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
Abstract. In this paper, we devoted the reliability analysis by combin-
ing 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 im-
prove 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 vio-
lation 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 un-
certainties 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
2 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 hy-
bridized algorithm combines the strengths of each of these algorithms to provide
the following advantages: (i) to produce better solutions, (ii) to provide solu-
tions 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 iden-
tify 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 heuris-
tics 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 investi-
gate 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 3
2 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 meth-
ods 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 situa-
tions 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 Fcalled 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 Xrandom 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 βas :
β=minX∈FsXi−µi
σiT
[R]−1Xi−µi
σi(2)
4 Hraiba et al.
Where µiand σiare respectively the mean and de standare deveiation for ran-
dom varibale X, and Rrepresente the correlation matrix. The probability can
estemated by:
Pf= 1 −φ(β) (3)
Where φ(.) is the cumulative distribution function of the standard normal vari-
able.
Based on the above assumptions, the following constrained optimization has
to be solved :
minimize q[n]T[R]−1[n]
Subject to:
G(X) = 0
(4)
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: Objective function f(x)
2: Initialization and random initial guess xt=0
3: while (stop criterion) do
4: Global exploration by randomization (e.g. Levy flights)
5: Evaluate the objectives and find a promising solution
6: Intensive local search via an efficient local optimizer
7: if (a better solution is found) then
8: Update the current best
9: end if
10: Update t=t+ 1
11: end while
A hybrid Egale Strategy with Moth-Flame Optimizer for Reliability Analysis 5
3.2 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 Fsimilar 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:
MF O = (I, P, T ) (5)
Iis 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 Pfunction, which is the main function, moves the moths around the search
space. This function received the matrix of M and returns its updated one even-
tually.
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 Pfunction is iteratively run until the Tfunction
returns true. The Pfunction 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 Miindicate the ith moth, fjindicates the jth flame, and Sis 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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