## 4 Computational Results
### Experimental Results for Larger Instances  
The experimentation considering large size instances was conducted in both versions of the algorithm. The complementary experimentation involves instances of up to 100 nodes. Tables 7-12 display the results of our computational experimentation. In Table 8, the minimum and maximum values for each objective are shown. According to the information obtained, for most of the instances, the MA-RK v2 reports better values for the total latency. In addition, the MA-RK v2 produces better values of tardiness for most of the instances. In summary, the selective version of the MA-RK clearly outperforms the MA-RK v1. Regarding the performance of the algorithms, it can be noticed that, for larger instances, the MA-RK v2 clearly outperforms the MA-RK v1. To better illustrate this, the fronts of Instances FNO8 and FNO10 are displayed in Figures 9 and 10. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Instance** & \multirow{2}{*}{**n**} & \multirow{2}{*}{**k**} & **MA-RK** & **MA-RK** \\
**Name** & & & **v1** & **v2** \\ \hline FNO3 & 20 & 3 & 8 & 5 \\ FNO4 & 22 & 8 & 8 & 6 \\ FNO5 & 75 & 4 & 9 & 11 \\ FNO6 & 75 & 7 & 5 & 18 \\ FNO7 & 75 & 10 & 7 & 13 \\ FNO8 & 75 & 7 & 10 & 5 \\ FNO9 & 75 & 8 & 13 & 11 \\ FNO10 & 100 & 5 & 8 & 12 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Quantity of non-dominated points for large-size instances. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Instance** & \multirow{2}{*}{**Type of**} & \multicolumn{2}{c}{**MA-RK v1**} & \multicolumn{2}{c}{**MA-RK v2**} \\ \cline{3-6}
**Name** & & **Objective** & **Min** & **Max** & **Min** & **Max** \\ \hline \multirow{3}{*}{**FNO3**} & Latency & 5327.35 & 7987.42 & **4852.72** & **7335.90** \\ \cline{2-6}  & Tardiness & 4853.59 & 19,782.90 & **3296.29** & **11,060.70** \\ \hline \multirow{3}{*}{**FNO4**} & Latency & **660.85** & 1102.98 & 759.182 & **975.53** \\ \cline{2-6}  & Tardiness & **233.65** & **1159.62** & 290.55 & 2072.82 \\ \hline \multirow{3}{*}{**FNO5**} & Latency & 8275.60 & **10,376.60** & **6788.84** & 10,460.50 \\ \cline{2-6}  & Tardiness & 27,515.3 & 66,396.30 & **23,903.30** & **51,934.70** \\ \hline \multirow{3}{*}{**FNO6**} & Latency & 7366.98 & **9251.54** & **6933.74** & 16,841.5 \\ \cline{2-6}  & Tardiness & 20,883.80 & 31,319.70 & **11,558.3** & **16,063.70** \\ \hline \multirow{3}{*}{**FNO7**} & Latency & 12,511.70 & 15,026.20 & **10,027.10** & **11,253.50** \\ \cline{2-6}  & Tardiness & 82,739.9 & **124,013.00** & **67,365.4** & 144,892 \\ \hline \multirow{3}{*}{**FNO8**} & Latency & 13,683.50 & 16,203.90 & **12,833.5** & **14,208.1** \\ \cline{2-6}  & Tardiness & 116,976.00 & 199,977.00 & **101,302** & **154,659** \\ \hline \multirow{3}{*}{**FNO9**} & Latency & 9617.94 & 14,949.70 & **9150.29** & **11,955.2** \\ \cline{2-6}  & Tardiness & 69,126.70 & 130,754.00 & **58,969.5** & **123,515.00** \\ \hline \multirow{3}{*}{**FNO10**} & Latency & 31,092.50 & 40,164.90 & **26,484.2** & **35,107.6** \\ \cline{2-6}  & Tardiness & 343,722.00 & 530,071.00 & **272,748.00** & **335,684.00** \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Minimum and maximum values for both objectives functions for large-size instances. Due to this, the metric of the execution time was evaluated to verify if any of the versions performs more quickly. Table 9 displays the elapsed CPU time for the best execution. Figure 10: Pareto fronts for Instance FNO10. Figure 9: Pareto fronts for Instance FNO8. As expected, the required time increases as the size of the instances increased. However, both versions of the metaheuristic were working within a similar computational performance range. As for the third metric, the hypervolume, the results of the algorithm are displayed in Table 10. There, we do not have enough evidence to confirm that any algorithm outperforms the other. What can be confirmed is that MA-RK v2 produced higher values of hypervolume for seven out of eight instances. However, for the instance where the MA-RK v1 obtained better values, the difference against the NSGA was small.