## 6 Numerical Results
### Performance of PBA vs. MILP
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \multicolumn{2}{c}{**Scenario**} & \multicolumn{4}{c}{**MILP**} & \multicolumn{4}{c}{**PBA**} & \multicolumn{4}{c}{**Relative Diff.**} \\ \hline \(\mu\) & \(R\) & \(\text{cols}^{\text{MILP}}\) & \(\boldsymbol{z}^{\text{MILP}}\) & \(\boldsymbol{\Delta}^{\text{MILP}}\) & \(\boldsymbol{T}^{\text{MILP}}\) & \(\text{cols}^{\text{PBA}}\) & \(\boldsymbol{z}^{\text{init}}\) & \(\boldsymbol{z}^{\text{PBA}}\) & \(\boldsymbol{T}^{\text{CG}}\) & \(\boldsymbol{T}^{\text{BB}}\) & \(\boldsymbol{T}^{\text{PBA}}\) & \(\boldsymbol{\Delta}^{\text{z}}\) & \(\boldsymbol{\Delta}^{\text{T}}\) \\ \hline
1 & 10 & 40,200 & 4 & 0\% & 4 & 1012 & 8 & 4 & 5 & 0.2 & 5 & 0\% & 0.7 \\  & 30 & 120,549 & 8 & 0\% & 32 & 2959 & 14 & 8 & 34 & 2 & 36 & 0\% & 0.9 \\  & 50 & 201,185 & 10 & 0\% & 216 & 5844 & 18 & 10 & 236 & 19 & 255 & 0\% & 0.8 \\  & 70 & 281,593 & 12 & 0\% & 1110 & 9413 & 20 & 12 & 984 & 15 & 999 & 0\% & 1.1 \\ \hline
2 & 10 & 40,752 & 4 & 0\% & 23 & 1141 & 8 & 4 & 6 & 0.6 & 7 & 0\% & 3.4 \\  & 30 & 122,144 & 6 & 0\% & 484 & 3136 & 14 & 6 & 31 & 3 & 34 & 0\% & 14.4 \\  & 50 & 204,028 & 13 & 46\% & 3600 & 6962 & 18 & 8 & 427 & 39 & 466 & 38\% & 7.7 \\  & 70 & 285,518 & 16 & 63\% & 3600 & 10,753 & 20 & 9 & 876 & 39 & 915 & 44\% & 3.9 \\ \hline
3 & 10 & 41,190 & 4 & 0\% & 71 & 1185 & 8 & 4 & 4 & 1 & 5 & 0\% & 14.2 \\  & 30 & 123,336 & 5 & 0\% & 2022 & 3226 & 14 & 5 & 37 & 4 & 41 & 0\% & 49.8 \\  & 50 & 206,012 & 14 & 63\% & 3600 & 7291 & 18 & 7 & 507 & 107 & 614 & 50\% & 5.9 \\  & 70 & 288,270 & 18 & 67\% & 3600 & 11,792 & 20 & 8 & 1486 & 192 & 1678 & 56\% & 2.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Comparison of the methods MILP and PBA in network MESH-20 assuming \(k=5\) and \(\gamma=0.2\); \(T\) in seconds. Figure 6 shows that MILP may achieve slightly better results than the PBA in smaller network scenarios (for \(R\leq 40\) in MESH-20 and \(R\leq 30\) MESH-38). Nonetheless, the PBA provides much better optimization results for larger problem instances, which is due to the complexity of solving MILP, also expressed by high optimality gaps on the level of some tens of a percent (see Figure 6). As shown in Figure 8, the relative difference in \(z\) increases with \(R\), up to 30% in MESH-20 and 45% in MESH-38 for \(R=70\). At the same time, the PBA runtimes, presented in Figure 7, are much shorter than in the MILP optimization approach. Figure 8: Relative difference in the MILP and PBA objective function values and computation times in networks MESH-20 and MESH-38 assuming a 1-h runtime limit. Figure 6: Comparison of the objective function values obtained using MILP and the PBA (lines) and the MILP optimality gaps (bars) in networks MESH-20 and MESH-38. Figure 7: Comparison of the MILP and PBA average computation times in networks MESH-20 and MESH-38 assuming a 1-h runtime limit. Here, \(\Delta^{\rm T}\) first increases, up to some tens of percentand subsequently decreases for larger \(R\), mainly due to the 1-h runtime limit reached in the MILP method. In summary, the presented results indicate a limited scalability of the MILP approach. Concurrently, the PBA significantly reduces the complexity of solving the MILP model, at the same time providing good-quality results in much shorter times, particularly in large network scenarios.