## 4 Results of Experiments
### Brandimarte Dataset  
For the rest of the experiments, the relative percentage deviation (\(RPD\)) and [PERSON]'s non-parametric test were employed to compare the GA-RRHC with the other methods [54]. The \(RPD\) is defined in Equation (7); \(BOV\) is the best-obtained makespan by each algorithm, and \(BKV\) is the best-known makespan for each problem.  
\[RPD=\frac{BOV-BKV}{BOV}\times 100 \tag{7}\]  
Table 5 provides the results of the GA-RRHC compared with the other methods for the BRdata dataset. In each case, the smallest obtained makespan is marked with *.  
\begin{table}
\begin{tabular}{l l l l l l l l l l} \hline \hline
**Instance** & \(\mathbf{n}\times\mathbf{m}\) & \(\mathbf{\beta}\) & **BKV** & \begin{tabular}{c} **BSO-** \\ **LAHC** \\ \end{tabular} & \begin{tabular}{c} **GA-** \\ **RRHC** \\ \end{tabular} & \begin{tabular}{c} **GLNSA** \\ **GRASP** \\ \end{tabular} & \begin{tabular}{c} **HA** \\ **HA** \\ \end{tabular} & \
\begin{tabular}{c} **IJA** \\ \end{tabular} &
\begin{tabular}{c} **TIPSO** \\ \end{tabular} \\ \hline MK01 & \(10\times 6\) & 0.2 & 36 & 40 * & 40 * & 40 * & 40 * & 40 * & 40 * & 40 * & 40 * \\ MK02 & \(10\times 6\) & 0.35 & 24 & 26 * & 26 * & 26 * & 26 * & 26 * & 27 & 26 * \\ MK03 & \(15\times 8\) & 0.3 & 204 & 204 * & 204 * & 204 * & 204 * & 204 * & 204 * & 204 * \\ MK04 & \(15\times 8\) & 0.2 & 48 & 60 * & 60 * & 60 * & 60 * & 60 * & 60 * & 60 * \\ MK05 & \(15\times 4\) & 0.15 & 168 & 173 & 172 * & 173 & 172 * & 172 * & 172 * & 173 \\ MK06 & \(10\times 15\) & 0.3 & 33 & 61 & 58 & 58 & 64 & 57 * & 57 * & 60 \\ MK07 & \(20\times 5\) & 0.3 & 133 & 141 & 139 * & 139 * & 139 * & 139 * & 139 * \\ MK08 & \(20\times 10\) & 0.15 & 523 & 523 * & 523 * & 523 * & 523 * & 523 * & 523 * \\ MK09 & \(20\times 10\) & 0.3 & 299 & 307 * & 307 * & 307 * & 307 * & 307 * & 307 * \\ MK10 & \(20\times 15\) & 0.2 & 165 & 204 & 198 & 205 & 205 & 197 * & 197 * & 205 \\ \hline \hline \end{tabular}
\end{table}
Table 5: BRdata dataset results.  
Figure 13: Number of best results per algorithm for the Kacem dataset.  
It can be seen in Table 5 that the GA-RRHC calculates the least makespan in eight cases, behind the HA and IJA, and with the same optimal results as the GRASP. Figure 14 shows the number of best results obtained for each algorithm in this dataset.  
Table 6 shows the ranking of each method based on the average \(RPD\), as well as the value \(p\) of the non-parametric Friedman pairwise test, comparing the GA-RRHC with each of the other algorithms. The GA-RRHC secured the second place among the compared algorithms, obtaining a value of \(p<0.05\) with the BSO-LAHC and GRASP.  
Figure 15 gives the signed difference between the average RPD of the GA-RRHC against the other methods in pairs. A negative difference means an inferior performance in contrast to the GA-RRHC.  
\begin{table}
\begin{tabular}{l c c c c c c c} \hline \hline
**Algorithm:** & **BSO-LAHC** & **GA-RRHC** & **GLNSA** & **GRASP** & **HA** & **IJA** & **TIPSO** \\ \hline Average \(RPD\): & 11.3881 & 10.6710 & 11.0121 & 11.4890 & 10.5289 & 11.2665 & 11.2017 \\ Rank: & 6 & 2 & 3 & 7 & 1 & 5 & 4 \\ \(p\)-value: & 0.0455 & - & 0.1573 & 0.0435 & 0.1643 & 0.0803 & 0.0833 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Algorithm ranking and Friedman test value for BRdata dataset.  
Figure 14: Number of best results per algorithm for the BRdata dataset.  
This analysis indicates that the GA-RRHC is statistically competitive with the other four best algorithms for optimizing the BRdata dataset.