## 3 Results and Discussions
### Validating the Hybrid ML Models  
Validation of the hybrid models was done using both training and testing datasets, as shown in Figure 6. The ideal regression lines are in discontinuous, whereas the R-values between predicted and actual critical buckling load are given in Table 6. The results are obtained after 1000 iterations for all hybrid ML models. The clouds of data points are given in color in order to better illustrate the concentration of points. The correlation results analysis shows that the R values vary from 0.927 to 0.962 for the training dataset and from 0.920 to 0.960 for the testing dataset. These results display a  
Figure 5: Optimization costs functions in terms of R using (**a**) RCSA, (**c**) CA, and (**e**) SFLA; in terms of MAE using (**b**) RCSA, (**d**) CA, and (**f**) SFLA. very good prediction performance of all the models, but ANFIS-SFLA is found to be the best predictor compared with other models. Figure 6: Graphs of regression between actual and predicted data, for the training dataset using (**a**) ANFIS-RCSA, (**c**) ANFIS-CA, and (**e**) ANFIS-SFLA; for the testing dataset using (**b**) ANFIS-RCSA, (**d**) ANFIS-CA, and (**f**) ANFIS-SFLA. A detailed summary of the prediction capability of ANFIS-RCSA, ANFIS-CA, and ANFIS-SFLA is presented in Table 6, whereas \"Error Mean\" is the mean of all relative errors, \"Error StD\" is the standard deviation of relative errors and \"Slope\" represents the slope between the ideal regression line and the linear fit given by the models. The results also confirm that ANFIS-SFLA is superior to ANFIS-RCSA and ANFIS-CA in predicting the critical buckling load of I-shaped beams. Histograms of the scaled critical buckling load of I-shaped beams of both actual and predicted values issued from ANFIS-RCSA, ANFIS-CA, and ANFIS-SFLA algorithms is presented in Figure 7 for both training and testing datasets The results show that a high concentration of the values is found in the [0, 0.5] interval, which emphasize the values around 0.1. This is also confirmed by the red color from Figure 6. It is worth noticed that the better correlation between actual and predicted critical buckling load is represented by the height of bars. An equally height of bars for a given value of critical buckling load means a perfect correlation (R = 1). More precisely, the error histograms between actual and predicted critical buckling load simulated by ANFIS-RCSA, ANFIS-CA, and ANFIS-SFLA are presented in Figure 8. The errors are found in the [-0.1, 0.1] range, which speaks for about 10% of error. ANFIS-SFLA is again proved as the best predictor, following by ANFIS-CA and ANFIS-RCSA, respectively. This could be easily observed by the highest value centered on 0 (ANFIS-SFLA) and the narrow curves (ANFIS-SFLA and ANFIS-CA) of about 2% of error. In general, in terms of six error criteria, three models perform well, but ANFIS-SFLA is shown to be superior to ANFIS-RCSA and ANFIS-CA for predicting the critical buckling load of I-shaped beams. On the basis of a comparison with the previous published study by [PERSON] et al. [31] using the same datasets but different ML model, namely ANN, it can be observed that although the R values of ANFIS-SFLA model in this study (0.96) is smaller than those of ANN model (0.9999), the results of ANFIS-SFLA model are more reasonable for predicting the critical buckling load of I-shaped beams as the R value of 0.9999 of ANN model in the previous study indicated the serious overfitting problem which should be avoided in predictive problems. Moreover, the advantage of the ANFIS-SFLA model is that it used an optimization technique, namely SFLA, to optimize the parameters of ANFIS model for avoiding the overfitting and improve the accuracy of the predictive model [65].