## 7 Experimental Results
### Three-Objective Approach  
Table 6 shows average hypervolume for MO-VNS*, MO-VNS, MO-FA, MO-GSA, and MOEA/D, solving the three-objective RNPP for each test case and stop condition. In this table, hypervolumes in bold correspond to results never before published. Note that the hypervolumes for NSGA-II, SPEA2, and MO-ABC were not shown to simplify the table because they reported significantly lower hypervolumes than the other algorithms, instead, we refer readers to [47]. Analyzing Table 6, we verify that the algorithms show a homogeneous growth with an asymptotic trend in 400,000 evaluations, as for the bi-objective approach. Hence, the stop condition is representative to analyze the performance of the algorithms. Higher hypervolumes in Table 6 are shaded, reaching that some algorithms seem to outperform others. The differences observed are analyzed following the same methodology as for the bi-objective study. After removing possible outliers, we checked that data do not follow a normal distribution, and then we considered the same hypothesis as before for the Wilcoxon-Mann-Whitney's test. As a result, Table 7 shows the percentage of test cases where a metaheuristic is significantly better than any other according to the p-values obtained with a significance level of 0.05. Analyzing Table 7, for \(50\times 50\) instances, we check that MO-FA provides the best behavior with up to 13.36%, followed by MOEA/D and MO-ABC with up to 9.92% and 9.54%, respectively. For \(100\times 100\) instances, we check that MO-VNS provides the best behavior with up to 11.55%, followed by MO-VNS* and MO-FA with up to 10.92% and 8.61%, respectively. For \(200\times 200\) instances, we check that MO-FA provides the best behavior with up to 12.50%, followed by MOEA/D and SPEA2 with up to 11.00% and 7.00%, respectively. For \(300\times 300\) instances, we check that MO-FA provides the best behavior with up to 11.90%, followed by MOEA/D and MO-ABC with up to 11.17% and 8.46%, respectively. For all instances, MO-FA provides the best behavior with up to 11.62%, followed by MOEA/D and MO-VNS with up to 9.74% and 6.37%. From this analysis based on hypervolume, we check that MO-FA is the best algorithm in general. For small and large instances, MO-FA is also the best algorithm. For medium size instances, MO-VNS is the best algorithm. Table 8 shows the average set coverage metric of an algorithm compared to the others for the median Pareto front. Analyzing this table, for \(50\times 50\) instances, MO-GSA is the best algorithm with up to 81.78% followed by MO-FA and MO-VNS* with up to 72.00% and 67.86%, respectively. For \(100\times 100\) instances, MO-VNS* is the best algorithm with up to 73.19%, followed by MO-VNS and MO-FA with up to 72.36% and 65.68%, respectively. For \(200\times 200\) instances, MO-FA is the best algorithm with up to 83.71%, followed by MO-VNS* and MO-VNS with up to 49.37% and 48.77%, respectively. For \(300\times 300\) instances, MO-ABC is the best algorithm with up to 63.12%, followed by MO-FA and MO-VNS with up to 51.12% and 40.80%, respectively. For all instances, MO-FA is the best algorithm with up to 67.52% followed by MO-VNS and MO-VNS* with up to 50.80% and 49.61%, respectively. From this study, we check that MO-FA is the best algorithm on average term, followed by MO-VNS and MO-VNS*. Focusing on instance size, MO-GSA is the best algorithm for small instances, MO-VNS* is the best algorithm for medium size instances, and MO-FA is the best algorithm for large instances.