## 3 Materials and Methods
### Mathematical Model Formulation
\begin{table}
\begin{tabular}{c c} \hline \hline
**Notation** & **Description** \\ \hline Parameters & \\ \(q_{r}\) & quantity of baggage associated to request \(r\in R\) \\ \(c_{ijk}\) & traveling distance between the arc \((i,j)\in A\) using the vehicle \(k\in K\) \\ \(\tau_{ij}\) & traveling time between the arc \((i,j)\in A\) \\ \(E_{i}\) & earliest time that the request \(r\in R\) can be picked up at the location \(i\in N\) \\ \(L_{i}\) & latest time that the request \(r\in R\) can be delivered at the location \(i\in N\) \\ Variables & 1 if vehicle \(k\in K\) travels through the arc \((i,j)\in A\), \\ \(x_{ijk}\) & 0 otherwise. \\ \(y_{ijkr}\) & 1 if request \(r\in R\) is transported by vehicle \(k\in K\) through the arc \((i,j)\in A\), 0 otherwise. \\  & auxiliary variable used for sub-tour elimination, \\ \(u_{i}\) & indicating the rank order in which \\  & node \(n\in N\) is visited (see [PERSON] et al. [23]) \\ \(a_{ik}\) & arrival time for vehicle \(k\in K\) at location \(i\in N\) \\ \(b_{ik}\) & departure time for vehicle \(k\in K\) at location \(i\in N\) \\ \hline \hline \end{tabular} \[Min\sum_{k\in K}\sum_{(ij)\in A}c_{ijk}\cdot x_{ijk}\] \[\sum_{(i,j)\in A}x_{ijk}-\sum_{(j,i)\in A}x_{ijk}=0\qquad\forall\;k\in K,\; i\in N\] (1) \[\sum_{(o,j)\in A}x_{ojk}=1\qquad\forall\;k\in K\] (2) \[\sum_{(j,a)\in A}x_{jok}+\sum_{(o^{\prime},j)\in A}x_{o^{\prime}jk}=0 \qquad\forall\;k\in K\] (3) \[\sum_{(j,o^{\prime})\in A}x_{jo^{\prime}k}=1\qquad\quad\forall\;k\in K\] (4)  
\end{table} 
Table 1: Parameters and variables of the PDP-TW model. Figure 2: An illustrative example of the PDP-TW. \[\sum_{(i,j)\in A}\sum_{k\in K}x_{ijk}=1\hskip 28.452756 pt\forall\;i\in N \tag{5}\]  
\[u_{i}-u_{j}+n\cdot\sum_{k\in K}x_{ijk}\leq n-1\hskip 28.452756 pt\forall\;i,\;j\in P\cup D \tag{6}\]  
\[y_{ijkr}\leq\;x_{ijk}\hskip 28.452756 pt\forall\;(i,j)\in A,\;k\in K,\;r\in R \tag{7}\]  
\[\sum_{r\;\in\;R}q_{r}\cdot y_{ijkr}\leq C\cdot x_{ijk}\hskip 28.452756 pt\forall\;(i,j )\in A,\;k\in K \tag{8}\]  
\[\sum_{k\in K}\sum_{(j,i)\in A}y_{ijkr}=1\hskip 28.452756 pt\forall\;r\in R,\;i=d_{r} \tag{9}\]  
\[\sum_{(i,j)\;\in\;A}y_{ijkr}-\sum_{(j,i)\;\in\;A}y_{ijkr}=0\hskip 28.452756 pt \forall\;k\in K,\;r\in R,\;i\in N\backslash\{p_{r},\;d_{r}\} \tag{10}\]  
\[\sum_{(i,j)\in A}\sum_{k\in K}y_{ijkr}=1\hskip 28.452756 pt\forall\;r\in R,\;i=p_{r} \tag{11}\]  
\[b_{ik}+\tau_{ijk}-a_{jk}\leq M\Big{(}1-x_{ijk}\Big{)}\hskip 28.452756 pt\forall\;(i,j)\in A,\;k\in K \tag{12}\]  
Figure 2:0  
Figure 2:1  
Figure 2:2  
Figure 2:3  
Figure 2:4  
Figure 2:5  
Constraints (1)-(5) ensure the vehicles' flow conservation, guaranteeing that each vehicle departs from the original depot, visits all necessary nodes, and concludes its route at the destination depot. Set of constraints (6) avoid solutions with subtours, based on the Miller-Tucker-Zemlin (MTZ) formulation [23]. Constraints (7) connect the vehicles' flow with the requests' flow, ensuring that a request can only be transported if the corresponding route is traveled by a vehicle. The set of constraints (8) guarantees that the total baggage carried by a vehicle does not exceed its capacity \(C\). Sets (9)-(11) ensure the requests flow conservation, ensuring that each request is picked up and delivered to its origin and destination, respectively. The set of constraints (12)-(14) address time windows, imposing that the arrival and departure times at each node respect the specified limits. Finally, Constraints (15)-(18) refer to the variables' domain.