## 5 Results and Discussion  
To test the proposed methodologies, we have used the 60 problem instances introduced in [PERSON] et al. (2016). These instances differ in the number of retail centers (\(r\in\{10,15,20,25\}\)), online customers (\(c\in\{20,50,75,100,150\}\)), and also in the inventory scenarios of the retail centers (tight, relaxed, and abundant). Therefore, for each inventory scenario, 20 different problem combinations are generated. The maximum tour length of the routes and the vehicle capacity are set to 8 h and 100 weight units, respectively. While the BRLH is guided by a single parameter, \(\beta\), the simheuristic approach is also controlled by the maximum running time \(time_{max}\), the acceptance margin of worst solutions \(m\), and the number, \(q_{short}\) and \(q_{long}\), of simulation repeats in short and long simulation runs, respectively. The stochastic travel times for each edge are set by the log-normal distribution parameters, \(\mu\) and \(\sigma\). Table 2 summarizes the setup of the parameters employed during the computational experiments. For calibrating these parameters, we have used the methodology proposed in [PERSON] et al. (2016), which is based on a general and automated statistical learning procedure. Regarding the maximum run time \(time_{max}\), the value was set depending on the size of the instance \((c+r)\times 0.342\), which leads to a maximum execution time of 60 s in the case of the largest instance. This stopping criterion is employed in both the multi-start and simheuristic strategies. The algorithms were coded in Java and all tests were performed on an Intel Core i7-8550U processor with 16 GB of RAM. Note that three different values have been considered for the \(\sigma\) parameter, which is used to modify the deterministic travel times. Since the maximum tour length is fixed independently of the size of the instances, small-sized instances are more likely to generate short routes. Therefore, a larger value for \(\sigma\) introduces more variability in the travel times, which increases route failure rates. On the other hand, large-sized instances are often composed of larger routes, then a small value for \(\sigma\) is introduced. In particular we have, \(\sigma=2.5\) for small instances (composed of 25 customers), \(\sigma=1.55\) for large instances (composed of 150 customers), and \(\sigma=1.9\) for the remaining medium-sized ones. This approach ensures that small, medium, and large instances each have similar levels of difficulty with respect to the risk of route failure. The initial analysis aims to compare the solutions generated by our greedy heuristic (LH) and by our MSBRLH-in which \(\beta\) is (uniformly) randomly selected in the interval \([0.45,0.75]\)-with the solutions obtained by the two-phase heuristic (AH) and multi-ant colony metaheuristic (MAC) proposed by [PERSON] et al. (2016). Their methodologies were performed on four 2.1 GHz processors with 16-cores each and a total of 256 GB RAM. That is, we initially focus on comparing each algorithm in terms of deterministic travel cost. Tables 3-5 present the results obtained for tight, relaxed, and abundant inventory scenarios, respectively. For each problem instance (I), we present results for: the cost of the best-found solution obtained by the different methodologies; the average cost of our MSBRLH; the CPU time (in seconds) required by each methodology; and their percentage gaps. The best results returned by the solution methodologies are highlighted in bold. Figures 4 and 5 present how both the gap and the cost of the solutions, i.e., the objective function (OF) value, behave according to the employed solution approach and inventory scenario, respectively. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(\beta\) & \(m\) & \(\mu\) & \(\sigma\) & \(q_{short}\) & \(q_{long}\) & \(time_{max}\) \\ \hline \([0.45,0.75]\) & 20\% & 0 & \(\{1.55,1.9,2.5\}\) & 100 & 1000 & \((r+c)\times 0.342\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameter setup.