## 6 Conclusions  
With the goal of determining the optimal number and location of automated parcel locker (APL) systems in a multi-period time horizon, this paper has proposed the use of an integrated simulation-optimization approach combining system dynamics with exact optimization and Monte Carlo simulation. We propose this integrated model as a decision support tool for future APL implementations as a last-mile distribution scheme. The analysis is based on a real-world case study where service requirements are considered as random variables that evolve over time. First, a system dynamics simulation model is  
Figure 10: Optimal solutions evaluated in terms of costs and reliability.  
designed to determine the 36-month performance of parameters such as APL users and number of deliveries. Then, these results feed a multi-period facility location model that provides the optimal number of APLs. To deal with the demand uncertainty, different scenarios are considered and solved with precise methods. The solutions associated with each scenario are then sent to a Monte Carlo simulation to estimate both their costs and reliability level.  
The model provides an optimal number of APLs, taking into account the expectations of user demands. We have considered three scenarios \(S1\), \(S2\), and \(S3\) for 50%, 60%, and 70% of the e-shopper rate. The results for the number of deliveries (units) after 36 months show a wide range of shipments from about 277,000 in \(S1\) to nearly 400,000 in \(S3\). We used our CFLP to evaluate ten APL network configurations (\(k=1,...,10\)) with increasing demand in relation to each scenario. Obviously, there is a strong impact on the number of APLs that the city needs. After 36 months, the number of APLs increases from 99 in the case of the lowest demand to 165 at maximum demand. Interestingly, the number of APLs stabilizes from month 24 in all configurations. Thus, we can conclude that the effect on APLs appears linear in relation to the potential users of APL with no obvious scale effects. From a stochastic environment, we assumed that the demand per district is uncertain and follows a known probability distribution. Whenever the demand follows either a uniform or a triangular distribution, the most expensive configurations always reach a reliability level of 100.0% regardless of the scenario. The configuration \(k=6\), with total costs of 748,660E, already locates a suitable number of APLs. However, if the budget is lower, our approach offers other alternatives for decision makers.  
All in all, the work illustrates the potential of combining different simulation and optimization techniques to correctly address complex optimization problems in real urban logistics, where uncertainties must also be taken into account. The following research lines are still open for the future: (i) increasing the level of detail on the demand side, taking into account correlated and individual customers' demands instead of aggregated ones--which will significantly increase the size of the problem; (ii) develop a metaheuristic-based approach for the optimization phase, as this will be a necessary step when larger instances are to be analyzed; and (iii) extend the approach to a fully simheuristic algorithm, so that the feedback provided by the Monte Carlo simulation can be reused to guide the metaheuristic search.  
Conceptualization, M.R., J.G.-F., and J.C.-V.; methodology, J.C.-V. and R.D.T.; software, J.C.-V. and R.D.T.; validation, J.C.-V. and R.D.T.; formal analysis, M.R. and J.G.-F.; writing--original draft preparation, J.C.-V. and R.D.T.; writing--review and editing, M.R. and [PERSON]. All authors have read and agreed to the published version of the manuscript.  
This work has been partially supported by the German Academic Exchange Service (DAAD) Research Grants--Doctoral Programmes in Germany, 2017/18.  
Not applicable.  
Data Availability Statement: Data are available upon reasonable request to the corresponding author.  
The authors declare no conflict of interest.