 of 33
Contents
1
2
3
4
Restatement of the problem .................................................................................. 3
Assumptions ......................................................................................................... 3
Notations ................................................................................................................. 3
Model ....................................................................................................................... 4
4.1 Model introduction............................................................................................ 4
4.2 The initial probability of containment (POC) ................................................. 5
4.3 Probability of Detection (POD) ...................................................................... 6
4.3.1 Model of POD .......................................................................................... 6
4.3.2 Sweep width ............................................................................................. 7
4.4 Safety margin of different types of planes to be searched .............................. 8
4.5 Safety margin of different types of planes to be searched .............................. 9
4.5.1 Types of search planes based on electronics and sensors equipped ......... 9
4.5.2 Distribution scheme based on the two states of suspected objects .......... 9
5 Distribution strategies ........................................................................................ 11
5.1 Fallout strategy................................................................................................ 11
5.1.1 Refresh the probability of containment(POC) ....................................... 11
5.1.2 The flight path of SRU in a cell ............................................................. 12
5.1.3 The actual sonar detecting depth ............................................................ 13
5.1.4 Multi-objective programming of the fallout strategy............................. 14
5.2 Drift strategy ................................................................................................... 16
5.2.1 Refresh the probability of containment(POC) ....................................... 17
5.2.2 The flight path of SRU in a cell ............................................................. 18
5.2.3 ‘Nuggets-Gold miners Model ’ .............................................................. 18
5.2.4 Multi-objective programming of the drift strategy ................................ 18
6 Model solving with a particular case ................................................................. 19
7 Result analysis ..................................................................................................... 28
8 Error Analysis ..................................................................................................... 30
8.1 Probably Error ............................................................................................... 30
8.2 Navigation Error ........................................................................................... 30
9 Strengths and Weaknesses .................................................................................. 31
9.1 Strengths ....................................................................................................... 31
9.2 Weaknesses ................................................................................................... 31
References ................................................................................................................... 31
Report.......................................................................................................................... 32
Team #34025
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1 Restatement of the problem
The optimal search plan is one of the hot issues in nowadays, the development of
the optimal search plan keeps going all the time. Due to the characteristic of lost plane,
it is necessary to build model to optimize plan.
For a generic mathematical search model, we should consider the various planes,
searching ability limit their search area. Then we can establish a Multi-objective
programming which takes the maximum POS and the minimum expense in per unit
time as objective function, by employing intelligent algorithm to solve the
programming. The obvious difference of terrain and climate among various oceans
has contribution to the actual detecting depth. We also can get the actual detecting
depth through intelligent algorithm. For the various plane which are searched , their
types and size, color are determined the POS.
2 Assumptions






Ignore the plane’s length
POC of the lost plane is evenly distributed in every cell
Ignore the time in landing and taking off
The speed of the plane is uniform
Ignore the time in transferring course
Ignore the curvature of earth
3 Notations
Table 1:
Notations
T
POC
POD
POS
E
v0
W

Pdrift Pfall
,
M, N
D
 k ,  k , k
Notations and Descriptions
Descriptions
time unit
probability of containment
probability of detection
probability of success
initial probable error
velocity of drift target
sweep width
safety margin
the settlement probability, the drifting
probability
the number of planes target at the
suspected drifts ,fallouts.
characteristic matrix of the type of plane
0-1 variables
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d

the distance of each cell
reduction coefficient
ideal sonar detecting depth
actual sonar detecting depth
cost
day-night coefficient
h
ck

the effective work time
Te
4 Model
4.1 Model introduction
4.1.1 Parameter introduction
For the Search and Rescue optimal planning strategy, we divide this strategy into
three parts: Ocean system, the search planes and the lost plane.
In Ocean system, we discuss about hydrological environment (wave, current, tide,
sea ice),meteorological environment(surface wind, precipitation) and biological
environment
For the search planes, we discuss about electronics, night vision, navigation, speed,
the fuel load and the maximum mileage.
For the lost planes, we discuss about the size, color and settlement probability of
plane.
4.1.2 Process introduction
In order to make our model clearly, we give a process diagram as shown in
Figure1 which contains the main procedures.
Figure 1:
flow chart of the whole model
Team #34025
Page 5 of 33
4.2 The initial probability of containment (POC)
In a long searching time horizon Ttotal , our ultimate goal is to maximize the
Probability of Success (POS). For this discussion, time will be discrete so as to obtain
a dynamic distribution scheme of search efforts. A time unit T denotes the actual
search time in one day.
First of all, we make a square area as the search region which we look the flight
path as its diagonal. First of all, we make a square area as the search region which we
look the flight path as its diagonal. Then we divide it into 120 120 cells. In each cell,
its distribution accord to uniform distribution. So, we need to have the initial
distribution for the search region,
As for the problem of searching target objects in the large range of sea, we use
Multi-mode hybrid method to simulate the actual probability distribution for the
initial target object at sea. Target object accord to uniform distribution N 2 and the
moving path of target object accord to track distribution N1  L  , namely we randomly
select points Di  i  1, 2,…, n  in the flight path of target object ,the circle of center
Di and of radius Ri contains the probability P of circular normal distribution
which Di is center. According to LiHao [2011], the probability P can be described:
P  1  e E
2
/2
E  X 2  Y 2  De 2
(1)
(2)
The E is initial probable error, the X is initial position error, the Y is search
equipment error and the De is only decided by target object location. We can
through ‘International Search and Rescue Manual’ know the value X and Y .Hence
our initial probability distribution weighted by the two distributions, thus we can get
POC in the whole search region.
Then we use a Monte-Carlo based simulator for developing probability
distributions for the location of objects missing at the sea. We establish the
rectangular coordinate which looks the moving track of target object as the x-axis, the
center of the moving track as the origin, the Monte-Carlo analog image as shown in
Figure 2:
Team #34025
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Figure 2:
initial probability distribution
4.3 Probability of Detection (POD)
4.3.1 Model of POD
Kratzke T M et al. [2010]recommend for the path of the Search and Rescue Unit
(SRU) performing the search, each time the planner considers area for placement of
the SRU, it uses the exponential detection function
POD  1  ec
(3)
to compute probability of detection(POD) given the area is in the total area.
The initial exponential function only applied to the stationary target. Nevertheless,
the objects drift in the sea under the influence of wind, ocean current, etc. Hence
ZhouTao [2011] advanced the detection function, the drift objects are mainly
influenced by wind and current. Wind will push objects and become wind driven
current. Current can divided into wind driven current, sea current, tidal current, river
current, swell, etc.
LiHao[2011]introduce that leeway is decided by drift target’s size, direction and
shape, the acreage of leeway above the sea and in the sea are also necessary. We
define ve as the leeway drift velocity of target, ve can be described :
ve  0.068
a
* vw
b
(4)
Where ve denotes the average velocity of wind, where a denotes the acreage of
leeway above the sea and b is the acreage of leeway above the sea.
In order to simplify calculation, when the plane in full load condition:
a 2
b
(5)
When the plane in no-load condition:
a  1.3
b
(6)
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Then we use vector superposition method to make main impact factors into a
combined vector as the velocity of drift target. The combined vector as shown in
Figure 3:
Figure 3 :
combined vector of drift target in common condition
We assume that the objects drift towards any direction in some area, Due to the
direction is random. Therefore we consider the possible existence of target as a circle
which center is starting point, and radius continuously expand. With the time going,
the circle continuously become larger , the search area become larger as well. The
circle area is decided by the time and velocity of target. We assume temporarily the
velocity of target and drift direction is constant, the acreage of circle search area A can
be described:
A    voto 
2
(7)
Where v0 denotes velocity of drift target, where t0 denotes time of drift target.
Then POD can described:
Z
A
POD  1  e  1  e  1  e
c

Wvd
  voto 
2
T
(8)
Where W denotes the sweep width.
4.3.2 Sweep width
Different detectors have different sweep width, ZhouTao[2011] introduce W is
influenced by plenty of factors, the main factors contain the characteristic of objects,
the detecting way of search equipment and natural environment. These three factors
are also influenced by several factors. The factors schematic diagram is shown in
Figure4. When we try to obtain the weight of the three aspects of the first-level
evaluation, the weight of the several second-level evaluation criteria and the weight of
several third-level evaluation component, subjective judgment is ill-considered.
Therefore we choose the AHP as the method to solve the weight coefficients.
Team #34025
Page 8 of 33
Figure 4 : hierarchy structure model of sweep width

Establish hierarchy structure model

Construct the pairwise comparison matrix.
We use the pairwise-comparison method and saaty 1–9 method of AHP to
construct the comparison matrix .

Calculate weight vector and check the consistency of matrix.

Calculate the combination weight vector.
4.4 Safety margin of different types of planes to be searched
It’s generally accepted that the state of suspected objects are infected by many
small factors. We define the random variable X as the state of the suspected object,
Therefore based on law of large numbers, X obeys normal distribution N (0， 2 ) ),
Pdrift =2

0
t2
 2
1
e 2  dt
2 
(9)
Where  denotes the safety margin inspired by QIN Shengping[2002].  varies
with different planes.
Naturally
Pf a ll 1  P
d r i f t
(10)
Here Pdrift and Pfall indicate the settlement probability and the drifting probability
of an object. With Pdrift and Pfall , two search team come into being.
Team #34025
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M  Pdrift Z
Where
Z
,
N  Pfall Z
(11)
denotes the total number of planes available to be assigned
; M, N
denote the number of planes target at the suspected drifts and the suspected fallouts.
4.5 Distribution scheme base on different types of search planes
4.5.1 Types of search planes based on electronics and sensors equipped
There are abundant types of planes,therefore it is necessary to know how to make
the best use of their own superiorities.In this essay,we discuss about five types of the
plane,e.g. patrol aircraft( D1 ),water plane( D2 ),reconnaissance aircraft( D3 ),
electronic-jamming aircraft ( D4 )and anti-submarine warfare aircraft( D5 )
Among them the advantages of patrol aircraft are radar,sonar and farther voyage.
Water plane can land on sea,reconnaissance aircraft has radar,infrared camera,aerial c
amera,thermal imaging.moreover,electronic-jamming aircraft’s radar technology is be
tter.In addition,anti-submarine warfare aircraft has the better sonar. According to the
se characteristics of planes and SRU, Plane need have the following five kinds of elec
tronics :aerial camera( E (1) ),radar( E (2) ),infrared camera( E (3) ),sonar( E (4) ),therma
l imaging( E (5) ).
We set a row vector to represent them [ E(1), E(2), E(3), E(4), E (5)]
E(s), s  1, 2
5 is 0-1 variable. Where 1 represents the plane carry this
electronic, otherwise, 0 represents the plane doesn’t carry this electronic.
Then we can easily indicate the characteristic matrix D of each type of plane.
D1  [0,1,0,1,0]T D2  [1,0,0,0,0]T D3  [1,1,1,0,1]T
D4  [0,1,0,0,0]T D5  [0,0,0,1,0]T
4.5.2 Distribution scheme based on the two state of suspected objects
In the light of the two state of suspected object: fallen or drifting, so as to let the
special characteristic of each type of plane play their role as much as possible, we can
divide search effort into two team: one target at the suspected fallouts, the other target
at the suspected drifts.
Here comes the concrete assignment programming of types of planes
th
correspondingly. If we assign the k th plane to the i cell. Then
P O S P O
1 o, S
p t i ma l
PO
2 , S opt i mal
(12)
Team #34025
Page 10 of 33
POS2,optimal
POS1,optimal
where
and
indicate the optimal success probability of fallout
strategy and drift strategy.
Now we construct a 0-1 programming to compute the plan of distribution search
effort. In order to specify the assignment, we introduce 0-1 variables  k and  k to
describe the k th plane go to a cell of or not. 1 represent go, 0 represent not go.
Our target is to achieve the maximum success probability in the specified time
unit. And it must satisfy the following constraints:




A plane can’t be assigned to the suspected drifts and fallouts simultaneously.
The planes assigned to the fallouts should have equipped sonar.
The planes assigned to the drifts should have equipped other electronics .
The number of planes assigned is no more than limit(M,N) respectively.
Mathematically:
POS
T
 k   k
 (1  E (4))  0
k
 k
 k (1  Ek (1))(1  Ek (2))(1  Ek (3))(1  Ek (5))  0
M  N

s.t.    k  M
 k 1
M  N
  k  N
 k 1
 ,   0  1
 k k
max
th
Where tk denotes the actual flight time of the k plane.
We adopt the elitist strategy to optimize the distribution plan  k ,  k
.
(13)
Team #34025
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Figure 4:
flow chart of the elitist strategy
Run this circle, we can obtain the concrete plane assignment of type and number
of the two team.
5 Distribution strategies
As is discussed previously, we divide our search strategy into two parts on
account of the state of suspected objects. We name them fallout strategy and drift
strategy.
5.1 Fallout strategy
The overall look of fallout strategy is shown in the figure below
Figure 5:
flow chart of fallout strategy
5.1.1Refresh the probability of containment (POC)
The dynamic probability of containing (POC) of fallout varies whenever a
unsuccessful search take place. This is a kind of feedback mechanism, if a
unsuccessful search has taken place in the i th cell, then Poc(i) will reduce and POC
Team #34025
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of the rest cells will rise.
According to the Bayesian Law, the updated POC is determined by equation(14)
Poc(i )(1  pod (i))

 Poc(i)  (1  Poc(i) pod (i))  Poc(i)


Poc( j )
 Poc( j ) 
 Poc( j )  j  i

(1  Poc(i) pod (i))
(14)
5.1.2 The flight path of SRU in a cell
When we assume the plane as a partical, LiJie[2011] introduce that generally we
use the extend square search method (Figure6(a) )and the accordion search method
(Figure6(b) )to develop the search plan.
(a) extend square search method
(b)the accordion search method
Figure 6: the schematic diagram of search method in a cell
The extend square search method is suitable that SRU can arrive search region
fleetly, we look the center of cell as starting point and we follow the path of extend
concentric square to search drift. The former two search path length is equal to the
sweep depth W , then the search path length need to add one times sweep depth per
two stages .The accordion search method is suitable that the search objects locate at
search area and the location of the search objects is uncertain. We look the vertex of
cell as starting point, then we consider the sweep width as the distance of the adjacent
parallel search path, we follow the long side of cell to develop search.
In the two methods, as result of we neglect the width and length of plane, so between
both distances d is equal. d can be described:
d
S
W
(15)
Where S denotes the acreage of a cell and W denotes the sweep width.
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5.1.3 The actual sonar detecting depth

the reduction coefficient
Aimed at fallout strategy in the open ocean, we choose sonar as the main
electronic for searching. As is proposed in Liu Yan, Pan Wengliang [2012],the
oceangraphic has an obvious restriction on sonar performance. In particular,
hydrological environment (wave, current, tide, sea ice), meteorological
environment (surface wind, precipitation) and Biological environment.
We define ideal sonar detecting depth as , the reduction coefficient as  ,
Such that the actual sonar detecting depth is h   .

Reduction coefficient based on BP Neural Network
Artificial neural network is adaptive and have a high self-learning ability, we
let the artificial neural networks learn the seven factors on oceangraphic, thus
outlet the actual sonar detecting depth.
Based on the typical samples of the seven factors in news online, we can grade
the degree of them into five priorities 1-5. We define a sample as
[a, b, c, d , e, f , g ]
The specific process is as follows:
ⅠData preprocessing:
Since the S-shaped activation function is very gentle outside the range (0,1),
discrimination is too small, And the neural network’s output range is limited, therefore,
we normalize the data, using
X 
2  X  MIN 
 MAX  MIN 
X
(16)
to normalize the data to [-1, 1].
Ⅱ Select the activation function
Here we choose the hyperbolic tangent S-shaped function as the activation
function,
2
 1  1  f  x   1 (17)
1  e2 n
Using these data to train a 4-input, 3 output BP neural network.
f  x 
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Ⅲ Input-output mapping
g： a, b, c, d, e, f,g
Ⅳ Network construction
Using the data to train a 7 input,1 output BP neural networks, there are three
layers in the hidden layer, respectively, formed by 5,5,1 neurons.
Figure 7:
diagram of neural networks
5.1.4Multi-objective programming of the fallout strategy
 Multi-objective programming
I
M
i
k 1
the Probability of Success of fallout strategy POS1  ik Pod (k ) Poc(i )
Now we construct a multi-objective programming to compute the scheme of
distribution search effort. One target is to achieve the maximum success probability in
the specified time unit, the other target is to reduce the total cost to the largest extent.
And it must satisfy the following constraints:
(1) The POD of each cell is less than 1.
(2) In terms of all the planes been assigned, the actual flight time of all searching
trips is no more than a time unit.
(3) In terms of all the planes been assigned, the actual flight time of each searching
trip is no more than duration of flight.
(4)the actual sonar detecting depth is large than the ocean depth
max
POS1
T
N
min  tk ck
k 1
 ik Pod (k )  1  i  1, 2, , I
 k
 2d 1  d 2
ik
ik

 T  k  1, 2, , N

vk
s.t.  i
 2di1  di2
k
  k  k  1, 2, , N
 k
 vk
  h
, I ; k  1, 2,
i , ocean  i  1, 2,
 i k
,N
(18)
Team #34025
Where
Page 15 of 33
ck denotes the cost in a time unit of the k th plane assigned; 
ik
variable determines whether the
is a 0-1
k th plane search the i th cell or not; d 1 , d 2 denote the
ik
ik
th
one-way distance of the k plane to the i th cell and the distance of searching of the
of the k
th
plane respectively;  k denotes the duration of flight of the k
th
plane;
hi ,ocean denotes the ocean depth of the i th cell.
 Model solving based on PSGA
The general solutions to multi-objective programming problem are the linear
weighting method and the layered method. The former method lacks credibility in
weights, while the latter is also rough in evaluating the priority.
On account of the weaknesses of the two solutions, we adopt the genetic
algorithm focused on Pareto optimal into Multi-Objective Optimization mentioned in
HU Guiqiang[2008].we apply parallel selected genetic algorithm(PSGA).It’s basic
idea is: divide all the individuals of the population into groups by the number of
objective functions equally ,each group assigned a objective function, each group
operate selection independently corresponding to its own objective function and select
some highest fitness individuals to form a new group, then the newly generated
groups merge into a complete generation, then operate crossover and mutation in it,
thus creating the next population. Run the "segmentation - parallel selection - merge"
mechanism constantly, we can work out the optimal solution of multi-objective
optimization problems in the end.
Now we are going to using binary encoding PSGA to optimize ik
.
The specific steps are as follows:
 Coding
Convert the variables to be optimized into binary strings.
 Grouping and Individual adaption evaluation
Divide all the individuals of the population into two equal size groups.
Compute the fitness value of the first group with the first objective
(1)
i
function, fitness

I
M
i
k 1

ik
Pod (k ) Poc(i )
T
,
(19)
By the same token, the fitness value of the second group is
N
fitnessi(2)   tk ck
k 1
(20)
 Selection and merge
Here we adopt the roulette method to generate selection probability, in order to
ensure that the higher the fitness, the more likely to be inherited.
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Both group operate independent selection with correspond to the fitness function,
select out the highest fitness individuals, then merge into a complete generation.
 Crossover
Here we adopt 1-point crossover, the crossover loci creates at random.
Exchange
the right part of the crossover loci to get the new chromosome.
 Mutation
Produce random numbers as many as the total number of each generation and
number them, pick the ones smaller than the mutation probability, then inverse the
gene value , namely 0
1.
Therefore, new population comes into being after a round of parallel-selection,
crossover and mutation with ik been adjusted. Cycle by this process, We can obtain
the ideal solution that meets the condition through finite steps.
Figure 8:
flow chart of PSGA
5.2 Drift strategy
The overall look of drift strategy is shown in the figure below:
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Page 17 of 33
Figure 9:
flow chart of drift strategy
5.2.1 Refresh the probability of containment(POC)
For a dynamic drift strategy ,the timely probability of containing (POC) is
rectified by three aspects, the posterior probability of containing, the new found
floaters and the motion of the floaters.
 the posterior probability
This aspect has already been discussed in 5.1.1.
 the new found objects
According to the discussion of the initial POC, we adopt Multi-mode hybrid
method to describe the probability distribution of the objects. Inspect of the track
distribution N1  L  the uniform distribution N 2 and there exists another distribution
point distribution N3  x  ,which is a two-dimensional normal distribution centered by
the location of the new found object X.
Therefore the containing probability distribution is revised to
Poc  w1 N1  w2 N 2  w3 N3 ,
w1 , w2 , w3  0, w1  w2  w1  1

,
the motion of the objects
As is discussed in part 5.1.1, the velocity of drift object is defined
by v0 ,concerning the current, wind, etc. However, it is only the directional drift of its
motion. Additionally, the motion is slightly influenced by the stochastic surge caused
by wave, as is shown in figure 9. The velocity of wave is non directional drift which
obeys random motion. So we use Monte-Carlo method to simulate the motion of the
objects.
Figure 10 :
the motion of object
On the basis of the three revise steps above, we can refresh the POC by every
time unit.
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5.2.2 The flight path of SRU in a cell
The same circumstance has been discussed in 5.1.2.
5.2.3 ‘Nuggets-Gold miners Model ’
Aimed at the scheme of distribution of search effort on drift strategy, we
introduce our ‘Nuggets-Gold miners Model’ creatively.
The ‘Nuggets’ are those who are assigned to precise search in the maximum
probability cells, while the ‘Gold miners’ are those who are assigned to search
traverse the whole area and revise POC.
That’s to say, it’s a kind of game between the existent and the potential
maximum probability. If there are only ‘Nuggets’, the search task will come into a
dead end keeping searching by the prior distribution, however, with the help of
‘Gold miners’, we can generate posterior distribution in instance another object is
found. By contrast, most existing models only target at the suspected cells, our
model can also find new suspected cells constantly. Owing to the Gold miners, our
model stand out.
5.2.4 Multi-objective programming of the drift strategy
Owing to the planes equipped with electronics that have night vision such as
radar( E (2) ),infrared camera( E (3) ),thermal imaging( E (5) ),We introduce the
day-night coefficient  =
Tnight
T
.
 is determined by the proportion of night time varies from the location.
Therefore the actual work time can be extended to the effective Te .
Te  [1   (1  (1  Ek (2))(1  Ek (3))(1  Ek (5)))]T
I
M
i
k 1
the Probability of Success of drift strategy POS2  ik Pod (k ) Poc(i )
The multi-objective programming model of drift and fallout are almost the same.
The only difference is in the second constraint where the actual work time be replaced
by the effective work time.
POS2
max
T
M
min  tk ck
k 1

 ik Pod (k )  1  i  1, 2, , I
 k
 2d 1  d 2
i
ik
s.t.  k
 Te  k  1, 2, , M
v
i
k

 2d 1  d 2
ik
 ik
  k  k  1, 2, , M
 vk
(20)
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With the two strategy solved respectively, we obtain the total probability
of success POS  POS1  POS2 .
6 Model solving with a particular case
We simulate our generic model with a particular case: a airbus 321 plane crashed in the
Indian Ocean.

Initial probability of containment
Figure 11 :
rectangular search area in the Indian ocean
Take the uniform distribution and track distribution of the plane, we obtain the
initial POC of the crash area.
Figure 12 : initial probability of containment
 The probability of detection
We adopt AHP to calculate the combination weights of sweep width W .
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the pairwise comparison matrix which measures the weight
of the three aspects of the first-level evaluation:
1

1 4

A  4 1

1
2
2

1
2

2

1

Check the consistency of matrix A:
We can obtain the largest eigenvalue and its weight vector correspondingly
 A  3.0000  (A)  (0.2182 0.8729 0.4364)
The consistency index is CI (A) 
 A  mA
mA  1
0
From Table 2, when mA  3 ,the random consistency index RI  0.58
m
RI
1
0
2
0
Table 2: The Quantitative Values of RI
3
4
5
6
7
0.58 0.90 1.12 1.24 1.32
Then, we can obtain consistency ratio CR
(A)
8
1.41
9
1.45
10
11
1.49 1.51
CI (A)

 0  0.1
RI
Therefore, we can safely draw the conclusion that the inconsistent degree of
matrix A is in a tolerable range, and we can take its normalized eigenvector 0 =
（0.1428 0.5715 0.2857）as weight vector .
Namely, W  0.1428 a1 +0.5715 a2 +0.2857 a3
In the same, the pairwise comparison matrix between the second and the third level,
third and the forth level are as follows:
1

1 1 5 


1

B1  1 1

5


 5 5 1 



1 2

1
B2  
1
2

 3 6

B1  3.0000、 B 2  3.0000、 B3  2.0000
1
3

1
6

1


1

1

B3 
2


2 1 
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The consistency index : CI (B1)  0、 CI (B2)  0、 CI (B3)  0;
the consistency ratio : CR(B1)  0、 CR(B2)  0、 CR(B3)  0,less than0.1,pass the
consistency check.
the weight vector   B1  (0.1925 0.1925 0.9623)、   B 2  (0.3123 0.1562 0.9370)、
  B3  (0.4472 0.8944)
normalization: 1(1) （0.1429 0.1429 0.7142）, 1(2) （0.2222 0.1111 0.6667）, 1(3) 
（0.3333 0.6667）
then
=0.1429
a1
b1
+0.1429
b2
+0.7142
b3
,
a2
=0.2222
b4 +0.1111 b5 +0.6667 b6 , a3 =0.3333 b7 +0.6667 b8
1

1
1

C4   3
1
2

 1
2
1
3
2
1
3
2
1
1
2
1
3
1
2
1
2
2 1
2 1
2

2
1


1
2

1

1  C7  
3

1

1 

4
3
1
1
2

4

2
1

 C8   1

1
7

7

1 

he largest eigenvalue: C 4  5.0133 C 7  3.0183 C 8  2.0000
the weight vector:
 C 4  (0.6143 0.6143 0.1831 0.3254 0.3254)
 C 7  (0.9154 0.3493 0.1999)  C 8  (0.9899 0.1414)
the consistency index : CI (C 4)  0.0033、 CI (C 7)  0.0092、 CI (C 8)  0;
the consistency ratio : CR(C 4)  0.0029 、 CR(C 7)  0.0158 、 CR(C 8)  0; less than
0.1,pass the consistency check.
(4)
Normalization: 2  (0.2978 0.2978 0.0888 0.1578 0.1578)
(8)
2( 7 ) (0.6250 0.2385 0.1365) 2  (0.8750 0.1250)
then
b4 
c
c
c
c
c
0.2978 1 +0.2978 2 + 0.0888 3 + 0.1578 4 + 0.1578 5
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b7  0.6250 c6 + 0.2385 c7 + 0.1365 c8
b8  0.8750 c9 + 0.1250 c10
Thus, we can compute the combination weights

W = 0 . 0 2b01 4
0 . 0 6 b35 5
0 .b02
204
0 . b3 681 0
b03. 1 0 2 0c 1 0 . 0 3c7 82
0 .c0 3 37 82 0 c0 0. 04 01 .1032c 0 005 . 0
c0 6
. 0 5 9 5 c 7 0 . 0 2 c2 78
0 . c0 139 0 38c0 . 11 60 6 7
The time-varying fallout probability of wreckage
Based on the information of airbus 321 plane, we can obtain the safety margin of the
crash plane, the overall time-varying fallout probability of wreckage is provided.
1
0.9
0.8
0.7
pfall
0.6
0.5
0.4
0.3
0.2
0.1
0
0
Figure 14:
1
2
3
4
5
6
time/month
7
8
9
10
the time-varying fallout probability of wreckage
In consideration of the ratio of fallouts and drifts shown in Figure 14, we
obtain the number of planes assigned to the two strategies.
Now we are going to establish a two-month search scheme taking it into
account.
Assuming that we have five P-3 Orions, four Ilyushin-76s, six P-8 Poseidons,
three Hercules. The characteristic matrix of this planes are [0 0 1 1 1], [1 1 0 1 0],[0
0.02
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Page 23 of 33
0 1 1 1],[1 1 0 0 1].
Now we will provide a optimal scheme of day 60th.

Dynamic POC
To start with, we generate two teams of planes satisfied the constraints the 0-1
programming at random. The result shows that two P-8 Poseidons, a P-3 Orion and a
Ilyushin-76 are assigned to the fallout strategy while the rest of planes is assigned to
the drift strategy.
According to data of current and wind from Google Earth, on the basis of the
Bayesian Law, we adopt Monte-Carlo simulation to refresh the POC as discussed in
5.1.1 and 5.2.1.
Figure 15: the ocean climate (a) current (b) wind
We can obtain the dynamic refreshed POC of the fallout strategy and the drift
strategy. So as to achieve visualization in scientific computing, we offer three sub-graphs of
day 20th,40th and 60th,as shown in Figure 16 ,17.
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Page 24 of 33
Figure 16: the refreshed POC of the fallout strategy
Figure 17: the refreshed POC of the drift strategy
We can easily conclude from the two figures that the refreshed POC of fallout
strategy is almost along an obvious track, while the refreshed POC of drift strategy is
much more scattered. This credit to our ‘Gold miners’ who keep finding new
suspected cells while ‘ Nuggets’ are assigned to precise search in the maximum
probability cells. So our innovation in adopting ‘Nuggets-Gold miners Model’ makes
a remarkable difference.
Besides, for the fallout strategy, we determine the influence of the actual sonar detecting
depth. So we train the artificial BP Neural Network.
Import the typical samples to neural networks:
[5 4 3 4 5 3 4]  =0.18 ;[4 3 2 3 4 5 3]  =0.27;[3 3 4 1 2 1 3]  =0.53;
[1 2 3 2 1 2 1]  =0.89;[3 2 1 2 3 2 4]  =0.08
 Constraints of sonar
After normalizing the data. Corresponding parameter is: Maximum allowable
number of failures :20, training speed: 0.05,training times:1000, training
precision:10^(-6),the first hidden layer activation function is hyperbolic
S-shaped function, 5 neurons, the second hidden layer activation function is
hyperbolic S-shaped function, 5 neurons, the third hidden layer activation
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Page 25 of 33
function is hyperbolic S-shaped function, 1 neurons, Output layer use linear
activation function, 1 neuron.
Thus we can the reduction coefficient and obtain the actual detecting depth.
Then according to the data of ocean depth from Google Earth, we offer a map
of the area that the search plane can access (set a Ilyushim-76 as example).
Figure 18:
(a) the ocean depth
The red area in
(b) area accessible to search plane
Figure 18(b) illustrate where the plane can access while the
blue area illustrate where the plane can’t access.

The optimal distribution scheme of drift strategy and fallout strategy
Now we optimize the multi-objective programming by PSGA respectively in two
strategies, the optimal allocation of the planes are shown in Figure 19.
Figure 19: the optimal allocation of search planes in day 60th
(a)drift strategy (b)fallout strategy
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Page 26 of 33
Here in the figures,
illustrates the departure point of a search plane, + illustrates
the destination of it. Besides, the type of plane varies with the color of
, the blue
line illustrate the track of each plane.
 The total optimal distribution scheme
After many cycles by the elitist strategy, we get the optimal POS1 and the
optimal POS2 together with correspondingly allocation. Combine the allocation of
fallout strategy and drift strategy, finally we obtain the optimal POS and the whole
allocation strategy of day 60th as is shown in Figure 20.
150
100
Y/grid
50
0
-50
-100
-150
-150
-100
-50
0
X/grid
50
100
150
Figure 20： the allocation of search planes in day 60th
With our models, we can get the allocation of search planes of every day. Here is
two samples of allocation in day 20th and day 40th.
Team #34025
Page 27 of 33
Figure 21： the allocation of search planes (a) day 20th (b) day 40th
 The optimal result
Finally, we can easily obtain the optimal POS varies with time.
0.35
0.3
0.25
POS
0.2
0.15
0.1
0.05
0
0
10
20
Figure 22:
30
time/day
40
50
60
the time-varying total POS
The optimal POS for 60 days is 27.345%.
At the same time, we calculate out the optimal POS for a year is 62.751%.
The total cost is about $3,455,000,000 a year.
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Page 28 of 33
7 Result analysis

In the previous paper, we obtain the probability of success (POS) in any time
quantum. In order to enhance POS , we maybe dispatch as many as possible
planes to participate search plan. However, in fact it is necessary to consider
economic consumption for each search plan. Therefore we discuss about the
relationship of POS and quantity of planes. we obtain the hisgram as shown in
Figure 23.
0.35
0.3
0.25
POS
0.2
0.15
0.1
0.05
0
10
15
20
25
plane's number/unit
30
Figure 23: plane’s number/unit--POS
In this histogram, first we can clearly know with the quantity of planes increasing ,the
POS gradually also appear increasing trend, second when the quantity of planes range
from 10 to 20,the change rate of POS is obvious, thus we should consider POS firstly
although the expense become higher. Nevertheless, when the quantity of planes
range from 20 to 30, the change rate of POS become smaller. Thus we’re supposed to
combine the economic consumption and POS , so as to we can obtain the proper
quantity of planes.

In the previous paper, we also obtain the relationship of the search time unit
and the increment of the probability of success ( POS ) .
Team #34025
Page 29 of 33
0.03
0.025
△POS
0.02
0.015
0.01
0.005
0
30
60
90 120 150 180 210 240 270 300 330 360 390410 440
time/day
Figure 24: search time unit- POS
In this histogram, we can know when the time range from 30days to 90days, the
POS appear downward trend, because with time going by, the search objects maybe
sink or the search objects are pushed by wave, current etc. But when the time range to
150days or 240days, POS suddenly arrive at a high probability, we can conjecture
SRU find the suspected drift objects, hence the change rate of POS is high. when the
time range to 300days, the POS appear perfectly downward trend, we can conjecture
plenty of drift objects sink in the sea, however ,it is difficult to search and fishing
them. The curve contrast diagram of the drift strategy’ POS and the fallout strategy’
POS in a year as shown in Figure 25:
0.7
0.6
0.5
POS
0.4
0.3
0.2
0.1
0
Figure 25:
0
50
100
150
200
time/day
250
300
350
400
Comparison chart of POS for drifts and fallouts
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Page 30 of 33
In this diagram, the blue curve represent the relationship of search time and drift
strategy’ POS, the green curve represent the relationship of search time and fallout
strategy’ POS. From the diagram our conjecture in previous paper is correct. With
time going by, the POS of the two strategy both increasing, the POS of fallout strategy
lower than the POS of drift strategy during all the time.
8 Error Analysis
8.1Probably Error
In previous paper, the initial probability of the object accord with normal
distribution, thus the change of probably error E will lead to the change of initial
probability, according to normal distribution table and formula, we can obtain the
relationship of the POC and E as shown in Figure 25:
1
0.95
0.9
0.85
POC
0.8
0.75
0.7
0.65
0.6
0.55
0.5
1
1.5
2
2.5
E
Figure 26:
the relationship between probably error and POC
From Figure 26, we can know the increasing trend of POC gradually smaller,
when the E range to 2.5 E , the POC almost equal to 100%.
8.2 Navigation Error
In previous paper, we refer the flight path of SRU in a cell. The flight distance d in a
cell will change in the result of actual navigation capability, we assume the motion of
plane weighted by random motion and accordion motion. The navigation coefficient
(Coe) is 0 to 1.The navigation coefficient influence POS obvious. We simulate and
give a table to describe the relationship as Table 3.
Team #34025
Time
POS
Page 31 of 33
30 days
60 days
90 days
120 days
0
0.265
0.307
0.335
0.350
0.3
0.246
0.286
0.314
0.339
Coe
Table 3: the relationship between Navigation Error and POS
We can know that POS obviously changes lower when the navigation coefficient
higher.
9 Strengths and Weaknesses
9.1 Strengths
 Excellent Timeliness. The function of the model is related to time, we can obtain
the search plan in any day. Therefore this model has a high accuracy. Moreover
the model is accordance with reality.
 Comprehensive Analysis. We consider all aspects of the search plan. We discuss
not only the Ocean system ,but also the various parameters of lost plane and
the search plane.
9.2 Weaknesses
 Difficult to determine weight. In order to obtain an accurate model, the model
needs to determine weights carefully. However, it is difficult to make proper
weights.
References:
[1]LiHao, Study on the probability of containment at sea.[2011] June
[2]Kratzke T M, Stone L D, Frost J R. Search and rescue optimal planning
system[C]//Information Fusion (FUSION), 2010 13th Conference on. IEEE, 2010:
1-8.
[3]ZhouTao,Research on POD in Sea Searches,2011,6,26
[4]QIN Shengping. Time variant reliability analysis of ship strucyures considering
corrosion [D], Shanghai Jiao Tong University, 2002
[5]Liu Yan, Pan Wengliang, Effect of oceanographic environment on sonar
performance,Meteorological,Hydrological and Marine Instruments,No.3 Sep.2012
[6] HU Guiqiang,The Research and Implementation of Genetic Algorithm for
Multi-Objective Optimization,Journal of Chongqing University of Arts and
Sciences(Natural Science Edition),Vol 27 No 5 Oct,2008
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A report for future searching plan
The search task of the lost Malaysian flight MH370 has been conducting for
almost a year since March 8th, 2014. With time going by, the hope of rescue the
victims has dying out. The search task progresses more and more slowly. As a
spokesman for airline, we show deeply apology into this and we ensure that we will
never forgive searching until the day we find the wreckages and find out the truth
behind the accident. And we would give account to the relatives of the victims and do
the aftermath settlement as much as we can.
On the occasion of the first anniversary of the crash, we hold a press conference
to summarize the existing search and announce our advanced plan for future searches.
For the search problem of a lost plane feared to have crashed on the open water,
we build a generic mathematical search model on the foundation of search theory
taking various types of search planes and crash planes into account. The basic
conception of our search task is to achieve the maximum the probability of success
(POS) and reduce the total cost if possible in a time unit . Of course, the principle is to
rescue.
In order to search successfully, first of all, we employ Multi-mode hybrid methods
to obtain the initial probability of containment (POC). Then we consider the
synthesized detecting ability of search planes, so as to evaluate the probability of
detection (POD).The next procedure is to optimize the assign distribution scheme of
quantity and type of search planes, which is the most essential part. We divide our
search strategy into two categories: drift strategy and fallout strategy. We allocate the
quantity of planes according to the curve of the settlement probability.
Aimed at the two search strategies we establish dynamic POC model on the
foundation of Bayesian. Owing to the influence of ocean environment and electronics,
we establish programming with multiple objectives and multiple stages. Especially, in
fallout strategy, we solve the problem of detecting depth of sonar adopting Neural
Network. In drift strategy, we create a new model named ‘Nuggets-Gold miners’; The
‘Nuggets’ are those who are assigned to precise search in the maximum probability
cells, while the ‘Gold miners’ are those who are assigned to search traverse the whole
area and revise POC, thus enhancing the POC obviously.
So with the newly-established model, we can increase the POS to the maximum!