## 5 Results and Discussion
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{\(I\)} & \multirow{2}{*}{\(|R|\)} & \multirow{2}{*}{\(|C|\)} & \multicolumn{2}{c}{**1**} & \multicolumn{2}{c}{**2**} & \multicolumn{2}{c}{**3**} &  
By analyzing Tables 3-5, we can observe that our MSBRLH algorithm is able to improve previous results (from LH) by 9-12%, on average (column _gap (2)-(1)_). When comparing the MSBRLH with the AH heuristic (column _gap (3)-(2)_), our approach is able to reduce solution costs by up to 26% in short computational times (about 18 s on average). On the other hand, when comparing our results with those generated by the MAC approach (column _gap (4)-(2)_), our solutions are between 9% and 21% worse, on average. Particularly, in the abundant inventory scenario, MSBRLH's results are only 9% worse than MAC. Notice, however, that the processing time required by MAC is substantially larger in all inventory scenarios. By analyzing Figure 4, it is evident that MSBRLH performs better in the abundant inventory scenario, being able to find one better solution and several others with a maximum gap of 8%. Moreover, we can observe a variability of around 10% in the gap between MSBRLH and MAC on average. This variability is reduced to around 8% for the relaxed and abundant scenarios. These results demonstrate the robustness of our solution approach for the deterministic case. When analyzing Figure 5, which presents the overall performance of each solution approach for each inventory scenario, we can observe that our multi-start strategy is more efficient than both LH and AH  
Figure 5: Comparison of the solutions cost (OF value) from each solving approach, for each inventory scenario. heuristics, by generating solutions with a lower cost. To complement these box-plots, an ANOVA test was run for each inventory scenario. The \(p\)-values associated with the tight, relaxed, and abundant inventory scenarios were, respectively, of 0.001, 0.000, and 0.000. Also, the [PERSON]'s LSD test suggests significant differences in all tree scenarios between MAC and AH, between MS-BRLH and AH, as well as between MAC and LH. However, cost differences between MAC and MS-BRLH were not significant in any scenario, despite the fact that MAC employed a noticeably higher amount of computing time than MS-BRLH. Next, in Figure 6, we present the convergence of three problem instances' solutions, including one from each different inventory scenario (instances \(b\theta\), \(b26\), and \(b46\)), by comparing the MS\({}_{\text{BRLH}}\) with the best-known solutions, in terms of gap. As introduced, these instances require approximately 15 s of processing time, given their magnitude. As we can see in Figure 6, the instances demonstrate the same convergence behavior as solution time increases. It is noticeable that the convergence rate is abrupt during the first few seconds. However, contrary to the tight and relaxed inventory scenarios, where the solutions continue improving over time, the search demonstrates more stable convergence during the remaining execution time in the abundant scenario. Being more efficient in the more flexible inventory case, our multi-start approach quickly achieves its best solutions. Next, we want to compare the quality of the solutions generated for the deterministic scenario (MS\({}_{\text{BRLH}}\)) against the solutions generated for the stochastic scenario (Sim\({}_{\text{BRLH}}\)). Since the only difference between the deterministic and stochastic scenarios is that stochastic delays are added to edge traversal times, we can consider the deterministic cost of the best solutions generated by MS\({}_{\text{BRLH}}\) as a lower bound (LB) of the stochastic travel times of the best Sim\({}_{\text{BRLH}}\) solution. Moreover, since MS\({}_{\text{BRLH}}\) does not account for stochastic travel times, we can consider the stochastic travel time of the best MS\({}_{\text{BRLH}}\) solution as an upper bound (UB) for the stochastic travel time of the best Sim\({}_{\text{BRLH}}\) solutions.