## 4 Conclusions  
Accurately predicting the critical buckling load is crucial in the field of structural engineering applications. In this study, different hyrbid ML models, namely ANFIS-RCSA, ANFIS-CA, and ANFIS-SFLA, i.e., a combination of ANFIS with metaheuristic optimization techniques such as RCSA, CA and SFLA, respectively, were used to predict the critical buckling load of I-shaped cellular steel beams with circular openings. A total of 3645 FE configurations on buckling behavior of I-section beams with circular openings were extracted to generate the training and testing datasets. Out of these, a total of eight input variables, namely the beam length, flange width, flange thickness, web thickness, inter-opening distance, opening diameter, section height, and beam end-opening distance, as well as one output variable of the critical buckling load of I-shaped cellular steel beams were used in the modeling. Various validation criteria, namely R, RMSE, and MAE were used to validate and compare  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Input Variable**} & \multirow{2}{*}{**Notation**} & **Linear Correlation** & **Influence Level** & **Classification** \\  & & **Coefficient between** & **Obtained by** & **Order by** \\  & & **Variables and Target** & **ANFIS-SFLA** & **ANFIS-SFLA** \\ \hline Beam length & L & 0.667 & 0.429 & 1 \\ Beam end-opening distance & d\({}_{0}\) & 0.023 & 0.003 & 8 \\ Opening diameter & D & 0.107 & 0.046 & 6 \\ Inter-opening distance & d & 0.072 & 0.047 & 5 \\ Section height & H & 0.092 & 0.008 & 7 \\ Web thickness & t\({}_{\text{web}}\) & 0.332 & 0.121 & 4 \\ Flange width & w\({}_{\text{flange}}\) & 0.375 & 0.215 & 2 \\ Flange thickness & t\({}_{\text{flange}}\) & 0.209 & 0.130 & 3 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Summary of sensitivity analysis.  
Figure 10: Graphs of sensitivity analysis obtained by (a) linear statistical correlation and (b) ANFIS-SFLA model.  
the models. In addition, a sensitivity analysis was also done to evaluate the importance of input variables for predicting the critical buckling load of I-shaped cellular steel beams using the ML models.  
The results showed that the three hybrid proposed models performed well but ANFIS-SFLA is the best predictor as compared with other models (ANFIS-CA and ANFIS-SFLA). The sensitivity results showed that the length of beam is a dominant factor for predicting the critical buckling load of I-shaped cellular steel beams using the ML models, followed by the flange width, flange thickness, web thickness, inter-opening distance, opening diameter, section height, and the beam end-opening distance, respectively.  
Overall, the hybrid ML technique developed in this study offers a powerful tool to accurately predict the critical buckling load of I-shaped cellular steel beams. However, the application of the ML model in engineering practice and design is not often an appropriate way. In further research, the current ANFIS-SFLA prediction function could be exploited to derive an explicit empirical equation for use in practical engineering design and analysis. For this purpose, the sensitivity analysis previously presented could play an important role in deriving the desired formula by searching, first, the empirical coefficients between the output and the most relevant inputs.  
Conceptualization, H.-B.L., T.-T.L., and B.T.P.; methodology, H.-B.L., T.-T.L., and B.T.P.; validation, H.-B.L., T.-T.L., and B.T.P.; formal analysis, H.-B.L., T.-T.L., and B.T.P.; data curation, L.M.L., V.Q.T., V.M.L., H.-L.T.V., and Q.H.N.; writing--original draft preparation, H.-B.L., T.-T.L., L.M.L., V.Q.T., V.M.L., H.-L.T.V., Q.H.N., and B.T.P.; writing--review and editing, H.-B.L., T.-T.L., and B.T.P.; visualization, L.M.L., V.Q.T., V.M.L., H.-L.T.V., and Q.H.N.; supervision, H.-B.L., T.-T.L., and B.T.P.; project administration, H.-B.L.; funding acquisition, T.-T.L.  
This research received no external funding.  
The authors declare no conflicts of interest.