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 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 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 4 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) 6 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) 8 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 10 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 12 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 References 1. J-M Bourinet, François Deheeger, and Maurice Lemaire. Assessing small failure probabilities by combined subset simulation and support vector machines. Structural Safety, 33(6):343–353, 2011. 2. Jin Cheng. Hybrid genetic algorithms for structural reliability analysis. Computers & Structures, 85(19-20):1524–1533, 2007. 3. François Deheeger. Couplage mécano-fiabiliste: 2 SMART-méthodologie d’apprentissage stochastique en fiabilité. PhD thesis, Université Blaise PascalClermont-Ferrand II, 2008. 4. Armen Der Kiureghian, T Haukaas, and K Fujimura. Structural reliability software at the university of california, berkeley. structural safety, 28(1-2):44–67, 2006. 5. Mohamed Abd El Aziz, Ahmed A Ewees, and Aboul Ella Hassanien. Whale optimization algorithm and moth-flame optimization for multilevel thresholding image segmentation. Expert Systems with Applications, 83:242–256, 2017. 6. Charles Elegbede. Structural reliability assessment based on particles swarm optimization. Structural Safety, 27(2):171–186, 2005. 7. Paul Lemaı̂tre. Analyse de sensibilité en fiabilité des structures. PhD thesis, Bordeaux, 2014. 8. BK Low and Wilson H Tang. New form algorithm with example applications. In Proceedings of the Fourth Asian-Pacific Symposium on Structural Reliability and its Applications, Hong Kong, 2008. 9. Rosario Nunzio Mantegna. Fast, accurate algorithm for numerical simulation of levy stable stochastic processes. Physical Review E, 49(5):4677, 1994. 10. Seyedali Mirjalili. Moth-flame optimization algorithm: A novel nature-inspired heuristic paradigm. Knowledge-Based Systems, 89:228–249, 2015. 11. Dimitrios I Papadimitriou, Zissimos P Mourelatos, and Zhen Hu. Reliability analysis using second-order saddlepoint approximation and mixture distributions. Journal of Mechanical Design, 141(2):021401, 2019. 12. Iason Papaioannou, Karl Breitung, and Daniel Straub. Reliability sensitivity estimation with sequential importance sampling. Structural Safety, 75:24–34, 2018. 13. Bruno Sudret and Armen Der Kiureghian. Comparison of finite element reliability methods. Probabilistic Engineering Mechanics, 17(4):337–348, 2002. 14. David H Wolpert, William G Macready, et al. No free lunch theorems for optimization. IEEE transactions on evolutionary computation, 1(1):67–82, 1997. 15. Xin-She Yang. Engineering optimization: an introduction with metaheuristic applications. John Wiley & Sons, 2010. 16. Hongbo Zhao, Zhongliang Ru, Xu Chang, and Shaojun Li. Reliability analysis using chaotic particle swarm optimization. Quality and Reliability Engineering International, 31(8):1537–1552, 2015.
