## 3 Variable Importance Analysis
### Random Forest-Based Variable Importance Quantification  
The variable importance measurement in random forest is illustrated in Figure 7. This study employs the out-of-bag (OOB) error-based approach, which leverages the inherent bootstrap sampling mechanism of the algorithm. During forest construction, each decision tree is trained on a bootstrap sample from the original dataset. Approximately 37% of instances are naturally excluded from each bootstrap sample, forming the OOB validation set for that particular tree. Variable importance is quantified by systematically perturbing variables in these OOB sets. For a target variable, its values are randomly permuted within the OOB data, and the resulting change in prediction accuracy is recorded. Critical variables will induce significant accuracy degradation (manifested as increased OOB error) when permuted, while non-informative variables will show negligible impact [33]. The formal importance score \(I\) for a variable is computed as  
\[I=\frac{1}{T}\sum_{i=1}^{T}(OOB_{permuted}^{i}-OOB_{original}^{i}) \tag{1}\]  
where \(T\) denotes the total number of trees, and \(OOB_{permuted}^{i}\) and \(OOB_{original}^{i}\) represent the OOB error rates before and after variable permutation for the \(i\)-th tree, respectively. The orthogonal experimental design generated 49 samples through an \(L_{49}(7^{6})\) orthogonal array, systematically covering six design variables across seven levels. These 49 instances were fully leveraged for random forest training. In accordance with the algorithm's bootstrap sampling mechanism, each decision tree was constructed using approximately 63% of the data (31 samples), while the remaining 37% (18 samples) inherently formed the OOB validation set. Leveraging the inherent OOB validation mechanism, the model \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Factor** & **Unit** & **Level 1** & **Level 2** & **Level 3** & **Level 4** & **Level 5** & **Level 6** & **Level 7** \\ \hline \(\delta_{S}\) & mm & 0.30 & 0.50 & 0.70 & 0.90 & 1.10 & 1.30 & 1.50 \\ \(b_{r}\) & mm & 3.00 & 3.33 & 3.67 & 4.00 & 4.33 & 4.67 & 5.00 \\ \(b_{nr}\) & mm & 3.00 & 3.40 & 3.80 & 4.20 & 4.60 & 5.00 & 5.40 \\ \(h_{s}\) & mm & 15.00 & 16.67 & 18.33 & 20.00 & 21.67 & 23.33 & 25.00 \\ \(l_{m}\) & mm & 30.00 & 35.83 & 41.67 & 47.50 & 53.33 & 59.17 & 65.00 \\ \(R_{mt}\) & mm & 37.50 & 39.17 & 40.83 & 42.5 & 44.17 & 45.83 & 47.50 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Six experimental factors and their levels. Figure 7: Random forest modeling principle diagram. achieved stabilization with 0.0181, 0.0197 and 0.0181 RMSE values for respective objectives as tree numbers increased (Figure 8a), confirming prediction reliability. This approach eliminates the requirement for explicit data partitioning, ensuring that all experimental data contribute synergistically to both model construction and validation. The importance of each design variable for \(T_{av}\), \(\eta\) and \(\Delta T\) is shown in Figure 8b-d, respectively. As can be seen from Figure 8b-d, the importance scores of \(\delta_{g}\) and \(R_{m}\) are all negative, indicating that \(\delta_{g}\) and \(R_{m}\) are not important for \(T_{av}\), \(\eta\) and \(\Delta T\), and therefore are not selected as key variables. The importance scores of \(b_{r}\) are negative for both \(T_{av}\) and \(\eta\), but the importance score for \(\Delta T\) is notably positive, indicating that although it is not important for \(T_{av}\) and \(\eta\), its effect on \(\Delta T\) should not be ignored, so \(b_{r}\) is classified as a key variable. The importance scores of \(b_{m}\) and \(l_{m}\) on \(T_{av}\), \(\eta\) and \(\Delta T\) are all high and positive, indicating that they have a high impact on all optimization objectives and are therefore all classified as key variables.