## 3 Optimization Algorithms
### Grasshopper Optimization Algorithm (GOA)
In the present study to have a good comparison among the four models, in addition to RMSE, Pearson correlation coefficient (R), mean absolute percentage error (MAPE), mean absolute error (MAE), mean absolute bias error (MBE), scatter index (SI), and relative square error (RSE) are employed, which are formulated as follow:

\[R=\frac{Q\sum_{i=1}^{Q}M_{q}V_{q}}{(Q\sum_{i=1}^{Q}M_{q}^{2}-(\sum_{i=1}^{Q}M_ {q})^{2})(Q\sum_{i=1}^{Q}V_{q}^{2}-(\sum_{i=1}^{Q}V_{q})^{2})} \tag{17}\]

\[MAPE=\frac{100}{Q}\sum_{i=1}^{Q}\frac{V_{q}-M_{q}}{V_{q}} \tag{18}\]

Figure 5: Architecture of (a) hybridized ANN with GA (ANNGA), (b) hybridized ANN with SSA (ANNSSA), and (c) hybridized ANN with GOA (ANNGOA). Figure 6: M5P tree model. \[MAE=\frac{1}{Q}{\sum_{i=1}^{Q}{\left|{{M_{q}}-{V_{q}}}\right|}} \tag{19}\]

\[MBE=\frac{1}{Q}{\sum_{i=1}^{Q}{\left({{V_{q}}-{M_{q}}}\right)}} \tag{20}\]

\[SI=RMSE/\overline{M} \tag{21}\]

\[RSE=\frac{{\sum_{i=1}^{Q}{\left({{V_{q}}-{M_{q}}}\right)^{2}}}}{{\sum_{i=1}^{Q} {\left({\overline{M}-{M_{q}}}\right)^{2}}}} \tag{22}\]

where \(\overline{M}\) is mean value of measured results, and other parameters are explained in the previous section. MBE indicates that the model overestimates (MBE\(>\) 0) or underestimates (MBE \(<\) 0). SI determines that the performance of the model is \"excellent\" (0 \(\leq\) SI \(<\) 0.1), \"good\" (0.1 \(\leq\) SI \(<\) 0.2), \"fair\" (0.2 \(\leq\) SI \(<\) 0.3), or \"poor\" (0.3 \(\leq\) SI). Table 3 represents these indicators. According to Table 3, RMSE of ANNSSA is the lowest with the value of 2.734 MPa, while the M5P has the highest RMSE, which is over 2.5 times bigger than that of ANNSSA. The AANNSA and ANNGOA have the second and third least RMSE, respectively. In the case of MAE, the ANNSSA M5P tree has the highest cost by the value of 5.502 MPa followed by the ANNGOA of which MAE is lower than that of the M5P tree by almost 60%, the same value of ANNGA and ANNSSA are less than that of the M5P tree by 62% and 66%, respectively. According to RSE, the ANNSSA is the most accurate model with the RSE value of 0.017 while the M5P tree model is the least accurate one with the RSE value of 0.149, and ANNGA and ANNGOA are in between. The ANNSSA and ANNGA have almost the same MAPE while this value of the ANNGOA and the M5p tree are more than that of the ANNSSA by 25% and 245%, respectively. MBE indicates that the ANNSSA, the ANNGOA, and the M5P tree overestimate the compressive strength while the ANNGA underestimates that. According to SI, the M5P tree performs well, and other models performances are excellent. The R-values of all models are 0.99, which show a great correlation between measured and predicted compressive strength, except the R-value of M5P tree, which is 0.94. Figure 7 illustrates the predicted vs measured compressive strength, in which it can be seen that the scatter around the base line (linear regression) in ANNSSA is more than others. Moreover, Figure 8 compares MAE, RMSE, and MAPE in order to sort the models based on their efficiency. According to this figure, the ANNSSA is the most efficient model followed closely by the ANNGA and the ANNGOA respectively. Furthermore, the M5P tree is the least efficient model and has the highest error values. This order is true for MAE, RMSE, and MAPE. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Models** & **RMSE** & **MAE** & **RSE** & **MAPE** & **MBE** & **SI** & **R** \\  & **(MPa)** & **(MPa)** & & **(\%)** & **(MPa)** & & \\ \hline ANNSSA & 2.734 & 1.890 & 0.017 & 6.216 & 0.156 & 0.075 & 0.992 \\ ANNGA & 2.944 & 2.086 & 0.019 & 6.797 & \(-\)0.227 & 0.081 & 0.990 \\ ANNGOA & 3.027 & 2.227 & 0.020 & 7.790 & 0.037 & 0.083 & 0.990 \\ M5P tree & 7.098 & 5.502 & 0.149 & 21.458 & 0.179 & 0.196 & 0.947 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Statistic indicator of models. Figure 8: (**a**) A bar chart and (**b**) a radar chart to compare models’ results overlay. Figure 7: Predicted vs measured compressive strength for (**a**) ANNSSA, (**b**) ANNGA, (**c**) ANNGOA, and (**d**) MSP three.