Fuzzy reliability analysis using a new alpha level set optimization approach based on particle swarm optimization

Abstract

In real world application of structural reliability analysis some random variables contains two types of random and epistemic uncertainty, while in classic methods of structural reliability only the random uncertainty is considered completely. Therefore in order to have a reliable estimation of structural safety, random variables should cover both random and epistemic uncertainty. In this paper, modeling of epistemic uncertainty of random variables has been brought in to focus. Hybrid random variables are simulated using fuzzy numbers. A new alpha level set optimization approach applying particle swarm optimization technique was addressed in order to determine the minimum and the maximum members of reliability index interval. Importance sampling technique was used for reliability analysis to decrease the computational effort. Three numerical examples were given to illustrate the accuracy and efficiency of the proposed method. Results showed that the proposed method was more efficient compared to the alternative search approaches thorough low computational burden.

Keywords

Fuzzy RandomVariable (FRV)alpha level set optimization ParticleSwarm Optimization (PSO)Importance Sampling (IS)

1 Introduction

Engineers are inevitably faced with risks and uncertainties in their professional work. Therefore, uncertainties are an integral part of engineering models. Engineering models are an outlined issue of mathematical properties of a system. Applying models engineers can forecast that how subsystems will perform under specified conditions. Structural engineering models relate one or more of desired quantity. For example reliability index of a structure to a set of input variables such as material properties and geometry. The nature of uncertainties and their modeling approaches have been considered by many researchers in recent years [1 –3].

Generally two types of random and epistemic uncertainty exist. Random uncertainty generally refers to the inherent changes of a system or a phenomenon over time. This type of uncertainty represents the variability or randomness in input variables of the mathematical model. For example, the properties of steel or concrete that must be applied as inputs in a model to calculate the reliability index of a structure might be uncertain. This is because many factors such as human errors and curing conditions affect the properties of material that are not measurable. Hence, random uncertainty cannot be reduced [4].

Epistemic uncertainty results from a lack of information on the nature of the specific system or phenomenon. In structural reliability, it is usually the result of ambiguities associated with the definition of variables and parameters of statistical distribution (e.g. mean and standard deviation). It can also be caused in the process of mathematical modeling and simplification of the system. With an increase in the amount of data/information of the variable or problem of concern, the level of epistemic uncertainty decreases [5].

In structural reliability analysis, random uncertainty is modeled by assigning probability distributions to random variables [2]. Methods such as First-Order Reliability (FORM) [6, 7] Second-Order Reliability (SORM) [8, 9] Monte Carlo Simulation (MCS) [10, 11] Importance Sampling (IS) [12, 13] and simulation methods [14, 15] have been proposed for safety evaluation of structures.

Due to the inherent ambiguity in statistical properties of some random variables in real world applications, a combination of the two types of uncertainties exists which is not taken in to account in classic methods completely [16]. In order to have a reliable estimation of structural safety, random variables should be included with both uncertainties. Although in recent years the problem of random uncertainty in reliability assessment of structures has been brought into focus using probability theory, modeling of epistemic uncertainties in this field is still a challenge faced by researchers. Recently, useful studies have been conducted on this challenge and some methods were proposed for analysis and safety assessment of structures to joint modeling of random and epistemic uncertainties.

Möller and Beer proposed a gradient based alpha level set optimization approach for structural analysis involving uncertain parameters [17]. On the basis of fuzzy set theory a general method for fuzzy structural analysis was formulated in terms of alpha level set optimization method.

Farkas et al. proposed a fuzzy finite element analysis method based on reanalysis technique [18]. They introduced the reanalysis-based FEM with the purpose of reducing computational cost of the repeated deterministic FE solutions that arise in a fuzzy FE analysis. Reviews of interval FE based structural reliability assessment can be found in Ref. [19].

Serafinska et al. proposed a method for consideration of uncertain quantities in the multi-objective optimization problems in order to account the uncertainties in engineering problems [20].

In addition, some studies were also conducted to model epistemic uncertainty in the form of fuzzy random variables applied to structural reliability analysis. Karen et al. proposed a filtered fuzzy relation algorithm which utilizes fuzzy set theory to analyze the qualitative data methodologically. The results yielded a safety function which showed the degree of belonging for each level of safety reduction from original design reliability [21].

Möller et al. proposed a method of fuzzy probabilistic safety assessment [22]. The proposed safety concept was formulated as a development of conventional probabilistic approaches applying alpha level set optimization [17].

Zhang et al. compared the uncertainty models in reliability analysis of offshore structures applying alpha level set optimization proposed by Möller et al. [23]. Results showed that Non-traditional models possessed certain advantages in the case of limited information and associated difficulties of a traditional probabilistic modeling.

Giuseppe proposed a fuzzy based version of classic Cornell reliability index [24]. The study was focused on a new and efficient definition for reliability index applying credibility theory with application to real structures. Results showed that it is an effective approach in the class of the non-probabilistic methodologies.

In structural reliability analysis, in order to model random variables in the form of fuzzy sets, several methods such as extension principle [25], sampling [26], vertex [27], function approximation [28] and alpha level set optimization [17 , 29] have been proposed. From among, it is only the alpha level set optimization method which is appropriate for engineering problems and is not restricted to a certain case [26]. The main drawback of this method is highly computational effort in non-linear reliability problems. Moreover, it is faces with some challenges such as the integration, especially with an increased in number of variables, discontinuity of some functions and being trapped in local optima instead of global ones.

In contrast of gradient-based methods are population-based methods [30, 31]. Bagheri et al. proposed an alpha level set optimization approach applying a robust genetic algorithm in order to address the drawbacks of gradient-based alpha level set optimization method [32]. Results indicated the accuracy of their proposed method and showed that classical methods cover only special case of the proposed method.

Since in the past several years PSO has gets better results in a faster and cheaper way compared with genetic algorithm [31, 34 and 36], in this paper a new alpha level set optimization approach based on swarm movement has been proposed. PSO has been applied to perform the optimization task and IS method has been used for reliability analysis. For this purpose, in the following some useful concepts about the fuzzy analysis are presented. Afterwards a brief review of classical formulation of IS method, PSO and alpha level set optimization method is illustrated respectively. Specifically, the proposed method has been identified. Finally, an illustrative and three numerical examples to demonstrate its accuracy and efficiently are presented.

2 Fundamentals of fuzzy analysis

2.1 Fuzzy set

Fuzzy sets which have been introduced by Zadeh [25] are an extension of classical set theory and are used in fuzzy logic. Let X be a nonempty set. A fuzzy set $\tilde{A}$ in X is characterized by its membership function μ_A(x) : U → [0, 1]. Where μ_A(x) is interpreted as the degree of membership of element x in fuzzy set $\tilde{A}$ for each x ∈ X.

2.2 Support

Let $\tilde{A}$ be a fuzzy subset of X, the support of $\tilde{A}$ denoted supp (A) is the crisp subset of X whose elements all have nonzero membership grades in A. $supp (A) = {x \in X | μ_{A (x)} > 0}$ (1)

2.3 α-Cut

An α-level set of a fuzzy set $\tilde{A}$ of X is a non-fuzzy set denoted by x_{α
_K} and is defined by: $x_{α_{K}} = {_{cl supp (A) if α = 0}^{{x \in X | μ_{A (x)} \geq 0} if α > 0}$ (2)where cl supp (A) denotes the closure of the support of A.

2.4 Crisp subspace

The same α-cuts of fuzzy variables are crisp sets that constitute a spatial shape which is named crisp subspace X_{α
_k}.

3 Importance sampling method

Structural probability of failure is calculated by integrating the random joint probability density function on its failure region which is expressed as: $P_{f} = \int_{G (X) \leq 0} f_{X} (X_{n}) {dX}_{n}$ (3)where X_n is a vector of random variables including uncertain quantities such as loads, geometrical specifications, and mechanical properties. Moreover f_X (X) is the random joint probability density function and G (X) is the Limit State Function (LSF). Direct integration of Equation (3) is difficult due to the increase in dimension of integration with an increase in complexity, non-linearity or multi-dimensionality of the f (X) function [3]. Hence, approximation and simulation methods which were mentioned in the literature have been developed in structural reliability domain. IS method is an example of such methods [12 , 33]. In IS method, sampling is focused on the failure region and a more precise probability of failure is achieved with fewer samples. In this method Equation (3) is expressed as follows: $P_{f} = \int_{G (X) \leq 0} \frac{f_{X} (X_{n})}{h_{X} (X_{n})} h_{X} (X_{n}) {dX}_{n}$ (4)

Where h_X (X_n) is the new sampling density function. Equation (4) also can be expressed as follows: $P_{f} = \frac{1}{N} \sum_{j = 1}^{N} I (X_{j}) \frac{f_{X} (X_{j})}{h_{X} (X_{j})}$ (5)

Where I (X) is the indicator function of failure which is defined according to Equation (6). $I (X_{j}) = {_{1 ifG (X_{j}) \leq 0}^{0 ifG (X_{j}) > 0}$ (6)

Basic problem in IS method is selection of sampling density function which reduces the number of samples. In IS method, it is basically assumed that the sampling density function is near the most probable failure point (MPP). Figure 1 shows the sampling density functions around the MPP in the standard normal space in MCS and IS methods. Since IS method requires MPP coordinates, in this paper MPP provided by the HL-RF method [2] is used as the mean of the sampling probability density function.

4 Determination of PSO parameters

PSO is an optimization technique with ability to solve problems whose solution is a single point or surface in an n-dimension space. PSO was originally conceptualized in 1995 by Eberhart and Kennedy [34]. In PSO, a series of particles are formed and are assigned with an initial velocity. As the position of particles is updated in accordance with their fitness, the set of particles is led to the optimal solution. In PSO, every particle has a unique velocity and direction. Equations (7) and (8) give the velocity and position vectors for the particle.

One of drawbacks of standard PSO is that the swarm may prematurely converge. The main reasons are the convergence of particles to a point between the global best and the local best positions and also the fast rate of information transmission between particles. In this paper, a popular approach proposed by Veeramachaneni et al. [35] was applied to avoid any prematurely convergence of particles. The procedure is based on division of the swarm into subgroups and to replace the global best with near neighbor best particle. Each particle is drawn to the best previous positions experienced by its neighbors in addition to the other movement mechanism of particle in standard PSO [35]. This position is called Neighbor Best and is selected thorough the maximizing of Fitness-Distance-Ratio (FDR) [35]. Using FDR, each velocity dimension is updated as follows in order to prevent the swarm of trapping in a local optimum. $\begin{matrix} \vec{v_{i}} (t) = φ \vec{v_{i}} (t - 1) + r_{1} c_{1} ({\vec{x}}_{{Pbest}_{i}} - \vec{x_{i}} (t - 1)) \\ + r_{2} c_{2} ({\vec{x}}_{Gbest} - \vec{x_{i}} (t - 1)) \\ + r_{3} c_{3} ({\vec{x}}_{Nbest} - \vec{x_{i}} (t - 1)) \end{matrix}$ (7)

$\vec{x_{i}} (t) = \vec{x_{i}} (t - 1) + \vec{v_{i}} (t)$ (8)

Where φ is the inertia coefficient and c₁,c₂ and c₃ are the acceleration factors. In addition r₁,r₂ and r₃ are randomly selected from the [0, 1] range to ensure the stability of the system [36]. t is the iteration counter and $\vec{x_{i}} (t)$ and $\vec{v_{i}} (t)$ denote the velocity and position vectors of the i-th particle respectively. ${\vec{x}}_{{Pbest}_{i}}$ is the direction vector for the best position in the history of the i-th particle. On the other hand, ${\vec{x}}_{Gbest}$ is the best direction in the history of all particles. The nearest position to the optimum and ${\vec{x}}_{Nbest}$ is the best previous positions visited by particle neighbors.

Convergence of PSO highly depends on coefficient φ. In this paper, a time-dependent function was used to obtain this coefficient [36]. As a result the permissable space was thoroughly searched and particles converge to the best optimums. In addition during the convergence, optimal neighboring points were also identified. The values of φ is linearly reduced from 0 to 1 according to the following equation: $φ (t) = φ_{final} - \frac{(φ_{final} - φ_{initial}) t}{\max_{t}}$ (9)where φ_final and φ_initial are the final and initial inertia coefficients respectively. $\max_{t}$ is also the total number of iterations of PSO.

In this paper fixed values 2.0 for both c₁ and c₂ is applied according to Ref. [34]. As the acceleration coefficients are multiplied by uniform random values, it gives a mean value of 1.0. Also a fixed value 4.0 is used for c₃. Multiplication by a uniform random value gives a mean value of 2.0 as it is the best value in the most of benchmark problems according to Ref. [35].

5 The alpha level set optimization method

Applying extension principle in engineering problems entails a large computational burden. Therefore alternative methods such as alpha level set optimization method as discussed in literature have been developed as a substitution.

If the structural reliability problem is formulated as β =∥ u^* ∥, where ∥ .∥ is the norm of random variables vector and $u^{*} = (u_{1}^{*}, . . ., u_{n}^{*})$ is the MPP coordinates in U space. The elements of α-cut set corresponding to the fuzzy reliability index can be obtained in the case of convex fuzzy random variables [17].

On the other hand, the interval knowing the minimum and maximum elements would be specified. Typically in the alpha level set optimization method two optimums in the crisp subspace are determined which yield the minimum and maximum elements of the β_{(α_k)} interval. Thus determining interval vertices turns into a constraint optimization problem aims at finding two optimums in the crisp subspace. The objective functions of the optimization problem are defined as: $β = ∥ u^{*} ∥ \to Max | (β_{1}, . . ., β_{n}) \in β_{α_{k}}$ (10) $β = ∥ u^{*} ∥ \to Min | (β_{1}, . . ., β_{n}) \in β_{α_{k}}$ (11)

6 New alpha level set optimization approach

In the proposed method, the corresponding crisp subspace is obtained by creating identical α-cuts on the all fuzzy random variables. If the reliable problem has two, three or n fuzzy random variables, the crisp subspace is in the form of a rectangle, cube, and an n-dimension hypercube respectively. After creating the crisp subspace by choosing equal intervals within the boundaries of the subspace, the partitions inside are located precisely. With an increase in the number of selected interval within the boundaries, the number of partitions identified in the subspace as well as precision of the solution will increase, see Fig. 2.

These partitions are used as the input for IS algorithm in order to determine the fuzzy reliability index. If the fuzzy random variables are convex, the crisp subspaces corresponding to the α_{k
_i} set is also convex [29]. The output of the IS algorithm gives a set of α-cut points corresponding to the fuzzy reliability index β_{α
_{k
_i}}. Assembling results membership function of the fuzzy reliability index μ (β) will be obtained, see Fig. 3.

In order to determine β_{α
_{k
_i}}, it is necessary to compute the smallest β_{α
_{k
_i
l}} and the biggest β_{α
_{k
_i
r}} value of interval. On the other hand for each α-cut, there are only two optimal points in the crisp subspace that give the smallest and biggest members of β_{α
_{k
_i}} interval. Therefore, in order to find these points the optimums x_{opt
_{α
_{k
_i
l}}} and x_{opt
_{α
_{k
_i
r}}} must be searched inside of the crisp subspace. The search process is restricted to the boundaries of the crisp subspace as a feasible region of movement of the swarm. Determination of optimums calls for using all the partitions in the crisp subspace as the input of the IS algorithm and accordingly determining β_{α
_{k
_i}} interval.

Considering challenges associated with existing gradient-based methods listed in literature, in the present method PSO is applied to perform the optimization task. A penalty function is assigned to the fitness operator so as to find the best optimums in the crisp subspace. The value of fitness function is reduced for finding x_{opt
_{α
_{k
_i
l}}} and it is increased to find x_{opt
_{α
_{k
_i
r}}}. In the proposed method, PSO algorithm starts with a set of random particles. Each point in the crisp subspace plays the role of a particle in the optimization process. In order to find the optimums, the search continues by updating the direction and velocity of particles. In each phase of movement of the swarm, every particle is updated with three best values. The first one is the best optimum in terms of fitness obtained separately for each particle which is called Personal best (Pbest). The second one is the best optimum value obtained for the entire particles and is called Global best (Gbest). While the last one is the best previous positions visited by particle neighbors which is called Near best (Nbest). In the proposed method the fitness of each particle is calculated using the reliability index resulted from IS algorithm. The new velocity and position of particles at each phase are updated using Equations (7) and (8). In each iteration the convergence conditions such as reaching maximum iterations or stability of the optimum solution with a specific number of iterations are examined.

7 Illustrative example

In order to examine the accuracy of the proposed method an illustrative example is presented here. The LSF of a steel beam with a compact section under bending moment which is taken from Ref. [2] is according to Equation (12). $G (Z, F_{Y}, M) = {ZF}_{Y} - M$ (12)

Where Z and F_Y are the plastic section modulus and the yield stress of steel respectively and M is the moment exerted on the beam. The statistical information of independent random variables are presented in Table 1.

Two Search Approaches (SAs) were adopted in this example. In the former (SA1), the crisp subspace was formed and whole partitions were used as the input of IS algorithm. In the case of n fuzzy random variables were discretized into the same m intervals, total number of (m + 1) ⁿ partitions were located in the corresponding crisp subspace. Since in this approach no optimization method was employed, the result could be offered as an exact solution [29]. While in the later, the crisp subspace was formed and the optimums were searched applying the proposed method. In the optimization process each α-cut was divided into five equal intervals. The values of β_{α
_{k
_i
l}} and β_{α
_{k
_i
r}} were also determined for five α-cuts. Here, SA1 was included in of (5 +1) ⁶ iteration for IS algorithm in each α-cut which entails in 48, 288, 960 samples. While the proposed method has applied the IS for 2355 times with total number of 2, 402, 100 samples. Figure. 4 shows the membership function of fuzzy reliability index resulted from the two SAs. Comparison the results suggests that the proposed method yields acceptable results in comparison with the exact solution thorough low computational effort.

8 Numerical examples

In this section three examples are provided. All the examples have explicit LSFs and are drawn from credible references. The purpose is to demonstrate the accuracy and efficiency of the proposed method in comparison with the MCS and SA1. The examples also demonstrate the applicability of the proposed method for nonlinear LSFs with non-normal PDFs. Since owning to their membership function, modeling the epistemic uncertainty of structural parameters via triangular fuzzy numbers is more realistic, the mean and standard deviation of random variables are modeled via triangular fuzzy numbers to the rate of 10 percent from the median of the fuzzy number by expert knowledge [29].

8.1 Example 1

Figure. 5 shows a simple elastic beam with a span length of 6 meters exposed to external load which is taken from Ref. [1]. The LSF is assumed according to Equation (13).

$G (P, L, E, I) = \frac{L}{100} - \frac{{PL}^{3}}{48 EI}$ (13)

In Equation (13), L is span length, P is external load, I is moment of inertia and E is modulus of elasticity. $\frac{L}{100}$ is the allowable deflection of the beam under dead load and $\frac{{PL}^{3}}{48 EI}$ is maximum beam deflection resulted from the external load. Random variables are normal and independent and their statistical information are shown in Table 2.

Here, the crisp subspace is a 6-Dimensional hyperspace. SA1, MCS and proposed method are compared in this example. Since there are 6 fuzzy random variables (fuzzy mean and fuzzy standard deviation), SA1 would include in solving total number of (5 +1) ⁶ distinct IS problems for each α-cut. Since in this example a single IS is conducted with 1600 samples, SA1 will require total number of 5 × 6⁶ × 1600 (373, 248, 000) samples. In order to verify the values obtained by the proposed method, the failure probability is calculated by crude MCS with 2 × 10⁴ samples. This gave the result P_f = 8.7 × 10^-4 (β = 3.1284). Applying MCS fuzzy reliability index has been calculated by total number of 6⁶ × 10⁴ (466, 560, 000) samples, which is a high computational burden. While application of the proposed method resulted in 2440 times of IS utilization and including 3, 904, 000 samples. The efficiently of the proposed method is completely obvious in comparison with the SA1 and MCS methods. The calculated lower and upper bounds for β and number of simulations are reported in Table 3. Results are also presented in Fig.6. The accuracy of the proposed method is negligible thorough low computational burden.

8.2 Example 2

In order to assay the capability of the proposed method for non-normal distributions, a nonlinear LSF is considered from Ref. [37]. Equation (14) defines the LSF including 6 random variables. $G (X_{1}, . . ., X_{6}) = X_{1} X_{2} X_{3} X_{4} - \frac{X_{5} X_{6}^{2}}{8}$ (14)

In Equation (14), random variable X₄ follows a uniform distribution with mean μ_(X₄) = 20 and standard deviation σ_(X₄) = 1. Whiles random variable X₆ is normally distributed which its mean and standard deviations are set as μ_(X₆) = 150 and σ_(X₆) = 10 respectively. The statistical information of other random variables are presented in Table 4. In this example all random variables are statistically independent and optimum points x_{opt
_{α
_{k
_i
l}}} and x_{opt
_{α
_{k
_i
r}}} are searched in a 8-Dimensional hyperspace. The purpose is to compare SA1, SA2 and MCS. Here, there are 8 fuzzy random variables. Therefore in SA1, (5 +1) ⁸ IS problems should be solved in each α-cut. On the other hand, each IS simulation of the LSF required about 1300 samples. Therefore, total number of 5 × 6⁸ × 1300 (1.09 × 10¹⁰) samples is needed which is too time consuming and inefficient.

According to Ref. [37] a single MCS of this example with 10⁶ samples gives the result of P_f = 3.54 × 10^-3 (β = 2.6920). Therefore determining of fuzzy reliability index by MCS would be included in the number of 5 × 6⁸ × 10⁶ (8.39×10¹²) samples, which is not applicable. It is noticeable that in this example epistemic uncertainty could not be modeled by existing methods as the huge computational burden. In the proposed method, IS has been recalled for 2527 times including 3, 285, 750 samples. The calculated lower and upper bounds for β and number of simulations of the proposed method are reported inTable 5.

8.3 Example 3

Figure. 7 shows a 10-bar aluminum truss subject to external load P. The structure layout and member sizes are adopted from Ref. [38]. The cross section of vertical, horizontal and diagonal members are denoted by A₁, A₂ and A₃ respectively. The LSF for horizontal displacement of the upper right node is defined by Equation (15), where E is modulus of elasticity. In Equation (16), the random variable B represents the uncertainty in the model. Moreover d₀ and L are deterministic whose values are set to 0.1m and 0.9m respectively. The fuzzy means, fuzzy standard deviations and other statistical information of independent random variables are listed in Table 6.

$G (A_{1}, A_{2}, A_{3}, B, P, L) = d_{0} - D$ (15)where $\begin{matrix} D = \frac{BPL}{A_{1} A_{3} E} {\frac{4 \sqrt{2} A_{1}^{3} (24 A_{2}^{2} + A_{3}^{2}) + A_{3}^{3} (7 A_{1}^{2} + 26 A_{2}^{2})}{D_{T}} \\ + 4 A_{1} A_{2} A_{3} \frac{20 A_{1}^{2} + 76 A_{1} A_{2} + 10 A_{3}^{2}}{D_{T}} \\ + 4 \sqrt{2} A_{1} A_{2} A_{3}^{2} \frac{25 A_{1} + 29 A_{2}}{D_{T}}} \end{matrix}$ (16)where $\begin{matrix} D_{T} = 4 A_{2}^{2} (8 A_{1}^{2} + A_{3}^{2}) + 4 \sqrt{2} A_{1} A_{2} A_{3} (3 A_{1} + 4 A_{2}) \\ + A_{1} A_{3}^{2} (A_{1} + 6 A_{2}) \end{matrix}$ (17)

Here, the crisp subspace is a 12-Dimensional hyperspace which is searched for optimums. Since there are 12 fuzzy random variables, SA1 would include in solving the total number of (5 +1) ¹² distinct IS problems for each α-cut, which is computationally infeasible. According to Ref. [38], a single MCS for this example requires 1.7 × 10⁹ samples for failure probability of P_f = 3.5 × 10^-6 (β = 4.4941). That means the fuzzy reliability index will be determined by the total number of (5 +1) ¹² × 1.7 × 10⁹ (3.7 × 10¹⁸) samples in MCS, which is not possible by existing PCs. However the proposed method has recalled IS algorithm only for 3508 times. The required number of sample applied by the proposed method is about 35, 075, 000. Table 7 presents the calculated lower and upper bounds of β with varying number of α-cuts and number of simulations of the proposed method. The efficiently of the proposed method is fully apparent compared with SA1 and MCS.

9 Conclusion

A new alpha level set optimization approach has been developed to compute the bounds for fuzzy reliability index interval where epistemic uncertainty of random variables is modeled as fuzzy triangular numbers. An illustrative and three numerical examples have been analyzed to evaluate the capability of the proposed method for nonlinear LSFs with non-normal PDFs. Two search approaches and MCS were adopted to demonstrate the efficiently and accuracy of the proposed method. In the first Search Approach (SA1), the entire points within the crisp subspace are used as the input of the IS algorithm. Since in this approach, no optimization method was employed, the result could be offered as an exact solution but it is not applicable for problems with high number of variables. In the second Search Approach, the crisp subspace was formed and the optimal points were searched applying the proposed method using PSO in the alpha level set optimization approach. The efficiently of the proposed method is completely apparent compared with SA1 and MCS.

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