## 5 Results and Discussion
### Analysis of Results among DE Variants
The results of Table 13 are summarized in Table 14. This last table displays the number of wins obtained by each DE variant per case study and indicates the total sum. Based on these results, it is confirmed that rand/1/exp was the best-performing variant in the solution of the dimensional synthesis problems for four-bar Grashof-type mechanisms with 22 wins, and this was followed in second place by best/1/bin with 15 wins. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Variant** & **Min \(J(\vec{p})\)** & **Max \(J(\vec{p})\)** & **Mean \(J(\vec{p})\)** & **Std \(J(\vec{p})\)** \\ \hline rand/1/bin & \(3.2919\times 10^{-2}\) & \(2.2387\times 10^{1}\) & \(1.0145\) & \(4.0762\) \\ rand/1/exp & \(4.7963\times 10^{-28}\) & \(1.5917\times 10^{-2}\) & \(1.3047\times 10^{-3}\) & \(3.3223\times 10^{-3}\) \\ best/1/bin & \(4.9180\times 10^{-4}\) & \(1.2401\) & \(1.2289\times 10^{-1}\) & \(2.5584\times 10^{-1}\) \\ best/1/exp & \(9.1777\times 10^{-5}\) & \(3.4345\) & \(1.5169\times 10^{-1}\) & \(6.2857\times 10^{-1}\) \\ current-to-rand/1 & \(2.6946\times 10^{-2}\) & \(1.5198\times 10^{1}\) & \(3.5729\) & \(4.2861\) \\ current-to-best/1 & \(1.6866\times 10^{-2}\) & \(6.2969\times 10^{1}\) & \(1.1066\times 10^{1}\) & \(1.4955\times 10^{1}\) \\ current-to-rand/1/bin & \(5.6907\times 10^{-2}\) & \(6.9120\) & \(2.7548\) & \(1.6211\) \\ rand/2/dir & \(2.1001\times 10^{-1}\) & \(4.2374\times 10^{1}\) & \(1.2032\times 10^{1}\) & \(1.2016\times 10^{1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Descriptive results for case study C1 concerning the value of the objective function \(J(\vec{p})\). Figure 3: Best trajectories generated by each DE variant for case study C2. Figure 2: Best trajectories generated by each DE variant for case study C1. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Variant** & **Min \(\mathbf{J(\vec{p})}\)** & **Max \(\mathbf{J(\vec{p})}\)** & **Mean \(\mathbf{J(\vec{p})}\)** & **Std \(\mathbf{J(\vec{p})}\)** \\ \hline rand/1/bin & \(8.9514\times 10^{-2}\) & \(4.8844\times 10^{-1}\) & \(2.0483\times 10^{-1}\) & \(1.1031\times 10^{-1}\) \\ rand/1/exp & \(1.4882\times 10^{-4}\) & \(8.6580\times 10^{-4}\) & \(3.5663\times 10^{-4}\) & \(2.2381\times 10^{-4}\) \\ best/1/bin & \(1.7205\times 10^{-4}\) & \(2.4649\) & \(2.2577\times 10^{-1}\) & \(6.3549\times 10^{-1}\) \\ best/1/exp & \(3.5591\times 10^{-1}\) & \(3.0170\times 10^{2}\) & \(7.6570\times 10^{1}\) & \(8.7209\times 10^{1}\) \\ current-to-rand/1 & \(3.2355\times 10^{-1}\) & \(1.3797\times 10^{2}\) & \(2.2160\times 10^{1}\) & \(3.0723\times 10^{1}\) \\ current-to-best/1 & \(1.5225\) & \(3.3413\times 10^{2}\) & \(9.9118\times 10^{1}\) & \(1.0246\times 10^{2}\) \\ current-to-rand/1/bin & \(3.2461\times 10^{-2}\) & \(2.5930\) & \(6.5177\times 10^{-1}\) & \(6.2819\times 10^{-1}\) \\ rand/2/dir & \(8.7032\) & \(2.0594\times 10^{2}\) & \(7.5678\times 10^{1}\) & \(5.3282\times 10^{1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Descriptive results for case study C3 concerning the value of the objective function \(J(\vec{p})\). Figure 4: Best trajectories generated by each DE variant for case study C3. Figure 5: Best trajectories generated by each DE variant for case study C4. \begin{table}
\begin{tabular}{c c c c c} \hline
**Variant** & **Min \(t\) (s)** & **Max \(t\) (s)** & **Mean \(t\) (s)** & **Std \(t\) (s)** \\ \hline rand/1/bin & 3.2880 & 3.5230 & 3.4037 & \(5.4166\times 10^{-2}\) \\ rand/1/exp & 3.5480 & 3.6490 & 3.6162 & \(2.8255\times 10^{-2}\) \\ best/1/bin & 3.5060 & 3.8940 & 3.7965 & \(1.0321\times 10^{-1}\) \\ best/1/exp & 2.8110 & 3.9540 & 3.2564 & \(3.5805\times 10^{-2}\) \\ current-to-rand/1 & 3.2950 & 3.4290 & 3.3674 & \(3.2500\times 10^{-2}\) \\ current-to-best/1 & 3.5020 & 3.6970 & 3.6184 & \(5.0301\times 10^{-2}\) \\ current-to-rand/1/bin & 3.3810 & 3.6050 & 3.4849 & \(5.8481\times 10^{-2}\) \\ rand/2/dir & 3.3790 & 3.4590 & 3.4094 & \(2.0647\times 10^{-2}\) \\ \hline \end{tabular}
\end{table} 
Table 10: Descriptive results for case study C3 concerning the execution time \(t\).