Parametric Investigation of Particle Swarm Optimization to Improve the Performance of the Adaptive Neuro-Fuzzy Inference System in Determining the Buckling Capacity of Circular Opening Steel Beams

In this paper, the main objectives are to investigate and select the most suitable parameters used in particle swarm optimization (PSO), namely the number of rules (nrule), population size (npop), initial weight (wini), personal learning coefficient (c1), global learning coefficient (c2), and velocity limits (fv), in order to improve the performance of the adaptive neuro-fuzzy inference system in determining the buckling capacity of circular opening steel beams. This is an important mechanical property in terms of the safety of structures under subjected loads. An available database of 3645 data samples was used for generation of training (70%) and testing (30%) datasets. Monte Carlo simulations, which are natural variability generators, were used in the training phase of the algorithm. Various statistical measurements, such as root mean square error (RMSE), mean absolute error (MAE), Willmott’s index of agreement (IA), and Pearson’s coefficient of correlation (R), were used to evaluate the performance of the models. The results of the study show that the performance of ANFIS optimized by PSO (ANFIS-PSO) is suitable for determining the buckling capacity of circular opening steel beams, but is very sensitive under different PSO investigation and selection parameters. The findings of this study show that nrule = 10, npop = 50, wini = 0.1 to 0.4, c1 = [1, 1.4], c2 = [1.8, 2], fv = 0.1, which are the most suitable selection values to ensure the best performance for ANFIS-PSO. In short, this study might help in selection of suitable PSO parameters for optimization of the ANFIS model.


Introduction
Circular opening steel beams have been increasingly acknowledged in structural engineering because of their many remarkable advantages [1], including their ability to bridge the span of a large aperture or their lighter weight compared with conventional steel beams. In general, the industrial approach to producing such a structural member is the rolled method, involving a single steel piece. This is then cut so that the two halves can be assembled, making an I-section, which is also called an H-section steel beam. Hoffman et al. [2] showed that the flexural stiffness and specific gravity per unit

Novelty and Significance of This Study
As reported in the introduction, the estimation of the buckling capacity of circular opening steel beams is important for the safety of structures under subjected loads. As instability is a complex (nonlinear) problem that is affected by various parameters, the determination of the critical buckling load remains challenge for researchers (engineers) in the fields of mechanics and civil engineering. Despite various experimental works having investigated this problem, it is not easy to derive a generalized expression that considers all the parameters that govern the instability of circular opening steel beams. To overcome this difficulty, the use of ML techniques, such as ANFIS optimized by the PSO algorithm proposed in this study, could be a good choice as a surrogate model. This soft computing method could help to explore the nonlinear relationships between the buckling capacity and the input variables, especially the geometrical parameters of the beams. In addition, the investigation of PSO parameters based on the Monte Carlo random sampling technique could contribute to better knowledge on selection of suitable parameters to achieve better performance with the PSO algorithm, which could be further recommended for other problems. Finally, the proposed ML-based model could be a potential tool for researchers or structural engineers in accurately estimating the buckling capacity of circular opening steel beams, which could (i) work within the ranges of values used in this study for the input variables and (ii) save time and costs in development of other numerical schemes (i.e., finite element models).

Database Construction
The database in this study was obtained by analyzing 3645 different configurations of circular opening steel beams (Figure 1). It should be noted that the database was extracted from a validated finite element model, which was previously proposed in the literature by Abambres et al. [34]. It consisted of 8 input parameters, namely the length of the beam (denoted as L), the end opening distance (denoted as d 0 ), circular opening diameter (denoted as D), the inter-opening distance (denoted as d), the height of the section (denoted as H), the thickness of the web (denoted as t web ), the width of the flange (denoted as w flange ), the thickness of the flange (denoted as t flange ), and the buckling capacity, which was considered as the target variable (denoted as P u ). It should be pointed out that the database was generated for one material type (with a typical Young's modulus of 210 GPa and Poisson's ratio of 0.3). The results of the statistical analysis of the Pu and the corresponding influential parameters are presented in Table 1.
Materials 2020, 13, x FOR PEER REVIEW 4 of 27 ranges of values used in this study for the input variables and (ii) save time and costs in development of other numerical schemes (i.e., finite element models).

Database Construction
The database in this study was obtained by analyzing 3645 different configurations of circular opening steel beams ( Figure 1). It should be noted that the database was extracted from a validated finite element model, which was previously proposed in the literature by Abambres et al. [34]. It consisted of 8 input parameters, namely the length of the beam (denoted as L), the end opening distance (denoted as d0), circular opening diameter (denoted as D), the inter-opening distance (denoted as d), the height of the section (denoted as H), the thickness of the web (denoted as tweb), the width of the flange (denoted as wflange), the thickness of the flange (denoted as tflange), and the buckling capacity, which was considered as the target variable (denoted as Pu). It should be pointed out that the database was generated for one material type (with a typical Young's modulus of 210 GPa and Poisson's ratio of 0.3). The results of the statistical analysis of the Pu and the corresponding influential parameters are presented in Table 1.
The input and target variables in this work were scaled in the range of [0, 1] to minimize the numerical bias of the dataset. After performing the simulation part, a transformation into the normal range was conducted to better interpret the obtained results. Concerning the development phase, the dataset was split into two parts, namely the training part (70% of the total data) and the testing part (the remaining 30% of the data), which served as the learning and validation phases of the proposed ANFIS-PSO model, respectively.    The input and target variables in this work were scaled in the range of [0, 1] to minimize the numerical bias of the dataset. After performing the simulation part, a transformation into the normal range was conducted to better interpret the obtained results. Concerning the development phase, the dataset was split into two parts, namely the training part (70% of the total data) and the testing part (the remaining 30% of the data), which served as the learning and validation phases of the proposed ANFIS-PSO model, respectively.

Adaptive Neuro-Fuzzy Inference System
Jang et al. [55] introduced the fuzzy adaptive system of adaptive neurology, called ANFIS, as an improved ML method and a data-driven modeling approach to evaluate the behavior of complex dynamic systems [56,57]. ANFIS aims to systematically generate unknown fuzzy rules from a given set of input and output data. ANFIS creates a functional map that approximates the internal system parameter estimation method [58][59][60]. Fuzzy systems are rule-based systems developed from a set of language rules. These systems can represent any system with good accuracy and are, therefore, considered to be universal approximators. Thus, ANFIS is the most popular neuro-fuzzy hybrid network used for the modeling of complex systems. The ANFIS model's main strength is that it is a universal approximator with the ability to request interpretable "if-then" rules [61]. In ANFIS, a Sugeno-type fuzzy system was used to construct the five-layer network.

Particle Swarm Optimization (PSO)
Eberhart and Kennedy developed the PSO algorithm in 1995. It is an evolutionary computing technique with a particular enhancement method, population collaboration, and competition based on the simulation of simplified social models, such as bird flocking, fish schooling, and swarming theory [62][63][64][65]. It is a biological-based algorithm that shapes bird flocking social dynamics large number of birds flock synchronously, suddenly change direction, iteratively scatter and group, and eventually perch on a target. The PSO algorithm supports simple rules for bird flocking and acts as an optimizer for nonlinear continuous functions [66]. PSO has gained much attention and has been successfully applied in various fields, especially for unconstrained continuous optimization problems [67]. Indeed, in PSO, a swarm member, also called a particle, is a potential solution, which is used as a search space point. The global equilibrium is known as the food position. The particle has a fitness value and a speed with which to change its flight path for the best swarm experiences to find the global optimum in the D-dimensional solution space. The PSO algorithm is easy to implement and many optimization problems have been empirically shown to perform well [68]. However, its performance depends significantly on the algorithm parameters described below.

Initial Weight (w ini )
The particle in the PSO is represented as a real-valued vector containing an instance of all parameters that characterize the problem of optimization. By flying a number of particles, called a swarm, the PSO explores the solution space. The initial swarm is generated at random, and generally consecutive iterations maintain a consistent swarm size. The swarm of particles looks for the optimum target solution in each iteration by referring to past experiences. Coefficient-c 1 ) PSO enriches swarm intelligence by storing the best positions that each particle has visited so far. Particles I recall the best position among those it met, called pbest, and the best positions of its neighbors. There are two variants, namely lbest and gbest, used to hold the neighbors in the best position. The particle in the local version keeps track of the best lbest location obtained by its neighboring local particles. For the global version, any particles in the whole swarm will determine the best location for gbest. Therefore, the gbest model is the lbest model's special case.

Social Learning Rate (Global Learning Coefficient-c 2 )
PSO starts with the random initialization in the search space of a population (swarm) of individuals (particles) and operates on the particles' social behavior in the swarm. Consequently, it finds the best global solution by simply adjusting each individual's trajectory to their own best location and to the best swarm particle in each phase (generation). Nevertheless, the trajectory of each particle in the search space is modified according to their own flying experience and the flying experience of the other particles in the search space by dynamically altering the velocity of each particle.

Number of Particles (Population Size-n pop )
The location and speed of the ith particle can be expressed in the dimensional search space. Every particle has its own best (pbest) location, according to the best personal objective value at the time t. The world's best particle (gbest) is the best particle found at time t in the entire swarm.

Velocity Limits (f v )
Each particle's new speed is determined as follows: where c 1 and c 2 are constants referred to as acceleration coefficients, w is referred to as the inertia factor, and r 1 and r 2 are two independent random numbers distributed evenly within the spectrum. The location of each particle is, thus, modified according to the following equation in each generation: a i,j (t + 1) = a i,j (t) + y i,j (t + 1), j = 1, 2, 3, . . . , d In the standard PSO, Equation (1) is used to calculate the new velocity according to its previous velocity and to the distance of its current position from both its own best historical position and its neighbors' best positions. The value of each factor in Y i can be clamped within the range to monitor excessive particles roaming outside the search area, then the particle flies toward a new location.

Monte Carlo Simulation
The Monte Carlo technique has been commonly used as a variability generator in the training phase of the algorithm, taking into account the randomness of the input space [69][70][71][72]. Hun et al. [73] studied the problem of crack propagation in heterogeneous media within a probabilistic context using Monte Carlo simulations. Additionally, Capillon et al. [74] investigated an uncertainty problem in structural dynamics for composite structures using Monte Carlo simulations. Overall, the Monte Carlo method has been successfully applied to take into account the randomness in the field of mechanics [75][76][77][78][79][80]. The key point of the Monte Carlo method is to repeat the simulations many times to calculate the output responses by randomly choosing values of the input variables in the corresponding space [81,82]. In this manner, all information about the fluctuations in the input space can be transferred to the output response. In this work, a massive numerical parallelization scheme was programmed to conduct the randomness propagation process. The statistical convergence of the Monte Carlo method reflects whether the number of simulations is sufficient, which can be defined as follows [83][84][85]: where m is the number of Monte Carlo iterations, S is the random variable considered, and S is the average value of S.

Quality Assessment Criteria
In the present work, three quality assessment criteria-the correlation coefficient (R), root mean squared error (RMSE), and mean absolute error (MAE)-have been used in order to validate and test the developed AI models. R 2 allows us to identify the statistical relationship between two data points and can be calculated using the following equation [86][87][88][89][90][91][92]: where N is the number of observations, y p and y are the predicted and mean predicted values, while y 0 and y are the measured and mean measured values of Young's modulus of the nanocomposite, respective j = 1:N. In the case of RMSE and MAE, which have the same units as the values being estimated, low value for RMSE and MAE basically indicate good accuracy of the models' prediction output [93,94]. In an ideal prediction, RMSE and MAE should be zero. RMSE and MAE are given by the following formulae [95][96][97][98][99]: In addition, the Willmott's index of agreement (IA) has also been employed in this study. The formulation of IA is given by [100,101]:

Description of Parametric Studies
In order to investigate the influence of PSO parameters on the performance of ANFIS, parametric studies were carried out by varying n rule , n pop , w ini , c 1 , c 2 , and f v , as indicated in Table 2. It is noteworthy that the proposed range was selected by considering both problem dimensionality (i.e., complexity) and computation time. As recommended by He et al. [102] and Chen et al. [48], the PSO initial weight should be carefully investigated. Therefore, a broad range of w ini was proposed, ranging from 0.1 to 1.2. The number of populations varied from 20 to 300 with a nonconstant step, whereas the coefficients c 1 and c 2 ranged from 0.2 to 2 with a resolution of 0.2. The number of fuzzy rules varied from 5 to 40. Finally, the f v ranged from 0.05 to 0.2.
The relationship between the number of fuzzy rules and the number of total ANFIS weight parameters is depicted in Figure 2. As can be seen, the relationship is linear, showing that as the number of fuzzy rules increases, the number of ANFIS weight parameters increases. For illustration purposes, the number of weight parameters increases from 50 to 370, while the number of fuzzy rules increases from 5 to 40. Additionally, the characteristics of the ANFIS structure are described in Table 3, showing that the Gaussian membership function was used to generate fuzzy rules.   Figure 2. Influence of the number of fuzzy rules on the number of total ANFIS weight parameters to be optimized by PSO.  Figure 3 presents the influence of nrule and swarm parameters on the computation time. It is worth noting that the running time was scaled with respect to the minimum value of the corresponding parameter. For instance, the computation time using nrule = 10 is two times larger than the case using nrule = 5. Additionally, in Figure 3, it is seen that nrule and npop exhibited the highest slope (about 0.75), confirming that these two parameters required considerable computation time. For all other parameters, the computation time remained constant when increasing the value of the parameter.   Figure 3 presents the influence of n rule and swarm parameters on the computation time. It is worth noting that the running time was scaled with respect to the minimum value of the corresponding parameter. For instance, the computation time using n rule = 10 is two times larger than the case using n rule = 5. Additionally, in Figure 3, it is seen that n rule and n pop exhibited the highest slope (about 0.75), confirming that these two parameters required considerable computation time. For all other parameters, the computation time remained constant when increasing the value of the parameter.

Computation Time
Materials 2020, 13, x FOR PEER REVIEW 9 of 27

PSO Stopping Criterion
In this study, 1000 iterations were applied as a stopping criterion in the optimization problem for the weight parameters of ANFIS. Figure 4 shows the convergence of statistical criteria in the function of nrule, whereas Figure 5 presents the convergence of these criteria regarding npop. For the evaluation of RMSE, MAE, and R over 1000 iterations in 6 cases for different nrule, the training parts are given in Figure 4a-c, whereas the testing parts are displayed in Figure 4d-f. It was observed that at least 800 iterations were required to obtain convergence results for RMSE, MAE, and R for all the cases. However, no specific trend could be deduced in order to obtain the best nrule parameter. Finally, it is worth noting that for all the cases of nrule, the values of RMSE, MAE, and R for the testing part were very close. Indeed, the values of RMSE for the testing part ranged from 0.038 to 0.043, the values of MAE for the testing part varied from 0.015 to 0.022, and those of R ranged from 0.95 to 0.97. The evaluation of RMSE, MAE, R over 1000 iterations in 9 cases of npope is shown ( Figure 5). Similar results were obtained as for nrule. At least 800 iterations were needed to obtain the convergence results.

PSO Stopping Criterion
In this study, 1000 iterations were applied as a stopping criterion in the optimization problem for the weight parameters of ANFIS. Figure 4 shows the convergence of statistical criteria in the function of n rule , whereas Figure 5 presents the convergence of these criteria regarding n pop . For the evaluation of RMSE, MAE, and R over 1000 iterations in 6 cases for different n rule , the training parts are given in Figure 4a-c, whereas the testing parts are displayed in Figure 4d-f. It was observed that at least 800 iterations were required to obtain convergence results for RMSE, MAE, and R for all the cases. However, no specific trend could be deduced in order to obtain the best n rule parameter. Finally, it is worth noting that for all the cases of n rule , the values of RMSE, MAE, and R for the testing part were very close. Indeed, the values of RMSE for the testing part ranged from 0.038 to 0.043, the values of MAE for the testing part varied from 0.015 to 0.022, and those of R ranged from 0.95 to 0.97. The evaluation of RMSE, MAE, R over 1000 iterations in 9 cases of n pope is shown ( Figure 5). Similar results were obtained as for n rule . At least 800 iterations were needed to obtain the convergence results.

PSO Stopping Criterion
In this study, 1000 iterations were applied as a stopping criterion in the optimization problem for the weight parameters of ANFIS. Figure 4 shows the convergence of statistical criteria in the function of nrule, whereas Figure 5 presents the convergence of these criteria regarding npop. For the evaluation of RMSE, MAE, and R over 1000 iterations in 6 cases for different nrule, the training parts are given in Figure 4a-c, whereas the testing parts are displayed in Figure 4d-f. It was observed that at least 800 iterations were required to obtain convergence results for RMSE, MAE, and R for all the cases. However, no specific trend could be deduced in order to obtain the best nrule parameter. Finally, it is worth noting that for all the cases of nrule, the values of RMSE, MAE, and R for the testing part were very close. Indeed, the values of RMSE for the testing part ranged from 0.038 to 0.043, the values of MAE for the testing part varied from 0.015 to 0.022, and those of R ranged from 0.95 to 0.97. The evaluation of RMSE, MAE, R over 1000 iterations in 9 cases of npope is shown ( Figure 5). Similar results were obtained as for nrule. At least 800 iterations were needed to obtain the convergence results.  4 5 6 7 8 9 10 11 12 13 14 15 Variable increment ratio

Statistical Convergence
In order to take into account variability in the input space, 200 random realizations were performed for each configuration. These realizations increased the influence of the probability density function of inputs on the optimization results. In terms of nrule, Figure 6a-c indicate the statistical convergence of RMSE, MAE, and R for the training part, whereas Figure 6d-f present the statistical convergence of the same parameters for the testing part, respectively. It can be seen that after about 100 random realizations, statistical convergence was reached, which was correct for all the tested cases. Similarly, Figure 7 shows the statistical convergence in terms of npop for both training and testing parts. Similarly, 200 random realizations were observed to be sufficient to achieve reliable results.
MAE for training part MAE for testing part R for training part

Statistical Convergence
In order to take into account variability in the input space, 200 random realizations were performed for each configuration. These realizations increased the influence of the probability density function of inputs on the optimization results. In terms of n rule , Figure 6a-c indicate the statistical convergence of RMSE, MAE, and R for the training part, whereas Figure 6d-f present the statistical convergence of the same parameters for the testing part, respectively. It can be seen that after about 100 random realizations, statistical convergence was reached, which was correct for all the tested cases. Similarly, Figure 7 shows the statistical convergence in terms of n pop for both training and testing parts. Similarly, 200 random realizations were observed to be sufficient to achieve reliable results. Materials 2020, 13, x FOR PEER REVIEW 12 of 27    The evaluation of RMSE, MAE, R, and IA in the function of nrule is presented in Figure 8a-d, respectively, for both training and testing parts. It can be seen that the accuracy of the ANFIS-PSO reduced when the number of nrule increased (i.e., RMSE and MAE increased, while R and IA decreased). It is worth noting that the higher the number of rules, the larger dimensionality of the problem (Figure 2). Therefore, regarding the total number of ANFIS weight parameters, the

Influence of Number of Rules (n rule )
The evaluation of RMSE, MAE, R, and IA in the function of n rule is presented in Figure 8a-d, respectively, for both training and testing parts. It can be seen that the accuracy of the ANFIS-PSO reduced when the number of n rule increased (i.e., RMSE and MAE increased, while R and IA decreased). It is worth noting that the higher the number of rules, the larger dimensionality of the problem ( Figure 2). Therefore, regarding the total number of ANFIS weight parameters, the computation time, and the average value of the statistical criteria (RMSE, MAE, R, and IA), n rule = 10 was considered as the most appropriate value.

Influence of Population Number (npop)
The evaluation of statistical criteria in the function of npop for RMSE, MAE, R, and IA is shown in Figure 9a

Influence of Population Number (n pop )
The evaluation of statistical criteria in the function of n pop for RMSE, MAE, R, and IA is shown in Figure 9a

Influence of Initial Weight (wini)
The evaluation of statistical criteria in the function of wini for RMSE, MAE, R, and IA is shown in Figure 10a-d, respectively, for both training and testing parts. It can be seen that poor prediction performance was obtained when wini was larger than 0.5 (i.e., an increase of RMSE and MAE values and a decrease of R and IA values). Regarding the statistical criteria (RMSE, MAE, R, and IA), a wini value range of between 0.1 and 0.4 was the most appropriate.

Influence of Initial Weight (w ini )
The evaluation of statistical criteria in the function of w ini for RMSE, MAE, R, and IA is shown in Figure 10a-d, respectively, for both training and testing parts. It can be seen that poor prediction performance was obtained when w ini was larger than 0.5 (i.e., an increase of RMSE and MAE values and a decrease of R and IA values). Regarding the statistical criteria (RMSE, MAE, R, and IA), a w ini value range of between 0.1 and 0.4 was the most appropriate.

Influence of Initial Weight (wini)
The evaluation of statistical criteria in the function of wini for RMSE, MAE, R, and IA is shown in Figure 10a-d, respectively, for both training and testing parts. It can be seen that poor prediction performance was obtained when wini was larger than 0.5 (i.e., an increase of RMSE and MAE values and a decrease of R and IA values). Regarding the statistical criteria (RMSE, MAE, R, and IA), a wini value range of between 0.1 and 0.4 was the most appropriate.

Influence of Personal Learning Coefficient (c1)
The evaluation of statistical criteria in the function of c1 for RMSE, MAE, R, and IA is shown in Figure 11a-d, respectively, for both training and testing parts. It can be seen that good prediction performance was obtained when c1 was in the range of [1, 1.4] for all statistical criteria (RMSE, MAE, R, and IA). Therefore, c1 = [1, 1.4] was the most appropriate value.

Influence of Personal Learning Coefficient (c 1 )
The evaluation of statistical criteria in the function of c 1 for RMSE, MAE, R, and IA is shown in Figure 11a-d, respectively, for both training and testing parts. It can be seen that good prediction performance was obtained when c 1 was in the range of [1, 1.4] for all statistical criteria (RMSE, MAE, R, and IA). Therefore, c 1 = [1, 1.4] was the most appropriate value.

Influence of Personal Learning Coefficient (c1)
The evaluation of statistical criteria in the function of c1 for RMSE, MAE, R, and IA is shown in Figure 11a-d, respectively, for both training and testing parts. It can be seen that good prediction performance was obtained when c1 was in the range of [1, 1.4] for all statistical criteria (RMSE, MAE, R, and IA). Therefore, c1 = [1, 1.4] was the most appropriate value.

Influence of Global Learning Coefficient
The evaluation of statistical criteria in the function of c 2 for RMSE, MAE, R, and IA is shown in Figure 12a-d, respectively, for both training and testing parts. It can be seen that good prediction performance was obtained when c 2 was higher than 1.8 for all statistical criteria (RMSE, MAE, R, and IA). Therefore, c 2 = [1. 8,2] was the most appropriate value.

Influence of Global Learning Coefficient
The evaluation of statistical criteria in the function of c2 for RMSE, MAE, R, and IA is shown in Figure 12a-d, respectively, for both training and testing parts. It can be seen that good prediction performance was obtained when c2 was higher than 1.8 for all statistical criteria (RMSE, MAE, R, and IA). Therefore, c2 = [1. 8,2] was the most appropriate value.

Influence of Velocity Limits
The evaluation of statistical criteria in the function of fv for RMSE, MAE, R, and IA is shown in Figure 13a-d, respectively, for both training and testing parts. It can be seen that no influence could be established regarding all statistical criteria (RMSE, MAE, R, and IA). Therefore, fv = 0.1 was finally chosen.

Influence of Velocity Limits
The evaluation of statistical criteria in the function of f v for RMSE, MAE, R, and IA is shown in Figure 13a Table 4 summarizes all of the optimal values, as identified previously. By using the optimal coefficient in Table 4, a regression graph between the real and predicted Pu (kN) is shown in Figure  14. The slope of the ideal fit was then used to measure the angle between the x-axis and the ideal fit, with angles closer than 45° showing better performance. Figure 14a shows the predictability when using the training set, whereas Figure 14b shows the same information applied to the testing set. In both cases, the angles generated by the predicted output had slopes close to that of the ideal fit. This showed that the performance of the proposed model was consistent. Figure 15 shows the error distribution graph using the training part, testing part, and all data. In short, using the selected number of fuzzy rules and PSO parameters, the prediction model gave excellent results (Table 5).   Table 4 summarizes all of the optimal values, as identified previously. By using the optimal coefficient in Table 4, a regression graph between the real and predicted P u (kN) is shown in Figure 14. The slope of the ideal fit was then used to measure the angle between the x-axis and the ideal fit, with angles closer than 45 • showing better performance. Figure 14a shows the predictability when using the training set, whereas Figure 14b shows the same information applied to the testing set. In both cases, the angles generated by the predicted output had slopes close to that of the ideal fit. This showed that the performance of the proposed model was consistent. Figure 15 shows the error distribution graph using the training part, testing part, and all data. In short, using the selected number of fuzzy rules and PSO parameters, the prediction model gave excellent results (Table 5).

Sensitivity Analysis
The sensitivity analysis was performed in order to explore the degree of importance of each input variable using the ANFIS-PSO model. For this, quantile values at 21 points (from 0% to 100%, with a step of 5%) of each input variable were collected from the database and served as a new dataset for the calculation of critical buckling load. More precisely, for a given input, its value varied from 0% to 100%, while all other inputs remained at their median (50%). This variation of values following the probability distribution allows the influence of each input variable to be explored based on their statistical behavior. The results of the sensitivity analysis are indicated in Figure 16 in a bar graph (scaled into the range of 0% to 100%). It can be seen that all variables influenced the prediction of critical buckling load through the ANFIS-PSO model. The most important input variables were L, wflange, tweb, and tflange, which gave degree of importance values of 33.9%, 21.7%, 18.6%, and 10.6%, respectively. This information is strongly relevant and in good agreement with the literature, in which the length of the beam and geometrical parameter of the cross-section are the most important parameters [3][4][5]. However, it can be seen in Figure 16 that the height of the beam does not seriously affect the buckling capacity of the structural members. It should be noted that only three independent values of the section's height were used to generate the database; for example, 420, 560, and 700 mm. Consequently, the linear correlation coefficient between the section's height and the buckling

Sensitivity Analysis
The sensitivity analysis was performed in order to explore the degree of importance of each input variable using the ANFIS-PSO model. For this, quantile values at 21 points (from 0% to 100%, with a step of 5%) of each input variable were collected from the database and served as a new dataset for the calculation of critical buckling load. More precisely, for a given input, its value varied from 0% to 100%, while all other inputs remained at their median (50%). This variation of values following the probability distribution allows the influence of each input variable to be explored based on their statistical behavior. The results of the sensitivity analysis are indicated in Figure 16 in a bar graph (scaled into the range of 0% to 100%). It can be seen that all variables influenced the prediction of critical buckling load through the ANFIS-PSO model. The most important input variables were L, wflange, tweb, and tflange, which gave degree of importance values of 33.9%, 21.7%, 18.6%, and 10.6%, respectively. This information is strongly relevant and in good agreement with the literature, in which the length of the beam and geometrical parameter of the cross-section are the most important parameters [3][4][5]. However, it can be seen in Figure 16 that the height of the beam does not seriously affect the buckling capacity of the structural members. It should be noted that only three independent values of the section's height were used to generate the database; for example, 420, 560, and 700 mm. Consequently, the linear correlation coefficient between the section's height and the buckling

Sensitivity Analysis
The sensitivity analysis was performed in order to explore the degree of importance of each input variable using the ANFIS-PSO model. For this, quantile values at 21 points (from 0% to 100%, with a step of 5%) of each input variable were collected from the database and served as a new dataset for the calculation of critical buckling load. More precisely, for a given input, its value varied from 0% to 100%, while all other inputs remained at their median (50%). This variation of values following the probability distribution allows the influence of each input variable to be explored based on their statistical behavior. The results of the sensitivity analysis are indicated in Figure 16 in a bar graph (scaled into the range of 0% to 100%). It can be seen that all variables influenced the prediction of critical buckling load through the ANFIS-PSO model. The most important input variables were L, w flange , t web , and t flange , which gave degree of importance values of 33.9%, 21.7%, 18.6%, and 10.6%, respectively. This information is strongly relevant and in good agreement with the literature, in which the length of the beam and geometrical parameter of the cross-section are the most important parameters [3][4][5]. However, it can be seen in Figure 16 that the height of the beam does not seriously affect the buckling capacity of the structural members. It should be noted that only three independent values of the section's height were used to generate the database; for example, 420, 560, and 700 mm. Consequently, the linear correlation coefficient between the section's height and the buckling capacity was only −0.092.
On the contrary, the minimum value of the beam's length was 4000 mm (approximately 5.7 times larger than the maximum section's height) and five independent values were used to generate the database, ranging from 4000 to 8000 mm, with a step of 1000 mm. Thus, the linear correlation coefficient between the beam's length and the buckling capacity was −0.667 (approximately 7.25 times bigger than the linear correlation coefficient between the section's height and the buckling capacity). Consequently, a larger database should be considered in future studies to estimate the degree of importance of the section's height.
Materials 2020, 13, x FOR PEER REVIEW 20 of 27 capacity was only −0.092. On the contrary, the minimum value of the beam's length was 4000 mm (approximately 5.7 times larger than the maximum section's height) and five independent values were used to generate the database, ranging from 4000 to 8000 mm, with a step of 1000 mm. Thus, the linear correlation coefficient between the beam's length and the buckling capacity was −0.667 (approximately 7.25 times bigger than the linear correlation coefficient between the section's height and the buckling capacity). Consequently, a larger database should be considered in future studies to estimate the degree of importance of the section's height. The sensitivity analysis presented above demonstrates that the ML technique could assist in the design phase for circular opening steel beams. In addition to reliable prediction of the critical buckling load, the ANFIS-PSO model can also assist in the creation of input-output maps, as illustrated in Figure 17. In particular, as L, tweb, wflange, and tflange were the most important variables, they are used for map illustrations in this section. The values of the remaining variables were kept constant. In Figure 17, four maps of critical buckling load are presented (with the same color range), involving the relationship between Pu and L-wflange, L-tflange, L-tweb, and wflange-tflange, respectively. As can be seen from the surface plots, the input-output relationship exhibited nonlinear behavior, which cannot be easily identified from the database. Figure 17a shows that a maximum value for the critical buckling load can be obtained if L reaches its minimum and wflange reaches its maximum value. On the other hand, the critical buckling load reaches its minimum if L reaches its highest value and wflange reaches its lowest value. This map confirms the negative effect of L, as pointed out in the literature [4]. In Figure 17b,c, the same results are obtained as in Figure 17a. This observation again confirms that the geometrical parameters of the cross-section are highly important [1,5]. Such quantitative information allows the design and analysis recommendations to be explored, as well as for new beam configurations to be generated (within the range of variables considered in this present study). The sensitivity analysis presented above demonstrates that the ML technique could assist in the design phase for circular opening steel beams. In addition to reliable prediction of the critical buckling load, the ANFIS-PSO model can also assist in the creation of input-output maps, as illustrated in Figure 17. In particular, as L, t web , w flange , and t flange were the most important variables, they are used for map illustrations in this section. The values of the remaining variables were kept constant. In Figure 17, four maps of critical buckling load are presented (with the same color range), involving the relationship between P u and L-w flange , L-t flange , L-t web , and w flange -t flange , respectively. As can be seen from the surface plots, the input-output relationship exhibited nonlinear behavior, which cannot be easily identified from the database. Figure 17a shows that a maximum value for the critical buckling load can be obtained if L reaches its minimum and w flange reaches its maximum value. On the other hand, the critical buckling load reaches its minimum if L reaches its highest value and w flange reaches its lowest value. This map confirms the negative effect of L, as pointed out in the literature [4]. In Figure 17b,c, the same results are obtained as in Figure 17a. This observation again confirms that the geometrical parameters of the cross-section are highly important [1,5]. Such quantitative information allows the design and analysis recommendations to be explored, as well as for new beam configurations to be generated (within the range of variables considered in this present study).
Materials 2020, 13, x FOR PEER REVIEW 20 of 27 capacity was only −0.092. On the contrary, the minimum value of the beam's length was 4000 mm (approximately 5.7 times larger than the maximum section's height) and five independent values were used to generate the database, ranging from 4000 to 8000 mm, with a step of 1000 mm. Thus, the linear correlation coefficient between the beam's length and the buckling capacity was −0.667 (approximately 7.25 times bigger than the linear correlation coefficient between the section's height and the buckling capacity). Consequently, a larger database should be considered in future studies to estimate the degree of importance of the section's height. The sensitivity analysis presented above demonstrates that the ML technique could assist in the design phase for circular opening steel beams. In addition to reliable prediction of the critical buckling load, the ANFIS-PSO model can also assist in the creation of input-output maps, as illustrated in Figure 17. In particular, as L, tweb, wflange, and tflange were the most important variables, they are used for map illustrations in this section. The values of the remaining variables were kept constant. In Figure 17, four maps of critical buckling load are presented (with the same color range), involving the relationship between Pu and L-wflange, L-tflange, L-tweb, and wflange-tflange, respectively. As can be seen from the surface plots, the input-output relationship exhibited nonlinear behavior, which cannot be easily identified from the database. Figure 17a shows that a maximum value for the critical buckling load can be obtained if L reaches its minimum and wflange reaches its maximum value. On the other hand, the critical buckling load reaches its minimum if L reaches its highest value and wflange reaches its lowest value. This map confirms the negative effect of L, as pointed out in the literature [4]. In Figure 17b,c, the same results are obtained as in Figure 17a. This observation again confirms that the geometrical parameters of the cross-section are highly important [1,5]. Such quantitative information allows the design and analysis recommendations to be explored, as well as for new beam configurations to be generated (within the range of variables considered in this present study).
e b w f l a n g e t f l a n g e

Conclusions
PSO is one of the most popular optimization techniques used to optimize and improve the performance of machine learning models in terms of classification and regression. However, its effectiveness depends significantly on the selection of parameters used to train this technique. In this paper, investigation and selection of PSO parameters was carried out to improve and optimize the performance of the ANFIS model, which is one of the most popular and effective ML models, for prediction of the buckling capacity of circular opening steel beams. Different parameters (nrule, npop, wini, c1, c2, and fv) of PSO were tuned on 3645 available data samples to determine the best values for optimization of the performance of ANFIS.
The results show that the performance of ANFIS optimized by PSO (ANFIS-PSO) is suitable for determining the buckling capacity of circular opening steel beams, but is very sensitive under different PSO investigation and selection parameters. The results also show that nrule = 10, npop = 50, wini = 0.1 to 0.4, c1 = [1, 1.4], c2 = [1. 8,2], and fv = 0.1 are the most suitable selection settings in order to get the best performance from ANFIS-PSO. The sensitivity analysis shows that L, wflange, tweb, and tflange are the most important input variables used for prediction of the buckling capacity of circular opening steel beams.
In short, this study might help in selection of the suitable PSO parameters for optimization of ANFIS in determining the buckling capacity of circular opening steel beams. It also helps in suitable selection of input variables for better prediction of the buckling capacity of circular opening steel beams. However, it is noted that the optimal values of PSO parameters found in this study are suitable for the ANFIS model in determining the buckling capacity of circular opening steel beams. Thus, it is suggested that these parameters should be validated with other ML models applied in other problems. Finally, variation in the mechanical properties of material used should be investigated in further research, as this is important from a physics perspective.

Conclusions
PSO is one of the most popular optimization techniques used to optimize and improve the performance of machine learning models in terms of classification and regression. However, its effectiveness depends significantly on the selection of parameters used to train this technique. In this paper, investigation and selection of PSO parameters was carried out to improve and optimize the performance of the ANFIS model, which is one of the most popular and effective ML models, for prediction of the buckling capacity of circular opening steel beams. Different parameters (n rule , n pop , w ini , c 1 , c 2 , and f v ) of PSO were tuned on 3645 available data samples to determine the best values for optimization of the performance of ANFIS.
The results show that the performance of ANFIS optimized by PSO (ANFIS-PSO) is suitable for determining the buckling capacity of circular opening steel beams, but is very sensitive under different PSO investigation and selection parameters. The results also show that n rule = 10, n pop = 50, w ini = 0.1 to 0.4, c 1 = [1, 1.4], c 2 = [1. 8,2], and f v = 0.1 are the most suitable selection settings in order to get the best performance from ANFIS-PSO. The sensitivity analysis shows that L, w flange , t web , and t flange are the most important input variables used for prediction of the buckling capacity of circular opening steel beams.
In short, this study might help in selection of the suitable PSO parameters for optimization of ANFIS in determining the buckling capacity of circular opening steel beams. It also helps in suitable selection of input variables for better prediction of the buckling capacity of circular opening steel beams. However, it is noted that the optimal values of PSO parameters found in this study are suitable for the ANFIS model in determining the buckling capacity of circular opening steel beams. Thus, it is suggested that these parameters should be validated with other ML models applied in other problems. Finally, variation in the mechanical properties of material used should be investigated in further research, as this is important from a physics perspective.