## 5 Results and Discussion  
Findings of the simple regression analysis carried out in this study demonstrated that there is a need for predictive models to make precise estimations regarding rock TS. To this end, the present paper presents three hybrid models. In the proposed models, three neurons were utilized within the input layer, i.e., Rn, DD, and Is\({}_{50}\), while the BTS values were utilized in the output layer. Through a trial-and-error method, it was found that when the number of neurons within the hidden layer was set to five, the model performance was better than the other conditions. Aiming at choosing the best hybrid IWO-ANN, ABC-ANN, and ICA-ANN models, we developed numerous models on the basis of the results obtained from a number of parametric studies. At the final step, amongst the three hybrid models, the best one was chosen on the basis of the models' system error performance. In addition, more evaluations were conducted on the chosen models through comparing between the other PIs, i.e., R\({}^{2}\) and VAF. Table 4 presents the PI values of all hybrid models. Furthermore, Table 4 show the difference between the values achieved from the measured and predicted TS values of the rocks by means of IWO-ANN, ABC-ANN, and ICA-ANN, respectively. As can be concluded from Table 4, the IWO-ANN outperformed the other optimum hybrid models in terms of all the PIs. That is, IWO-ANN returned the most accurate values (i.e., the lowest RMSE with the maximum values of VAF and R\({}^{2}\)) for the training and testing datasets. For instance, in regard to testing the datasets, the IWO-ANN model with the RMSE of 0.0911, R\({}^{2}\) of 0.917, and a VAF of 91.731 estimated the BTS more accurately compared to ABC-ANN with the RMSE of 0.0955, R\({}^{2}\) of 0.904, and a VAF of 90.419. In addition, it presented a higher quality performance in comparison to ICA-ANN with an RMSE of 0.0978, R\({}^{2}\) of 0.901, and VAF of 90.134. Therefore, although all of the above-mentioned models were found to be capable of predicting the BTS value with an acceptable level of precision, the IWO-ANN model performed the best. Obviously, the IWO-ANN model for the rock TS prediction can be taken into account as an innovative predictive model in engineering fields of study. Finally, Figure 10 shows the prediction results for the IWO-ANN model for all the samples. In this figure, given that all the laboratory samples are in the new IWO-ANN system, a careful evaluation of the final accuracy is obtained. A linear relationship between the actual (parameter x) and the predicted (parameter y) value is also given. As expected, the chosen model presents as a new solution to the tensile strength of the rock samples in future works according to defined conditions for system design.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{\begin{tabular}{c} **Model** \\ **No.** \\ \end{tabular} } & \multirow{2}{*}{\
\begin{tabular}{c} **Seeds** \\ **No.** \\ \end{tabular} } & \multicolumn{4}{c}{**Network Result**} & \multicolumn{4}{c}{**Ranking**} & \multirow{2}{*}{
\begin{tabular}{c} **Total** \\ **Rank** \\ \end{tabular} } \\ \cline{3-3} \cline{5-10}  & & & **TR** & & **TS** & & **TR** & & **TS** & **Rank** \\ \cline{3-3} \cline{5-10}  & & & **R\({}^{2}\)** & **RMSE** & **R\({}^{2}\)** & **RMSE** & **R\({}^{2}\)** & **RMSE** & **R\({}^{2}\)** & **RMSE** & \\ \hline
1 & 5 & 0.898 & 0.0964 & 0.889 & 0.0971 & 5 & 5 & 4 & 4 & 18 \\
2 & 10 & 0.869 & 0.0998 & 0.881 & 0.0988 & 2 & 2 & 3 & 3 & 10 \\
3 & 15 & 0.861 & 0.1018 & 0.877 & 0.1004 & 1 & 1 & 2 & 2 & 6 \\
4 & 20 & 0.874 & 0.0987 & 0.869 & 0.1028 & 3 & 3 & 1 & 1 & 8 \\
5 & 25 & 0.923 & 0.0918 & 0.909 & 0.0943 & 7 & 7 & 6 & 6 & 26 \\
6 & 30 & 0.909 & 0.0935 & 0.922 & 0.0927 & 6 & 6 & 8 & 8 & 28 \\
7 & 35 & 0.928 & 0.0911 & 0.917 & 0.0936 & 8 & 8 & 7 & 7 & 30 \\
8 & 40 & 0.887 & 0.0979 & 0.894 & 0.0959 & 4 & 4 & 5 & 5 & 18 \\ \hline \hline \end{tabular}  
* TR: Training, TS: Testing.  
\end{table}
Table 3: Investigating effects of various seeds in predicting BTS.