## 4 Application in the City of Dortmund
### Multi-Period Facility Location Problem  
The FLP is a well-known optimization challenge where the typical goal is to find the minimum costs and location of facilities that must be open to meet customer requirements, either deterministically [30] or stochastically [31; 49]. If routing decisions are also included, the FLP turns into the so-called location routing problem [50; 51]. In general, FLPs are classified either as capacitated or uncapacitated. The former refers to the case where the facilities have a known limit to the demand they can meet. The latter is the case where the service capacity of each facility exceeds the total demand of customers. Figure 7 illustrates the capacitated FLP (CFLP) for the APL network in the city of Dortmund. Here, each district is a potential APL location (yellow square) and each APL is connected to each other in the APLs configuration (dashed lines). These connections are used to calculate the distance matrix between districts.  
A multi-period CFLP is taken into account in our work. Decisions made in a given period affect future periods over a time horizon of \(T\). In particular, as demand is expected to increase in future periods, we assume that whenever an APL is opened within a period \(t\in T\), it must remain open until the end of the time horizon, i.e., for all \(t^{\prime}\in T:t^{\prime}>t\). Similarly, third-party logistics providers indicate that a minimum percentage of \(m\in(0,1)\) of total installed capacity must be used. Therefore, with the set \(I\) of nodes representing all districts in the city, each district \(i\in I\) could contain no, one, or more APLs, each with a known capacity \(a_{i}>0\). Similarly, each district \(j\in I\) has an aggregated demand in the period \(t\in T\), \(d_{ji}>0\). For two districts \(i,j\in I\), the unit costs of assigning an APL located in the district \(i\) to a customer located in the district \(j\) is \(c_{ij}>0\). Similarly, the costs of opening an APL in district \(i\in I\) during the period \(t\in T\) is indicated as \(f_{ti}>0\). In this context, the binary variable \(x_{ijt}\) takes the value 1 if customers in the district \(j\in I\) are assigned to an APL in the district \(i\in I\) during the period \(t\in T\); otherwise, the value is 0. Similarly, the integer variable \(y_{it}\) represents the number of APLs that are open in district \(i\in I\) and in period \(t\in T\). Then, our multi-period CFLP can be formulated as follows.  
\[\begin{array}{ll}Minimize&\sum_{i\in I}\sum_{j\in I}\sum_{t\in T}c_{ij}d_{ ji}x_{ijt}+\sum_{i\in I}\sum_{t\in T}f_{ti}(y_{it}-y_{it-1})\end{array} \tag{1}\]  
Figure 7: Illustrative CFLP for APLs in the city of Dortmund.  
subject to:  
\[\sum_{i\in I}x_{jit}=1\quad\forall j\in I,\forall t\in T \tag{2}\]  
\[y_{it}\geq y_{it-1}\quad\forall i\in I,\forall t\in T\backslash\{1\} \tag{3}\]  
\[\sum_{j\in I}d_{ji}x_{jit}\leq a_{i}y_{it}\quad\forall i\in I,\forall t\in T \tag{4}\]  
\[\sum_{j\in I}d_{jt}\geq m\sum_{i\in I}a_{i}y_{it}\quad\forall t\in T \tag{5}\]  
\[x_{jit}\in\{0,1\}\quad\forall i\in I,\forall j\in I,\forall t\in T \tag{6}\]  
\[y_{it}\in\mathbb{Z}^{+}\quad\forall i\in I,\forall t\in T \tag{7}\]  
The expression (1) indicates the objective function that minimizes the total costs: The first term indicates the service costs of APLs, while the second represents the fixed costs of opening new APLs in the time horizon. Constraints (2) ensure that for each period \(t\in T\) and each district \(j\in I\) exactly one APL is assigned. Restrictions (3) ensure that once an APL is opened, it remains open until the end of the time horizon. Constraints (4) ensure that for each APL in district \(i\in I\) and time period \(t\in T\), the demand served by that APL does not exceed its capacity. Constraints (5) guarantee a minimum utilization percentage of the total installed capacity of APLs for each \(t\in T\) period. Finally, constraints (6) and (7) specify the ranges of the decision variables.