## 7 Experimental Results
### Bi-Objective Approach
For 100 \(\times\) 100 instances, MO-VNS provides the best behavior with up to 11.37%, followed by MO-FA and MO-ABC with up to 9.51% and 9.41%, respectively. For 200 \(\times\) 200 instances, MO-FA provides the best behavior with up to 10.77%, followed by MO-VNS and MO-VNS* with up to 10.03%. For 300 \(\times\) 300 instances, MO-FA provides the best behavior with up to 13.71% with up to 13.71%, followed by MO-ABC and MO-VNS* with up to 8.75% and 7.45%, respectively. For all instances, MO-FA provides the best behavior with up to 11.49%, followed by MO-ABC and MO-VNS* with up to 8.88% and 8.40%, respectively. From this hypervolume analysis, we conclude that MO-FA provides the best behavior solving the problem in general. Focusing on instance size, MO-FA is also the best algorithm solving small and large instances (50 \(\times\) 50, 200 \(\times\) 200, and 300 \(\times\) 300) and MO-VNS is better suited for medium size instances (100 \(\times\) 100). In Table 4, the behavior of MO-VNS and MO-VNS* is significantly different although their focus is almost the same, i.e., MO-VNS is better for small instances, while MO-VNS* is better for large ones. This fact is due to, for small instances, the perturbation mechanism in MO-VNS* penalizes the number of evaluations available for exploitation, while search space is not of concern, resulting in a better performance of MO-VNS. On the other hand, the perturbation mechanism in MO-VNS* is useful for exploring bigger search spaces in large instances, without being of concern the number of evaluations consumed by the process, resulting in a better performance of MO-VNS*. Table 5 shows the average set coverage metric of an algorithm compared to the others according to instance size. The metric was calculated using the median Pareto front obtained for each algorithm solving a given instance. Analyzing this table, for 50 \(\times\) 50 instances, we check that MO-ABC, MO-FA, and MOEA/D provide a similar behavior with up to 100.00%, followed by MOEA/D and MO-VNS with up to 90.00% and 88.28%, respectively. For 100 \(\times\) 100 instances, MO-VNS provides the best behavior with up to 87.56%, followed by MO-VNS* and MO-FA with up to 83.11% and 75.93%. For 200 \(\times\) 200 instances, MO-FA provides the best behavior with up to 77.52%, followed by MO-VNS* and MO-VNS with up to 68.92% and 66.62%. For 300 \(\times\) 300 instances, MO-FA provides the best behavior with up to 86.47%, followed by MO-VNS* and SPEA2 with up to 51.86% and 45.46%, respectively. For all instances, MO-FA is the best algorithm with up to 81.47%, followed by MO-VNS* and MO-ABC with up to 63.82% and 58.92%. From this set coverage analysis, we reach similar conclusions as for hypervolume. MO-FA provides the best behavior solving the problem in average term. If we focus on instance size, MO-VNS is the best algorithm for medium size instances and MO-FA is better suited for small and large ones. \begin{table}
\begin{tabular}{l||c|c|c|c|c} \hline \hline  & \multicolumn{5}{c}{Instance Size} \\ \cline{2-6}  & **50 \(\times\) 50** & **100 \(\times\) 100** & **200 \(\times\) 200** & **300 \(\times\) 300** & **All** \\ \hline
**NSGA-II** & 0.00\% & 2.35\% & 2.44\% & 2.27\% & 2.17\% \\
**SPEA2** & 0.00\% & 0.29\% & 1.11\% & 5.72\% & 2.52\% \\
**MO-VNS** & 6.31\% & 11.37\% & 10.03\% & 2.11\% & 7.16\% \\
**MO-VNS* & 4.85\% & 8.53\% & 10.03\% & 7.45\% & 8.40\% \\
**MO-ABC** & 9.71\% & 9.41\% & 8.55\% & 8.75\% & 8.88\% \\
**MO-FA** & 9.71\% & 9.51\% & 10.72\% & 13.71\% & 11.49\% \\
**MO-GSA** & 9.71\% & 5.00\% & 4.72\% & 5.83\% & 5.57\% \\
**MOEA/D** & 9.71\% & 3.53\% & 2.34\% & 4.16\% & 3.81\% \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Based on hypervolume, percentage of average test cases where a metaheuristic is significant better than any other solving the bi-objective RNPP.