## 5 Results and Discussion
### The Performance of the SIDE Variant
#### 5.3.2 Comparative Results of SIDE with the State-Of-The-Art Algorithms  
After determining the effectiveness of the SIDE concerning the other DE variants studied in this work, the results were contrasted with those obtained by algorithms in recent specialized literature for the dimensional synthesis of four-bar Grashof-type mechanisms. The works in [26; 35] were selected for this purpose. In [35], the dimensional synthesis of the four-bar mechanism was tackled with nine advanced optimization algorithms for six case studies, including those from the benchmark in [40]. On the other hand, the work in [26] proposed the dimensional synthesis of four-bar mechanisms through a parameterization based on a motion analysis with relative angles. This parameterization of the synthesis problem was solved through DE for five case studies, three of which corresponded to those of the benchmark in [40]. The best results obtained by the SIDE and by the algorithms used in [26; 35] are presented in Tables 20-23 for each case study. The columns of these tables show the values of the design variables calculated with each algorithm, while the value of the objective function \(J(\vec{p})\) achieved with them is noted at the bottom. The best synthesis of the four-bar mechanism for each case study is highlighted in gray. Based on the information in these tables, it is possible to observe that the results obtained by the SIDE were superior in all four case studies compared to those of [26; 35], except for case C4, where it was outperformed by the work on [26] but was still competitive. This highlights the value of this study as a basis for proposing advanced algorithms with a greater capacity to solve complex problems related to the synthesis of mechanisms. At this point, it is important to note that the configuration of the hyperparameters in the SIDE and in the algorithms of the papers [26; 35] are different. While in [26], the algorithms were set up to perform a similar number of evaluations of the optimization problem reported in the literature, the configuration of the algorithms in [35] were oriented to reduce this number considerably. The main aim of the SIDE was to provide a four-bar mechanism oriented to accurately develop the task, which would impact future engineering applications, i.e., accuracy was the most important criterion, thereby sacrificing the number of times that the performance function was evaluated. Of course, a reduced number of evaluations implies shorter execution times. However, the execution times achieved with the SIDE were relatively small (see Table 18) when contrasted with the effective syntheses of the obtained four-bar mechanism (see Table 15). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Case Study** & **Min \(\boldsymbol{J(\vec{p})}\)** & **Max \(\boldsymbol{J(\vec{p})}\)** & **Mean \(\boldsymbol{J(\vec{p})}\)** & **Std \(\boldsymbol{J(\vec{p})}\)** \\ \hline C1 & 1.9510 & 2.6320 & 2.3715 & 1.8188 \(\times\) 10\({}^{-1}\) \\ C2 & 2.1930 & 2.2840 & 2.2418 & 2.7821 \(\times\) 10\({}^{-2}\) \\ C3 & 3.6470 & 3.8430 & 3.7554 & 4.1968 \(\times\) 10\({}^{-2}\) \\ C4 & 4.0490 & 5.0110 & 4.6015 & 2.6940 \(\times\) 10\({}^{-1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 18: Results obtained with SIDE concerning the execution time \(t\) for each case study. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Case Study** & **Min \(\boldsymbol{e(\text{mm})}\)** & **Max \(\boldsymbol{e(\text{mm})}\)** \\ \hline C1 & 1.2041 \(\times\) 10\({}^{-4}\) & 1.4047 \(\times\) 10\({}^{-3}\) \\ C2 & 7.9865 \(\times\) 10\({}^{-2}\) & 6.1541 \(\times\) 10\({}^{-1}\) \\ C3 & 1.7193 \(\times\) 10\({}^{-4}\) & 6.7010 \(\times\) 10\({}^{-3}\) \\ C4 & 1.8661 \(\times\) 10\({}^{-1}\) & 6.9446 \(\times\) 10\({}^{-1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 19: Minimum and maximum errors obtained with the best mechanism synthesized by SIDE for all case studies.