## 5 Computational Results and Discussion
### Generating and Simulating Optimal Configurations  
As soon as our SDSM provides the number of deliveries (expected demand) for the three scenarios (\(s\in S\), where \(S=\{S1,S2,S3\}\)) under consideration of Table 2, are used to feed our CFLP model. We evaluate ten APL network configurations (\(k\in K\), where \(K=\{1,2,...,10\}\)) with the demand increasing proportional to \(k\), based on the scenario S2. Each configuration is obtained by optimally solving the CFLP model using the procedure described below. 1. Consider a uniformly distributed random demand \(D_{jtk}\) per district \(j\in J\) during the period \(t\in T\) for generating the configurations. 2. Define \(\mu_{jt}=E[D_{jtk}]\) and assume that \(\mu_{jt}\) is the medium demand corresponding to the scenario S2. 3. Define a factor \(\delta=0.01\) to increase the size of the uniform interval as we move forward into future periods. 4. Generate the random demand using Equation (8). The expression \(1+\frac{k-1}{|K|-1}\) is useful to increase \(\mu_{jt}\) proportionally to the value of \(k\). In this way, we guarantee that generated configurations differ in size. \[D_{jtk}\sim U\bigg{(}\bigg{[}1+\frac{k-1}{|K|-1}\bigg{]}(1-\delta t)\mu_{jt}, \bigg{[}1+\frac{k-1}{|K|-1}\bigg{]}(1+\delta t)\mu_{jt}\bigg{)}\quad\forall j \in I,\forall t\in T,\forall k\in K\] (8)  
The variable costs \(c_{ij}\) are proportional to the distance between each pair of districts. They were estimated using a web mapping service. The fixed costs are \(f_{it}=5500\)% for the first year and each district, and increase according to an average inflation rate of 2% per year. The capacity of each APL in a district \(i\in I\) is \(a_{i}=6000\) units per month, and the minimum utilization percentage is \(m=40\)%. Then, our CFLP model is solved with Cplex for all ten configurations. The number of resulting open APLs per month is shown in Figure 9 for three out of these configurations. The lowest and highest lines represent solutions for the lowest and highest demand, respectively. The rest of the solutions are in between. As the demand \(\mu_{jt}\) increases over time, the number of open APLs will behave the same regardless of the configuration. However, this consistent behavior does not extend beyond the year 1 for \(k=10\) and beyond the year 2 for \(k=1\) and \(k=5\), when the total installed APLs are sufficient to cover the total demand by the end of the planning horizon. Furthermore, there is a sharp increase in open APLs from months 11 to 12. This behavior is caused by two parameters: The annual growth of the fixed costs \(f_{it}\) drives the APLs that are opened when they are less expensive, but always limited by the minimum utilization percentage \(m\). Finally, the total number of APLs installed varies significantly from one scenario to another, for example, while 165 APLs are required for \(k=10\), only 99 APLs are installed in the configuration \(k=1\) at the end of the planning horizon. All configuration results are stored in Tables A4-A6 in Appendix B.  
Figure 8: Scenario comparison: APL users (**left**) and number of deliveries (right). Once all configurations have been generated, they are tested in a stochastic environment, assuming that the demand per district is uncertain and follows a known probability distribution. Consider a random demand \(D_{\mathit{jls}}\) whose mean and standard deviation are \(\mu_{\mathit{jls}}\) and \(\sigma_{\mathit{jls}}\), respectively, per district \(j\in J\) during the period \(t\in T\) for the scenario \(s\in S\). We assume that \(\mu_{\mathit{jls}}\) is the demand generated by our SDSM. Now, as the goal is to evaluate the performance of each configuration, they must be tested under the same demand conditions; therefore, the demand does not depend further on the configuration. Then, \(D_{\mathit{jls}}\) is simulated and each configuration is evaluated in terms of total costs (Equation (1)) and reliability. Studies on reliability in supply chains are found in [52; 53].