## 4 Results and Discussion
### Prediction Capability  
The effectiveness of ANFIS, ANFIS-SA, and ANFIS-BBO models is evaluated in this section. The results shown herein correspond to those obtained with ANFIS-SA at optimized iteration 72 and ANFIS-BBO at iteration 20. The prediction performance in a regression form is shown in Figure 5, whereas the corresponding summarized information is indicated in Table 3. It is noteworthy that a reverse transformation was previously applied to obtain the final critical buckling load from the predicted normalization space. As regards the training dataset, Figure 5a,c,e present outputs N\({}_{\text{u}}\) of ANFIS, ANFIS-SA, and ANFIS-BBO in function of the corresponding measured N\({}_{\text{u}}\) (in kN). Figure 5b,d,f present outputs N\({}_{\text{u}}\) of ANFIS, ANFIS-SA, and ANFIS-BBO in function of the corresponding testing data, associated with the testing part. Linear equations in the form \"predicted N\({}_{\text{u}}\) = slope * (measured N\({}_{\text{u}}\)) + intercept\" were also calculated in each case to estimate the goodness of fit between predicted and measured N\({}_{\text{u}}\) (slope and intercept values are indicated in Table 3). Figure 4: Optimization of ANFIS parameters using the SA technique for (**a**) R, (**c**) RMSE, and (**e**) MAE using the BBO technique for (**b**) R, (**d**) RMSE, and (**f**) MAE. With respect to the training ANFIS, ANFIS-SA, and ANFIS-BBO parts, the slopes of 0.889, 0.872, and 0.837 were extracted, respectively. The obtained values of statistical criteria were R = 0.943, 0.969, and 0.926, RMSE = 68.235, 52.552, and 77.692, MAE = 58.722, 40.793, and 60.437, respectively. It could be deduced that ANFIS-SA is the best model for the training data. This was also confirmed by the values of the standard deviation of error StD\({}_{\text{error}}\) (Table 3). For the training part, ANFIS-SA yielded the lowest value of StD\({}_{\text{error}}\) compared to ANFIS and ANFIS-BBO (i.e., StD\({}_{\text{error}}\) = 69.105, 53.190, and 78.553 for ANFIS, ANFIS-SA, and ANFIS-BBO, respectively). Figure 5: _Cont._  
Let us now consider the testing dataset. It was observed that the ANFIS-BBO model yielded the best prediction results with respect to R, RMSE, MAE, the mean of error m\({}_{\text{error}}\), and StD\({}_{\text{error}}\) (i.e., R = 0.896, 0.941, 0.960; RMSE = 111.830, 81.990, 66.558; MAE = 82.407, 65.594, 50.723; m\({}_{\text{error}}\) = -9.651, 16.445, 4.363; StD\({}_{\text{error}}\) = 114.842, 82.796, 68.459 using ANFIS, ANFIS-SA and ANFIS-BBO, respectively). Nonetheless, the slope of the fit linear equation using ANFIS-BBO was not the smallest deviation from the diagonal line. As indicated in Table 3, the slopes were 0.939, 0.897, and 0.916 (the corresponding slope angles were \(43.202^{\circ}\), \(41.883^{\circ}\), and \(42.482^{\circ}\)) using ANFIS, ANFIS-SA, and ANFIS-BBO, respectively. This indicates that ANFIS gave the best result in terms of linear slope. However, ANFIS-BBO was not far, with a slope angle value only 0.7\({}^{\circ}\) smaller. As observed (Figure 6a,b) for N\({}_{\text{u}}\) values in the function of sample index, the ANFIS-BBO model might be not best for the training dataset, but its prediction performance was highest for the testing dataset. This remark was confirmed by the histograms of the normalized errors for the training dataset (Figure 7a) and for the testing dataset (Figure 7b). The normalized errors (in %) were calculated by dividing the relative errors to the measured values of N\({}_{\text{u}}\). Figure 5: Graphs of regression results between measured Nu versus predicted Nu for the training part using (**a**) ANFIS, (**c**) ANFIS-SA, and (**e**) ANFIS-BBO; for the testing part using (**b**) ANFIS, (**d**) ANFIS-SA, and (**f**) ANFIS-BBO.