## 5 Computational Results and Discussion
### Generating and Simulating Optimal Configurations
We define the reliability \(R_{\mathit{ks}}\) of the configuration \(k\in K\) for the scenario \(s\in S\) as the probability that the stochastic demand of all districts in the city can be successfully satisfied, i.e.,  
\[R_{\mathit{ks}}=\left(1-\frac{b_{\mathit{ks}}}{n}\right)\cdot 100\%\quad \forall k\in K,\forall s\in S \tag{9}\]  
where \(b_{\mathit{ks}}\) is the total number of simulation runs where the configuration does not cover all district demands, and \(n\) is the total number of runs. In other words, if at least one APL in a configuration is not able to cover all assigned needs, that configuration will fail. In our experiments, a total of \(n=5000\) runs are performed for each combination of scenario \(s\) and configuration \(k\). Without losing generality, we assume that demand is independent of the customers' district, but our methodology can easily be adapted to take into account correlated demand. For the realization of the demand, three probability distributions have been tested:  
1. A uniform distribution, according to Equation (10). In this case, \(\sigma_{\mathit{jls}}=\frac{\sqrt{3}}{3}\delta t\mu_{\mathit{jls}}\). \[D_{\mathit{jls}}\sim U([1-\delta t]\mu_{\mathit{jls}},[1+\delta t]\mu_{ \mathit{jls}})\] (10)
2. A symmetric triangular distribution, according to Equation (11), i.e., the mode equals \(\mu_{\mathit{jls}}\). To obtain conditions similar to 1, the lower and upper limits of this distribution are calculated assuming that the standard deviation is equal. \[D_{\mathit{jls}}\sim T\Big{(}\Big{[}1-\sqrt{2}\delta t\Big{]}\mu_{\mathit{jls }},\mu_{\mathit{jls}}\Big{[}1+\sqrt{2}\delta t\Big{]}\mu_{\mathit{jls}}\Big{)}\] (11)
3. A lognormal distribution, according to Equation (12). Again, the standard deviation is the same as in the point 1 to preserve similar conditions. \[D_{\mathit{jls}}\sim\mathit{Lognormal}\big{(}\mu_{\mathit{jls}},\sigma_{ \mathit{jls}}\big{)}\] (12)  
Figure 9: Number of total open APLs along the planning horizon for three configurations (\(k=1,5,10\)). Figure 10 shows the main results of the simulation process for each configuration. Blue, orange, and green lines represent the results from the demand for uniform, triangular, or log-normal distribution. In addition, dotted, solid, and dashed lines represent the results for the scenarios \(S1\) (low demand), \(S2\) (medium demand), and \(S3\) (high demand), respectively. Each dot on each line represents a single configuration. In general, more expensive configurations result in higher reliability, because they include a larger number of installed APLs. When the demand follows either a uniform or a triangular distribution, the most expensive half of the configurations always achieve a 100.0% reliability level, regardless of the scenario. In other words, the configuration \(k=6\), with total costs of 748,660e, already locates a suitable number of APLs and eliminates the need to consider more expensive configurations. However, if the budget is lower, our approach offers other good alternatives for the decision makers. In general, configurations are less reliable when demand scenarios are increased. For example, configuration \(k=4\), with total costs of 661,100e, only achieves a reliability level of 14.0% under the high demand scenario and a log-normal distribution. Conversely, this configuration achieves a reliability level of 98.8% under the low demand scenario. Furthermore, the reliability is very sensitive to the probability distribution. Broadly speaking, a configuration fails if the demand is too high (Equation (9)). Therefore, configurations simulating a log-normal demand, which has no upper limit, are less reliable than those where the probability distribution is either uniform or triangular (Equations (10) and (11)). This fact underlines the relevance of integrating the study to determine the behavior of demand in the real case.