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LOD 2019 paper 99

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A hybrid Egale Strategy with Moth-Flame
Optimizer for Reliability Analysis
Aziz Hraiba1 , Achraf Touil1 , and Ahmed Mousrij1
Laboratory of Engineering, of Industrial Management and Innovation,
Faculty of Sciences and Technology, Hassan 1st University,
PO Box 577, Settat, Morocco
[email protected], [email protected], [email protected]
Abstract. In this paper, we devoted the reliability analysis by combining a two-stage eagle strategy (ES) with the first-order reliability method
(FORM). In the first stage of ES, we use the so-called levy flights to improve the global search ability. In the second stage, we use the moth-flame
optimizer (MFO) to intensify the local search. In the proposed method
named ES-MFO, FORM is used to evaluate the fitness of each moth. In
order to investigate the efficiencies of ES-MFO in reliability analysis, four
classic examples, as well as a roof truss model are employed. The results
are compared to four well-known heuristic algorithms. The results show
that reliability analysis by using ES-MFO is significantly better than the
current heuristic algorithms.
Keywords: Reliability Analysis · Eagle Strategy (ES) · Moth-Flame
Optimizer (MFO) · First-Order Reliability Method (FORM).
1
Introduction
The structural safety evaluation methods aim to evaluate the likelihood of a violation of the boundary condition by comparing probabilistic models active loads
and resistance of a component or structural system. A limit state is a condition
beyond which a structure exceeds a specified design requirement expressed in
mathematical form by a limit state function G (X). The probability of failure
(Pf) is defined as the probability of occurrence of failure (G (X)≤0), where X is
a random variables vector representing the uncertainties of the loads, as well as
on the material and geometrical properties of the structure. Although, the uncertainties are quantified in a probabilistic manner and the probability of failure
is used as the magnitude used as a basis for the safety measure.
There are several methods in the literature. The most famous is the Monte
Carlo simulation (MCS), which represents the reference for all other methods
[11]. In a paper by [7] describes that the first-order reliability method (FORM)
is more elegant and efficient than simulation methods. However, the previously
discussed methods depend on the possibility to calculate the value of G(X) for
a vector X. Sometimes these values require the results of other programs (finite
element), or the limit state function G is implicit. Or it might be ineffective to
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Hraiba et al.
link the iteration of the index of reliability with a non-linear dynamic analysis.
In addition the computational cost can be very high.
Currently, swarm intelligence algorithms are efficiently used to solve complex
optimization problems such as reliability analysis. They work effectively and have
many advantages over traditional deterministic methods and algorithms. It has
become evident that the researchers concentrated on using single metaheuristics.
However, there are some limitations. To overcome this problem, a wide variety
of hybrid approaches are proposed in the literature. The main idea of a hybrid
with two or more metaheuristics was inspired by the possibility that the new hybridized algorithm combines the strengths of each of these algorithms to provide
the following advantages: (i) to produce better solutions, (ii) to provide solutions in less time. In literature, a wide range of methods has been proposed by
combining the generic algorithm and Particle Swarm Optimization for reliability
analysis [2], [6]. Recently [16], they proposed a hybrid method based on particle
swarm otpimization combined with choatic theory in order to improve the global
search of standard PSO. The proposed method was tested on four examples as
well as a circular tunnel. The reported results shows that the proposed can identify the design point and compute the corresponding reliability index with high
accuracy.
Despite the merits of the above-mentioned works, the problem of local optima
entrapment still persists. In addition, there is a theorem in the field of heuristics called No Free Lunch [14] that says there is no optimization algorithm for
solving all problems. Since there are differents explicit and implicit state limit
functions. Hence, there are possibilities that one algorithm performs well on a
state limit set but worse on another. These reasons allow researcher to investigate the efficiencies of new algorithms in enhancing reliability analysis. This is
also the contribution of this study, in which the two-stage eagle strategy (ES)
recently proposed by [?] is proposed to be embedded to reliability analysis. In
this two-stage strategy, the first stage explores the search space globally by using
the so-called levy flight, if it finds a promising solution, then an intensive local
search is employed using a more efficient local optimizer such as hill-climbing.
Then, the two-stage process starts again with new global exploration followed
by a local search in a new region. One of the remarkable advantages of such
as a combination is to use a balanced tradeoff between global search (which
is generally slow) and a fast local search. To the best of our knowledge there
is no previous work that attempts to use ES in conjunction with Moth-Flame
Optimizer (MFO) [15] as a local optimizer for reliability analysis.
The rest of the paper is organized as follows. Section 2 describes the reliability
analysis. Section 3 provides the methodologies utilized in this paper. Section 4
reports the numerical results and discussion. Finally, our conclusions and future
work are presented in Section 5.
A hybrid Egale Strategy with Moth-Flame Optimizer for Reliability Analysis
2
3
Probabilistic modeling
Probabilistic modelling focuses on the system failure probability, this not query
to the phenomena that provoke them, but the frequency with which they occur.
Therefore, it is not a physical theory, but a theory of probabilities and statistics
[3]. Structural reliability is based on the probabilistic model and provides methods to quantify the failure probability. Several important contributions in this
area include the work developed by [3, 13, 4]
The reliability is defined as the probability that the performance function
G(X) is greater than zero. In other words, the theory of reliability assumes that
it is possible to estimate this event using a mathematical model, thus calculating
its failure probability [1]. The positive values of G(X) correspond to safety situations and the function negative values give the failure situations. Fig 1, illustrate
a general description of relibaility analysis.
Fig. 1: Probability of faillure
The reliability experiment is usually expressed in terms of equations (1),
Where F called failure events. The probability that the event F occurs is given
by the fact that the stress exceeds the resistance of the structure.
Pf = P r{G(X) < 0}
(1)
G (X) represente the performance function, Where X random variables noted
X = (X1 , ..., Xn ). These n random variables are called basic variables which
represent a physical uncertainty of the model.
Low and Tang [8] proposed a new algorithm for FORM by a new interpret
the Hasofer-Lind index, this approach admits the expansion of an ellipsoid in
the original space of the basic random variables and minimized the reliability
index β as :
s
T
Xi − µi
−1 Xi − µi
β = minX∈F
[ R]
(2)
σi
σi
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Hraiba et al.
Where µi and σi are respectively the mean and de standare deveiation for random varibale X, and R represente the correlation matrix. The probability can
estemated by:
Pf = 1 − φ(β)
(3)
Where φ(.) is the cumulative distribution function of the standard normal variable.
Based on the above assumptions, the following constrained optimization has
to be solved :

q
T
−1


[n] [ R] [n]
 minimize
(4)
Subject to:



G(X) = 0
where G(X) is the limited state function.
3
Two-stage Eagle Strategy based on MFO
This section describes our proposed two stage eagle strategy with moth flame
optimizer
3.1
Brief introduction to Eagle Strategy
Eagle strategy is a two-stage optimization strategy that was presented by [?].
This algorithm mimics the hunting behavior of eagles in nature. In fact, eagles
forage using two components: random search performed by flying freely and
intensive search to catch prey when sighted. In this two-stage strategy, the first
stage explores the search space globally by using a Levy flight; if it finds a
promising solution, then an intensive local search is employed using a more
efficient local optimizer. Then, the two-stage process starts again with new global
exploration, followed by a local search in a new region. The main steps of the
ES are outlined in Algorithm 1.
Algorithm 1 Eagle Startegy:
1:
2:
3:
4:
5:
6:
7:
8:
9:
10:
11:
Objective function f (x)
Initialization and random initial guess xt=0
while (stop criterion) do
Global exploration by randomization (e.g. Levy flights)
Evaluate the objectives and find a promising solution
Intensive local search via an efficient local optimizer
if (a better solution is found) then
Update the current best
end if
Update t = t + 1
end while
A hybrid Egale Strategy with Moth-Flame Optimizer for Reliability Analysis
3.2
5
Brief introduction to Moth-Flame Optimizer (MFO)
The Moth-flame optimization (MFO) is a new population-based stochastic search
algorithm recently proposed by [10]. The MFO is a newly developed optimization
technique to solve complex engineering optimization problems[5]. It is inspired
by the behavoir of moths for their special navigation methods in night.
In the MFO algorithm, the set of moths is represented in a matrix M . For
all the moths, there is an array OM for storing the corresponding fitness values.
The second key components in the algorithm are flames. A matrix F similar to
the moth matrix is considered. For the flames, it is also assumed that there is
an array OF for storing the corresponding fitness values.
The MFO algorithm is a three-tuple that approximates the global optimal of
the optimization problems and defined as follows:
M F O = (I, P, T )
(5)
I is a function that generates a random population of moths and corresponding
fitness values. The methodical model of this function is as follows:
I : Ø −→ {M, OM }
(6)
The P function, which is the main function, moves the moths around the search
space. This function received the matrix of M and returns its updated one eventually.
P : M −→ M
(7)
The T function returns true if the termination criterion is satisfied and false if
the termination criterion is not satisfied:
T : M −→ {T rue, F alse}
(8)
With I, P , and T , the general framework of the MFO algorithm is defined as
follows:
M = I();
while T (M ) is equal to false
M = P (M );
end
After the initialization, the P function is iteratively run until the T function
returns true. The P function is the main function that moves the moths around
the search space. As mentioned above the inspiration of this algorithm is the
transverse orientation. In order to mathematically model this behaviour, we
update the position of each moth with respect to a flame using the following
equation:
Mi = S(Mi , Fj )
(9)
where Mi indicate the ith moth, fj indicates the j th flame, and S is the spiral
function. A logarithmic spiral is defined for the MFO algorithm as follows:
S(Mi , Fj ) = Di × ebt × cos(2πt) + Fj
(10)
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Hraiba et al.
where Di indicates the distance of the ith moth for the j th flame, b is a constant
for defining the shape of the logarithmic spiral, and t is a random number in
[−1, 1].
Di is calculated as follows:
Di = |Fj − Mi |
(11)
where Mi indicate the ith moth, indicates the j th flame, and Di indicates the
distance of the th moth for the j th flame.
Equation (10) describes the spiral flying path of moths. From this equation,
the next position of a moth is defined with respect to a flame. The t parameter
in the spiral equation defines how much the next position of the moth should
be close to the flame ( t = −1 is the closest position to the flame, while t = 1
shows the farthest). A question that may rise here is that the position updating
in (10) only requires the moths to move towards a flame, yet it causes the MFO
algorithm to be trapped in local optima quickly. In order to prevent this, each
moth is obliged to update its position using only one of the flames in (10).
Another concern here is that the position updating of moths with respect to
different locations in the search space may degrade the exploitation of the best
promising solutions. To resolve this concern, an adaptive mechanism provided
the number of flames. The following formula is utilized in this regard:
f lame − no = round(N − l ∗
N −1
)
T
(12)
where l is the current number of iteration, N is the maximum number of flames,
and T indicates the maximum number of iterations. The gradual decrement in
number of flames balances exploration and exploitation of the search space.
3.3
Two-stage Eagle Strategy ES-MFO for reliability analysis
In this subsection, we explain the principal phases of the proposed to obtaining
optimal solution as follows.
Evaluation To evaluate the fitness of each moth of ES-MFO, the constrained
problem (4) should transformed to unconstrained problem by using the penaly
method proposed by [16]. Then the fitness of each moth is gievn as follows:
q
T
−1
F itness = [n] [ R] [n] + M × |G(X)|
(13)
Exploration phase using Levy flights In the first stage, ES uses the so-called
Levy flights, which represent a kind of non-Gaussian stochastic process whose
step sizes are distributed based on a Levy stable distribution to generate new
solutions. When a new solution is produced, the following Levy flight is applied:
Xit+1 = Xit + α ⊕ Levy(λ)
(14)
A hybrid Egale Strategy with Moth-Flame Optimizer for Reliability Analysis
7
Here, α is the step size that is relevant to the scales of the problem. The product
⊕ means entry-wise multiplications. Levy flights essentially provide a random
walk while their random steps are drawn from a Levy distribution for large steps:
Levy(λ) = u = t−λ , 1 ≤ λ ≤ 3
(15)
In this paper, we will use the algorithm proposed by [9], which is one of the most
efficient algorithms used to implement Levy flights.
Exploitation phase using ES-MFO We know that the MFO is global search
algorithm, but it can easily be tuned to do an efficient local search by limiting new
solutions locally around the most promising region. Since, such a combination
between ES and ES-MFO may produce better results than those using pure
MFO.
Framework of ES-MFO for reliability analysis The main steps of the
proposed ES-MFO for reliability analysis are described in Algorithm 2, where R
is the number of rounds, and Np is the population size.
4
Results and discussion
In this section, the proposed algorithm is compared with the choatic particle
swarm based reliability analysis proposed by [16]. The experiments were done
using MATLAB R2014a on PC with a 3.30 GHz Intel(R) Core (TM) i5 processor, 4GB of memory. In the paper by[16] describes a set of stat limit functions
which are presented in in the following example 1-4;
Example 1:
X1
X23
(16)
+ 0.00483 × (X1 + X2 − 20)4
(17)
G(X1 , X2 ) = 0.01846154 − 74.76923 ×
Example 2:
G(X1 , X2 ) = 2.5 − 0.2357 × (X1 − X2 )
Example 3:
G(X1 , X2 ) = e0.4(X1 +2)+6.2 − e0.3X2 +5 − 200
(18)
Example 4:
G(X1 , X2 , X3 ) = X1 −
X2
X3
(19)
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Hraiba et al.
Algorithm 2 The proposed algorithm for the reliability analysis
Initialization. Population of moth (Xi = 1, 2, ..., N ), N P , the maximum of iterations Gmax and rounds R;
while (R > r) do
Generate a set of moths X for global exploration using Levy flights implemented
by Mantegna alogirthm;
Compute the reliability index using the algorithm of Low and Tang.
Compute the fitness function using Eq.(13)
Update the best solution obtained so far
Inner loop :
Generate randomly a set of moths around this promising solution using using Levy
flights implemented by Mantegna alogirthm;
Local Search using MFO
while (g <= Gmax ) do
Update flame no using (12)
OM = Fitness(M ) using (15);
if (g == 1) then
F =sort(M );
OF =sort(OM );
else
F =sort(Mg − 1, Mg );
OF =sort(Mg−1 , Mg );
end if
for i = 1 : N do
for j = 1 : d do
Update r and t;
Calculate D using (11) with respect to the corresponding moth;
Update M (i, j) using (9) and (10) with respect to the corresponding moth;
end for
end for
Update g = g + 1
end while
if (a better solution is found) then
Update the current best
end if
End Inner loop
Update r = r + 1
end while
Report the best solution
A hybrid Egale Strategy with Moth-Flame Optimizer for Reliability Analysis
Example Variables Distribution Mean
X1
Normal
1000
(1)
X2
Normal
250
X1
Normal
10
(2)
X2
Normal
10
X1
Normal
1000
(3)
X2
Normal
250
X1
Normal
600
(4)
X2
Normal
1000
X3
Normal
2
STD
200
37.5
3
3
200
37.5
30
33
0.1
9
M
1000
1000
0.1
1000
Parameter
Value
Population size (N P )
1000
Maximum of generations (M axGen ) 20
Levy parameter
1.5
Number of rounds
5
(b) Parameters for Examples 1, 2, 3 & 4
(a) Given data for Examples 1, 2, 3 & 4
Example 1
Desing Points
Algorithms
RI Pf (10−2 )
X1
X2
ES-MFO 1118.5655 165.4647 2.3309 0.9879
CPSO
1125.5766 165.8097 2.3312 0.9871
Low& Tang 1118.5578 165.4640 2.3309 0.9879
MCS
0.9607
Example 2
Desing Points
Pf
Algorithms
RI
X1
X2
(10−2 )
ES-MFO
15.3034
4.6966 2.5000 0.62
CPSO
15.3334
4.7267 2.5001
0.62
Low& Tang 15.3034
4.6966 2.5000
0.62
MCS
RI: Reliability-Index; Pf : Probability of faillure
Example 3
Desing Points
Algorithms
RI
X1
X2
ES-MFO -2.5397
0.9453
2.7099
CPSO
-2.5407
0.9427
2.7099
Low& Tang -2.5397
0.9454
2.7099
MCS
2.685
Example 4
Desing Points
Algorithms
RI
X1
X2
X3
ES-MFO 555.6085 1029.0027 1.852028 2.2697
CPSO
553.2864 1023.0742 1.84909 2.2784
Low& Tang 555.6091 1029.0028 1.85203 2.2697
MCS
2.2490
RI: Reliability-Index;
(c) Comparison of results for Example 1 & 2
(d) Comparison of results for Example 3 & 4
Parameter Distribution Mean
q[N/m]
Normal
20000
l[m]
Normal
12
AS [m2 ]
Normal
9, 8210−4
AC [m2 ]
Normal
0,04
ES [N/m2 ]
Normal
1 × 1011
EC [N/m2 ]
Normal
2 × 1010
CV
7%
1%
6%
12%
6%
6%
(e) Variables of Roof truss model structure
Algorithms
MCS ES-MFO
q[N/m]
20476
l[m]
12.01
AS [m2 ]
96.95 × 10−4
AC [m2 ]
0.039096
ES [N/m2 ]
9.4 × 1010
EC [N/m2 ]
1.8 × 1010
−3
Pf (10 )
9.38
9.48
Parameters
(f) Comparison of results for Application
Table 1: Overall data and results
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Hraiba et al.
All variables are considered mutually independent normal distributions with
a given mean and standard deviation and the value of penalty coefficient (M)
are presented in Table 1. For the other parameters are given in Table 2.
For the example 1, the obtained results are listed in Table 3. It can be
seen that the exact failure probability is obtained using MCS with importance
sampling is 0.9607 × 102 , while as 0.9876 × 102 by using the proposed ES-MFO
which same value as Low and Tang method and better than the obtained with
CPSO. In addition, as shown in Fig 2 the proposed method converges quickly to
satisafied reliability index and design points after only a 50 generations which
represents a gain in computation time.
For the example 2, the obtained results are listed in Table 3 and Fig 3 . It
can be seen that the proposed ES-MFO obtained the same value as Low and
Tang method and better than the obtained with CPSO.
For the example 3, according to the obtained results listed in Table 4, the
proposed method can compute the reliability index with high efficiency and accuracy. In addition, as shown in Fig 4 the proposed method converges quickly to
satisafied reliability index and design points after only a 50 generations which
represents a gain in computation time. Finally, the obtained results for the example 4 are listed in Table 4 and in Fig 5. It can be seen that the proposed
ES-MFO are similar to previous Examples 12. This demonstrates that the application of the proposed method to reliability analysis is feasible, efficient, and
accurate.
Application
Figure 6 show the roof truss model, this model was considered by [12] in the
context of a sensitivity analysis. The limit state function is defined as equation
(20). The set of parameters are given as follows: q as a distributed load applied
to the structure, l is the length between the supports. Ac and As are the transverse cross sections, and Es , Ec are Youngs moduli of the steel and concrete
beams, respectively. The statistic models are listed in the table 5. Once more,
due to livelihood of the positive parameters, the normal distribution is not an
appropriate choice for modeling. However, in order to validate our algorithm,
We procedure the identical probabilistic model as in [12].
G(q, Ac , As , Ec , Es , l) = 0.03 −
1.13
ql2 3.81
(
+
)
2 Ac Ec
As E s
(20)
For the application, the obtained results are listed in Table 6. It can be
seen that the exact failure probability is obtained using MCS with importance
sampling is 0.9384 × 102 , while as 0.9486 × 102 by using the proposed ES-MFO.
In addition, as shown in Fig 7 the proposed method converges to satisafied
reliability index and design points after 60 generations which represents a gain
in computation time.
A hybrid Egale Strategy with Moth-Flame Optimizer for Reliability Analysis
11
(a) Roof truss model structure
Convergence Curves - Example 1
Convergence Curves - Example 2
ES-MFO
ES-MFO
6.5
5
Reliability-Index (Beta)
Reliability-Index (Beta)
6
4.5
2.33094
4
2.330935
3.5
2.33093
2.330925
3
2.33092
60
70
80
5.5
5
2.500045
4.5
2.50004
4
2.500035
3.5
90
2.50003
50
60
70
80
50
60
70
80
3
2.5
10
20
30
40
50
60
70
80
90
100
10
20
30
40
90
(b) Convergence curve of ES-MFO for example 1
(c) Convergence curve of ES-MFO for example 2
Convergence Curves - Example 3
Convergence Curves - Example 4
ES-MFO
ES-MFO
6.5
20
18
2.70996
14
2.70995
Reliability-index (Beta)
6
16
2.70994
12
2.70993
10
2.70992
2.70991
40
8
45
50
55
60
6
5.5
5
2.26975
4.5
2.269748
4
2.269746
2.269744
3.5
60
65
70
75
80
3
4
2.5
10
20
30
40
50
60
70
80
90
100
10
20
30
No->Generation
50
60
70
80
90
(e) Convergence curve of ES-MFO for example 4
Convergence Curves - Application
2.56
ES-MFO
2.54
2.52
2.365
2.5
2.48
2.36
2.46
2.44
2.355
2.42
2.4
2.35
50
60
70
80
90
50
60
70
80
90
2.38
2.36
10
40
No->Generation
(d) Convergence curve of ES-MFO for example 3
Reliability-Index (Beta)
Reliability-Index (Beta)
100
No->Generation
No-> Generation
20
30
40
No->Generation
(f) Convergence curve of ES-MFO for application
Fig. 2: Overall convergence cruves
100
100
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Hraiba et al.
5
Conclusion
A ES-MFO-based reliability analysis method was presented. The ES-MFO algorithm has strong global search capability. The proposed method can identify the
design point and compute the corresponding reliability index with high accuracy.
It does not require derivative information for the limited state function and is
fitted to an implicit limited state function. The method was applied to four classic examples and the reliability of a roof truss mode. The proposed method can
be used for reliability analysis in engineering with high efficiency and accuracy
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