## 4 Experiments and Results
### Analysis of Solutions Obtained with the Best Algorithm  
The previous experiment identified SMS-EMOA as the best-performing multi-objective optimization algorithm, achieving the highest hypervolume, with an execution time of 10.44 min. The solutions generated by this algorithm (shown in Figure 5) were subsequently used for sensitivity analysis. Figure 6 displays boxplots of the feature values corresponding to the best solutions identified by SMS-EMOA. It is observed that the \(x\) and \(y\) values for counterweights 4 and 5 are near zero. Furthermore, the thicknesses of these counterweights are very small, approaching the minimum bound (see Equation (15)); the average thicknesses are 0.0068 and 0.0059 for counterweights 4 and 5, respectively. This suggests that these counterweights contribute minimally to the optimization of the _ShM_ and **ShF** objective functions. Some features exhibit more significant variance than others; for instance, \(y_{3c}\) assumes both negative and positive values. Figure 4: Performance of single-objective optimization algorithms in terms of _ShM_ and **ShF**. Figure 5: Performance of multi-objective optimization algorithms in terms of _ShM_ and **ShF**. Figure 7 presents the feature values, categorized according to the corresponding \(ShM\) and **ShF** values. It is observed that the values of \(y_{\infty}\) play a critical role in determining whether the mechanism tends to optimize \(ShM\) or **ShF**: negative values correspond to \(ShM\) optimization, while positive values favor **ShF** optimization. A similar pattern is evident for \(y_{1c}\). A trend is also observed for \(t_{2c}\): lower thickness values tend to correlate with **ShF** optimization, whereas higher thickness values are more effective for \(ShM\) optimization. As shown in Figure 6, counterweights four and five tend to be minimized or effectively eliminated. Therefore, the SMS-EMOA algorithm was re-executed with modified parameters. First, counterweights four and five were removed while maintaining the original constraints. Subsequently, a second experiment was conducted in which counterweights four and five were removed, and the thickness constraint was increased, allowing a maximum value of 0.10. This adjustment was implemented because the thickness values of counterweights one, two, and three are close to the maximum allowable value (see Figures 6 and 7). \begin{table}
\begin{tabular}{l c c} \hline \hline
**Algorithm** & **Type** & **Hypervolume** \\ \hline GA & Single & 0.327 \\ PS & Single & 0.24 \\ DE & Single & 0.39 \\ ES & Single & 0.452 \\ SRES & Single & 0.441 \\ ISRES & Single & 0.448 \\ NSGA-II & Multi & 0.438 \\ NSGA-III & Multi & 0.405 \\ R-NSGA-II & Multi & 0.427 \\ R-NSGA-III & Multi & 0.387 \\ UNSGA-III & Multi & 0.201 \\ MOEA/D & Multi & 0.297 \\ AGE-MOEA & Multi & 0.454 \\ C-TAEA & Multi & 0.291 \\
**SMS-EMOA** & **Multi** & **0.458** \\ RVEA & Multi & 0.422 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Hypervolumes of different optimization algorithms. Bold letters indicate the best performance. \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Algorithm** & **Time per Repetition (min)** &
\begin{tabular}{c} **No. of** \\ **Repetitions** \\ \end{tabular} & **Total Time (min)** \\ \hline GA & 13.43 & 200 & 2686.0 \\ PS & 2.66 & 200 & 532.0 \\ DE & 13.3 & 200 & 2660.0 \\ ES & 26.62 & 200 & 5324.0 \\ SRES & 26.62 & 200 & 5324.0 \\ NSGA-II & 13.15 & 1 & 13.15 \\ NSGA-III & 10.86 & 1 & 10.86 \\ R-NSGA-II & 10.23 & 1 & 10.23 \\ R-NSGA-III & 21.09 & 1 & 21.09 \\ UNSGA-III & 10.32 & 1 & 10.32 \\ MOEA/D & 0.96 & 1 & 0.96 \\ AGE-MOEA & 10.56 & 1 & 10.56 \\ C-TAEA & 0.87 & 1 & 0.87 \\ SMS-EMOA & 10.44 & 1 & 10.44 \\ RVEA & 10.43 & 1 & 10.43 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Execution times of different multi-objective optimization algorithms. Figure 8 presents the results of these experiments. It can be observed that using only the first three counterweights and including all five yield comparable results in terms of \(ShM\) and **ShF** optimization.