## 5 Computational Results
### Computational Results of Metaheuristics  
In this section, the performance of VBIH algorithm is compared to the best-performing algorithms, IGRS and IGAL, from the literature. All algorithms are run five replications to solve the large VRF instances. In Table 6, we present average, minimum and maximum ARPD values for the CPU time limit \(T_{max}=15\times n\times m\) milliseconds. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Instance**} & \multicolumn{2}{c}{**NEH**} & \multicolumn{2}{c}{**NEH \({}^{*}\)**} & \multicolumn{2}{c}{**FRB5**} \\ \cline{2-7}  & **ARPD** & **CPU(s)** & **ARPD** & **CPU(s)** & **ARPD** & **CPU(s)** \\ \hline \(100\times 20\) & 5.82 & 0.00 & 5.82 & 0.01 & 2.45 & 0.10 \\ \(100\times  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**IGRS**} & \multicolumn{3}{c}{**IGALL**} & \multicolumn{3}{c}{**VBIH**} \\ \cline{2-10}
**Instance** & **Avg. ** & **Min** & **Max** & **Avg. ** & **Min** & **Max** & **Avg. ** & **Min** & **Max** \\ \hline \(100\times 20\) & 0.45 & 0.13 & 0.74 & 0.12 & \(-0.07\) & 0.33 & 0.00 & \(-0.2  
As seen in Table 6, VBIH generated better Avg, Min and Max RPD values on the overall average. On overall average, it was able to further improve the upper bounds up to \(-0.05\%\); its best overall performance was \(-0.18\%\) indicating that \(0.18\%\) of \(240\) instances are further improved and its worst-case performance was \(0.08\%\). In order to see if differences in ARPDs are statistically significant, we provide the \(95\%\) confidence interval plot of algorithms in Figure 7, where we can observe that differences in ARPD values are statistically significant on the behalf of VBIH against \(\mathrm{IG_{RS}}\) and \(\mathrm{IG_{ALL}}\) algorithms because their confidence intervals do not coincide. Computational results for Avg, Min and Max ARPD values with the CPU time limit \(T_{max}=30\times n\times m\) milliseconds are given in Table 7. As seen in Table 7, VBIH was able to generate better Avg, Min and Max ARPD values on the overall average. On overall average, it was able to further improve the upper bounds by \(-0.11\%\) in Avg value, \(-0.24\%\) of upper bounds are further improved on Min value and its worst-case performance was \(0.02\%\). However, as CPU times increased, the performance of \(\mathrm{IG_{ALL}}\) algorithm was also remarkable. Briefly, both VBIH and \(\mathrm{IG_{ALL}}\) outperformed \(\mathrm{IG_{RS}}\) in almost each problem set. In order to see if these results are statistically significant, we provide the \(95\%\) confidence interval plot of algorithms in Figure 8, where we can observe that differences in ARPD values are statistically significant on the behalf of both VBIH and \(\mathrm{IG_{ALL}}\) algorithms against \(\mathrm{IG_{RS}}\) algorithm because their confidence intervals do not coincide with \(\mathrm{IG_{RS}}\). In other words, VBIH and \(\mathrm{IG_{ALL}}\) algorithms were statistically equivalent but significantly superior to \(\mathrm{IG_{RS}}\). Figure 7: Interval plot at the \(95\%\) confidence level for large VRF instances. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**IGRS**} & \multicolumn{3}{c}{**IGALL**} & \multicolumn{3}{c}{**VBIH**} \\ \cline{2-10} \(n\times m\) & **Avg. ** & **Min** & **Max** & **Avg. ** & **Min** & **Max** & **Avg. ** & **Min** & **Max** \\ \hline \(100\times 20\) & 0.25 & \(-0.02\) & 0.54 & 0.03 & \(-0.11\) & 0.16 & \(-0.05\) & \(-  
Computational results for average, minimum and maximum RPD values with the CPU time limit \(T_{max}=45\times n\times m\) milliseconds are given in Table 8, where VBIH outperformed \(\text{IG}_{\text{RS}}\) and \(\text{IG}_{\text{ALL}}\) algorithms with respect to average, minimum and maximum RPD values on the overall average. On overall average, it was able to further improve the upper bounds by \(-0.25\%\) on the average value, \(-0.36\%\) on the minimum value, and its worst-case performance was \(-0.13\%\).