## 5 Multi-Objective Optimization Framework
### Pareto Solution Selection via Entropy-Weighted Distance  
To systematically prioritize high-quality solutions from the Pareto-optimal set, an entropy-weighted distance method was employed. This approach combines objective weighting based on information entropy with geometric distance metrics to quantify solution quality while preserving trade-off characteristics. The procedure consists of four steps:  
1. Data normalization: The performance values of Pareto solutions for each objective were normalized to unify optimization directions. For maximization objectives (\(T_{av}\), \(\eta\)), \[y_{ij}=\frac{x_{ij}-\min(x_{j})}{\max(x_{j})-\min(x_{j})}\] (9) For minimization objectives (\(\Delta T\)), \[y_{ij}=\frac{\max(x_{j})-x_{ij}}{\max(x_{j})-\min(x_{j})}\] (10) where \(x_{ij}\) denotes the original value of the \(j\)-th objective for the \(i\)-th solution, and \(y_{ij}\) represents the normalized value.
2. Entropy calculation: The information entropy \(E_{j}\) for each objective was computed to quantify its data variability: \[p_{ij}=\frac{y_{ij}}{\sum_{i=1}^{n}y_{ij}}\] (11)  
Figure 12: Pareto frontier images: (**a**) Pareto frontier 3D image; (**b**) projection of Pareto frontier on \(T_{av}\) and \(\eta\); (**c**) projection of Pareto frontier on \(T_{av}\) and \(\Delta T\); and (**d**) projection of Pareto frontier on \(\eta\) and \(\Delta T\).  
\[E_{j}=-\frac{1}{\ln n}\sum_{i=1}^{n}p_{ij}\ln p_{ij} \tag{12}\]  
where \(n\) is the number of Pareto solutions, and \(p_{ij}\) denotes the proportion of the \(i\)-th solution's contribution to the \(j\)-th objective.
3. Weight determination: The entropy weight \(w_{j}\) for each objective was derived based on its entropy value: \[w_{j}=\frac{1-E_{j}}{\sum_{k=1}^{m}(1-E_{k})}\] (13) where \(m\) is the number of objectives [40]. Lower entropy (\(E_{j}\)) corresponds to higher weight (\(w_{j}\)), reflecting greater decision information contained in the objective [31]. The result of entropy weight for \(T_{av}\), \(\eta\) and \(\Delta T\) is \(w_{1}=0.4010\), \(w_{2}=0.2589\) and \(w_{3}=0.3404\), respectively.
4. Entropy-weighted distance evaluation: The composite performance of each Pareto solution was assessed using a weighted Euclidean distance metric: \[D_{i}=\sqrt{w_{1}(1-T_{av,i}^{*})^{2}+w_{2}(1-\eta_{i}^{*})^{2}+w_{3}(0-\Delta T _{i}^{*})^{2}}\] (14) where \(T_{av,i}^{*}\), \(\eta_{i}^{*}\) and \(\Delta T_{i}^{*}\) are values of the objectives in Pareto solutions (normalized to [0, 1]). Solutions with smaller \(D_{i}\) values exhibit balanced performance closer to the ideal point \(Y^{*}=[1,1,0]^{T}\) in normalized objective space. Geometrically, this defines an ellipsoid in 3D objective space centered at \(Y^{*}\) (Figure 13). The point with the smallest \(D_{i}\) will be chosen as the final Pareto optimal solution.  
This methodology enables unbiased selection of optimal designs while retaining critical trade-off information, facilitating informed decision-making in multi-objective optimization scenarios. The final Pareto optimal solution with a minimal \(D_{i}\) of 0.3268 is presented in Table 7 after inverse normalization. Corresponding values of design variables are adopted to conduct a FEA validation in which the prediction and simulation values achieve a high degree of consistency (Table 8). As a result of multi-objective optimization, \(T_{av}\) and \(\eta\) show increases of 4.62% and 0.15% as \(\Delta T\) is suppressed by 10.48%.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \(b_{\tau}\) & \(b_{m}\) & \(l_{m}\) & \(T_{av}\) & \(\eta\) & \(\Delta T\) \\ \hline Prototype & 3.00 mm & 4.00 mm & 50.00 mm & 154.96 N-m & 98.05\% & 36.13 N-m \\ Optimized & 4.99 mm & 4.96 mm & 52.90 mm & 162.12 N-m & 98.20\% & 32.35 N-m \\ \hline \hline \end{tabular}
\end{table}
Table 7: Result of optimization.  
Figure 13: Relationship between 3D Pareto frontier and entropy-weighted distance.