## 5 Results and Discussion
### Analysis of Results among DE Variants  
In order to carry out the comparative study, 30 independent runs of each DE variant were performed on each of the previously described case studies. The above process was performed to collect enough information to describe the behavior of each variant. Each of the executions was performed in a PC with Intel(R) Core(TM) i9-10900K CPU @ 3.70 GHz and 32 GB of RAM. The algorithms were written in C++ programs. Tables 4-7 present the statistical results of the experiments performed concerning the value of the objective function \(J(\vec{p})\). Moreover, the statistics on the execution time \(t\) of all variants are summarized in Tables 8-11. The columns of these tables include, in addition to the name of the DE variant, the minimum, maximum, mean, and standard deviation of the objective function \(J(\vec{p})\) and the execution time \(t\), respectively, for the 30 independent runs. Moreover, the behavior of the best solutions to the dimensional synthesis problems obtained with each DE variant is shown in Figures 2-5 for each case study. The minimum and maximum tracking error \(e\) (i.e., the maximum and minimum distances between the precision points and the points generated with the mechanism) for those figures are reported in Table 12. In all tables, the values in gray background color denote the best results. Based on the above results, the main findings can be stated as follows:  
* In synthesis problems without prescribed timing related to the cases C1 and C3, the results shown in Tables 4 and 6 indicate that the variant rand/1/exp obtained unbeatable results based on the minimum value achieved in all descriptive indicators when compared to the rest of the variants. Its performance was significantly better than most of the DE variants based on the standard deviation and the search for the best solution. Only the variant best/1/bin performed competitively in the search for the minimum value. * In synthesis problems with prescribed timing related to the cases C2 and C4, the results shown in Tables 5 and 7 reveal that the variants current-to-rand/1/bin and rand/1/exp had the best performance in the case C2 and the case C4, respectively, based on the standard deviation and the search for the best solution. In the case C2, all variants, except for rand/1/bin and current-to-best/1, had similar performance results (i.e., they performed competitively) in terms of the minimum value of the objective function \(J(\vec{p})\). On the other hand, the variants best/1/bin, best/1/exp, and rand/2/dir performed comparably to rand/1/exp when considering the minimum value of the objective function \(J(\vec{p})\) in the case C4. This indicates that the second best promising algorithms in this kind of problem were the variants best/1/bin, best/1/exp, and rand/2/dir. This may be due to the exploratory capacity of these variants. * Based on the statistical information of the four case studies included in Tables 4-7, the DE variant rand/1/exp can be considered as the best alternative to solve the problem of synthesizing four-bar Grashof-type mechanisms. Since four-bar mechanisms with different syntheses can produce very similar trajectories [41], the success of the DE variant rand/1/exp may have been attributed in part to the known exploratory ability of the rand variants [42], which made it possible to explore different promising regions of the solution space (i.e., different ways of synthesizing the mechanism). In addition, the use of the exponential recombination \(exp\) in rand/1/exp may have increased its performance with respect to problems with a larger number of design variables [43], as in cases C1 and C3, where, in addition to the parameters for the synthesis of the mechanism, the input values that allowed for reaching each precision point were also searched. Here, it is worth noting that, in several of the applications requiring the dimensional synthesis of mechanisms, there is no prescribed time [44; 45; 46].