## 5 Comparative Experimental Study
### Statistical Analysis of the CHT Behavior through Metrics  
This work is also interested in knowing the search capabilities of metaheuristic algorithms with the constraint-handling techniques related to different performance metrics. Then, six performance metrics [62] found in the specialized literature were used. Each of these metrics is described below: \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline  & & \multicolumn{3}{c}{**Study Case 1**} & \multicolumn{3}{c}{**Study Case 2**} & \multicolumn{3}{c}{**Study Case 3**} \\  & **FR** & **SR** & \(e\)**C** & **PF** & **FR** & **SR** & \(e\)**C** & **PF** & **FR** & **SR** & \(e\)**C** & **PF** \\ \hline DER1B & 3 & 0 & 1 & 1 & 2 & 1 & 0 & 2 & 2 & 0 & 1 & 2 \\ DER1E & 1 & 0 & 1 & 1 & 2 & 2 & 0 & 1 & 1 & 0 & 1 & 2 \\ DEB1B & 2 & 0 & 0 & 2 & 0 & 0 & 0 & 2 & 0 & 0 & 2 \\ DEB1E & 3 & 0 & 2 & 1 & 2 & 0 & 0 & 2 & 3 & 0 & 1 & 1 \\ DECR & 2 & 0 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 0 \\ DECB & 1 & 0 & 1 & 2 & 2 & 0 & 0 & 1 & 2 & 0 & 3 & 0 \\ DECR1B & 3 & 0 & 2 & 1 & 1 & 3 & 0 & 1 & 2 & 1 & 1 & 0 \\ DECR1E & 2 & 0 & 0 & 2 & 2 & 1 & 0 & 2 & 2 & 0 & 3 & 1 \\ GA & 2 & 0 & 2 & 1 & 0 & 2 & 1 & 3 & 1 & 1 & 1 & 0 \\ PSO & 1 & 0 & 1 & 1 & 0 & 0 & 0 & 3 & 0 & 1 & 2 & 2 \\ \hline
**Total** & **20** & 0 & 11 & 13 & 12 & 10 & 1 & **16** & **16** & 4 & 14 & 10 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: The number of wins obtained by each algorithm through Friedman test for multiple comparisons using the execution best values as the sample. * The feasibility probability (FP) is obtained by dividing the number of feasible executions by the total number of executions. FP is in the range of \([0,1]\), where 0 represents that there are not feasible solutions, and 1 represents that all found solutions are feasible. * The convergence probability (P) is obtained by dividing the number of successful executions by the total number of executions. The successful solutions are defined as the solutions \(x\) close to the best solution \(x^{*}\) in the objective function space. Therefore, a satisfactory solution can be determined by \(|\bar{J}(x)-\bar{J}(x^{*})|\leq\bar{e}\), where the best executions per each study case is shown in Tables A7-A9 of Appendix C. The value of \(\bar{e}\) is related to the bias region in the synthesis problem where the mechanism executes a suitable rehabilitation routine. For the four-bar mechanism, \(\bar{J}(x^{*})=0.0021\) and \(\bar{e}=0.0018\) are considered; for the cam-linkage mechanism, \(\bar{J}(x^{*})=0.6540\) and \(\bar{e}=0.5931\) are chosen; and for the eight-bar linkage mechanism, \(\bar{J}(x^{*})=1.2054\times 10^{-4}\) and \(\bar{e}=0.0011\) are selected. The P metric value is in the interval \([0,1]\), where a higher value (the best P metric) implies that all executions converge to the region. The metric is related to the reliability of an algorithm to find successful solutions. * AFES is the average number of function evaluations required for each successful execution. Successful execution is considered when it finds the first solution in the bias region \(\bar{e}\). If successful executions are not found, Non-Successful Executions (NSE) are accounted for. The best result in the AFES metric is first related to the minimum value in NSE and then to the lower values of the AFES. This metric indicates a convergence speed measure to the bias region \(\bar{e}\). * Successful performance (SP) divides the AFES by P. The best result in the SP metric is first related to the minimum value in NSE and then to the lower values of the SP. * EVALS counts the number of evaluations an algorithm needs to find the first feasible solution in every execution. Lower evaluations mean a low computational cost to reach a feasible solution; therefore, this is preferred. This metric is presented with the use of statistics. * The Progress Ratio (PR) measures the improvement capability of an algorithm inside the feasible region in every execution.