## 7 Experimental Results
### Bi-Objective Approach  
Table 3 shows average hypervolume for MO-VNS*, MO-ABC, MO-VNS, MO-FA, MO-GSA, and MOEA/D, solving the bi-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 and SPEA2 were not shown to simplify the table because they reported significantly lower hypervolumes than the other algorithms, instead, we refer readers to [46]. If we analyze how hypervolumes change over stop conditions, we check that most algorithms show an homogeneous growth, reaching an asymptotic trend for 400,000 evaluations. That means that the set of stop conditions selected is representative to study convergence. On the other hand, if we analyze the tables focusing on shaded cells showing higher (better) hypervolumes, we note that some algorithms seem to outperform others. To check if the differences are significant, we analyze the data using the statistical methodology as follows. First, we remove possible outliers from the hypervolume distributions. Next, we analyze if data follow a normal distribution or not through [PERSON]'s and [PERSON]'s tests with hypothesis \(H_{0}\): data follow a normal distribution and \(H_{1}\): otherwise. As both tests provided p-values lower than 0.05 for all cases, we should consider a non-parametric test to compare the algorithms two by two. Specifically, as samples are independent, we consider [PERSON]'s test with hypothesis \(H_{0}\): \(\overline{Hyp_{i}}\leq\overline{Hyp_{j}}\), \(\forall i,j\in\{1=\text{NSGA},\ 2=\text{SPEA},\ 3=\text{MO-VNS},\ 4=\text{MO-VNS}^{*},\ 5=\text{MO-ABC},\ 6=\text{MO-FA},\ 7=\text{MO-GSA},\ 8=\text{MOEA}/\text{D}\}\), \(i\
eq j\) and \(H_{1}\): \(\overline{Hyp_{i}}>\overline{Hyp_{j}}\). Table 4 shows the percentage of test cases where a metaheuristic is significantly better than any other based on the p-values obtained before analyzed with a significance level of 0.05 and according to instance size. \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Parameter** & **2 Obj** & **3 Obj** & **Range** \\ \hline \multicolumn{4}{l}{**MO-FA**} \\ \(ps_{f}\) & **100** & **100** & [25,50,\(\ldots\),300] \\ \(r_{f}\) & **0.50** & **0.85** & [0.05,0.1,0.15,\(\ldots\)0.95] \\ \(\beta_{0_{f}}\) & **0.75** & **0.70** & [0.05,0.1,0.15,\(\ldots\)0.95] \\ \(\gamma_{f}\) & **0.05** & **0.60** & [0.05,0.1,0.15,\(\ldots\)0.95] \\ \(mut_{f}\) & **0.60** & **0.10** & [0.05,0.1,0.15,\(\ldots\)0.95] \\ \(when\_sc_{f}\) & **0.30** & **0.25** & [0.05,0.1,0.15,\(\ldots\)0.95] \\ \hline \multicolumn{4}{l}{**MO-GSA**} \\ \(ps_{ga}\) & **100** & **25** & [25,50,\(\ldots\),300] \\ \(mutqa\) & **0.40** & **0.20** & [0.05,0.1,0.15,\(\ldots\)0.95] \\ \(when\_sc_{ga}\) & **0.05** & **0.05** & [0.05,0.1,0.15,\(\ldots\)0.95] \\ \hline \multicolumn{4}{l}{**MOEA/D**} \\ \(CHIMinc_{m}\) & **1.30** & **1.30** & [1.00,1.05,\(\ldots\)2.00] \\ \(crow_{m}\) & **0.015** & **0.015** & [0.010,0.015,\(\ldots\)0.050] \\ \(neigh_{m}\) & **0.55** & **0.05** & [0.05,0.10,\(\ldots\)0.95] \\ \(crow_{m}\) & **0.15** & **0.15** & [0.05,0.10,\(\ldots\)0.95] \\ \(mut_{m}\) & **0.25** & **0.25** & [0.05,0.10,\(\ldots\)0.95] \\ \hline \hline \end{tabular}
\end{table} 
Table 2: _Cont._  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{MO-VNS*(Hpy)} & \multicolumn{4}{c}{MO-ABC(Hpy)} \\ \cline{2-11}  & \multicolumn{4}{c}{Evaluations (Stop Condition)} & \multicolumn{4}{c}{Evaluations (Stop Condition)} & \multicolumn{4}{c}{Evaluations (Stop Condition)} \\ \cline{2-11}  & **50.000** & **100.000** & **200.000** & **300.000** & **400.000** & **50.000** & 100.000 & 200.000 & 300.000 & 400.000 \\ \hline  
Analyzing Table 4, for 50 \(\times\) 50 instances, we check that MO-ABC, MO-FA, MO-GSA, and MOEA/D provide a similar behavior with up to 9.71%, followed by MO-VNS and MO-VNS* with up to 6.31% and 4.85%, respectively.