## 3 Computational Examples and Analysis
### Case Study I: Bi-Objective Optimization
#### 3.1.3 Results of Asphalt Mixture Proportion Optimization Using GPR-AWMOPSO  
(1) System Models Based on GPR  
As mentioned above, GPR is adopted to model objective functions based on ML methods, which would be trained using the collected data. There are three inputs and five outputs, the whole dataset includes several groups selected randomly from 19 orthogonal groups and is also separated into two datasets for training along with testing. Then we can predict the volumetric and physical performances of asphalt mixture containing basalt fibre from 25% of the selected dataset by using the training dataset from 75% of the selected dataset randomly. Besides, the root-mean-square error (RMSE) in Equation (21) is selected to assess the difference between observed values and predictive values and the squared \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{1}{c}{**Input Variables**} & \multicolumn{1}{c}{**Lower Bound**} & \multicolumn{1}{c}{**Medium**} & \multicolumn{1}{c}{**Upper Bound**} \\ \hline X\({}_{1}\) & \% & 0.25 & 0.40 & 0.55 \\ X\({}_{2}\) & mm & 3 & 6 & 9 \\ X\({}_{3}\) & \% & 6.4 & 6.6 & 6.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Range constrains of input variables. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Output Variables** & \(\mathbf{Y_{1}}\) & \(\mathbf{Y_{2}}\) & \(\mathbf{Y_{3}}\) & \(\mathbf{Y_{4}}\) & \(\mathbf{Y_{5}}\) \\ \hline Specification & Maximize & 2–5 & 3–4 & \(\geq\)17 & 75–85 \\ Unit & kN & mm & \% & \% & \% \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Volumetric and physical property constrains of output variables. correlation coefficient (R\({}^{2}\)) in Equation (22) is selected to evaluate the correlation between these two values [42]. \[\text{RMSE}=\sqrt{\frac{1}{m}{\sum\limits_{i=1}^{m}{(y_{i}^{*}-y_{i})^{2}}}} \tag{21}\]  
\[\text{R}^{2}=\frac{{(\sum\limits_{i=1}^{m}{(y_{i}^{*}-\overline{y}^{*})(y_{i}- \overline{y}))}^{2}}}{{(\sum\limits_{i=1}^{m}{(y_{i}^{*}-\overline{y}^{*})^{2} )(\sum\limits_{i=1}^{m}{(y_{i}-\overline{y})^{2})}}}} \tag{22}\]  
where \(y_{i}^{*}\) and \(y_{i}\) denote the predictive and observed values, \(\overline{y}^{*}\) and \(\overline{y}\) are the mean values of the observed values and predictive values. Figure 5 shows the reliable results of GPR model versus size of training dataset. It can be observed that the reliability of the GPR model is better with the size of training dataset increasing. Therefore, in the following study, the training dataset consisting of 8 groups would be selected randomly from the experimental groups for the GPR model, which possesses a good prediction effect. The prediction results for the volumetric and physical performances of asphalt mixture containing basalt fibre are listed in Table 4. Figure 6 shows the predictive values by the GPR model versus the experimental values for training and testing sets. The regression analysis indicates that the optimum GPR model was successful in learning the relationship between the input and output variables. There are lower RMSE values and higher R\({}^{2}\) values for both training and testing datasets. It can be indicated that the GPR model can accurately model the relationship between input and output variables. Thus, it could be confirmed that the optimal GPR model has been well trained based on the comparison between the volumetric and physical properties of training and testing datasets. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Output Variables**} & \multicolumn{2}{c}{**RMSE**} & \multicolumn{2}{c}{**R\({}^{2}\)**} \\ \cline{2-5}  & **Training** & **Testing** & **Training** & **Testing** \\ \hline \(Y_{1}\) & 0.242 & 0.145 & 0.984 & 0.992 \\ \(Y_{2}\) & 0.174 & 0.134 & 0.865 & 0.988 \\ \(Y_{3}\) & 0.096 & 0.154 & 0.996 & 0.995 \\ \(Y_{4}\) & 0.087 & 0.133 & 0.995 & 0.994 \\ \(Y_{5}\) & 0.135 & 0.154 & 0.996 & 0.996 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Root-mean-square error (RMSE) and R\({}^{2}\) values for volumetric and physical properties of basalt fibre reinforced asphalt mixture.