I am trying to add some more constraints to the flow-based ADVRP model in Almoustafa et al. (2013)1 (pp.4).
The mentioned model caps the travel distance, while I cap the travel time. Let $U$ represent the maximum travel time each vehicle can make. Additionally, there is a node-dependent time spent at each delivery node. For example, if 5 packages are delivered to location $j$, the truck will spend 25 minutes (assuming 5 min per package). I think, I can address this by embedding, $s_j:=\text{time spent at delivery node } j$, into (6) and (9). I don't add this into the objective because I only want to minimize the travel time without including the package drop time. Firstly, is the following formulation correct based on the assumptions? Consider the following set definitions: $I$ set of nodes, where $0$ represents the depot, and the parameter: $T_{ij}$ is the asymmetric travel time from node $i$ to $j$, and $T_{ii}=0$.
\begin{align} \min&\quad\sum_{i,j\in I, i\neq j}T_{ij} x_{ij}\tag1 \\\text{s.t.}&\quad\sum_{i\in I, i\neq j}x_{ij} = 1 \quad \forall j\in I\setminus\{0\}\tag2 \\&\quad\sum_{j\in I, i\neq j}x_{ij} = 1 \quad \forall i\in I\setminus\{0\}\tag3 \\&\quad\sum_{i\in I\setminus\{0\}}x_{0i} = m\tag4 \\&\quad\sum_{i\in I\setminus\{0\}}x_{i0} = m\tag5 \\&\quad\sum_{j\in I,i\neq j}z_{ij}-\sum_{j\in I,i\neq j}z_{ji}-\sum_{j\in I,i\neq j}\left(T_{ij}-s_j\right)x_{ij} = 0 \quad \forall i \in I\setminus\{0\}\tag6 \\&\quad z_{ij} \leq (U-T_{j0})x_{ij} \quad \forall i\in I,j\in I\setminus{\{0\}}, i\neq j\tag7 \\&\quad z_{i0} \leq U x_{i0} \quad \forall i\in I \setminus{\{0\}}\tag8 \\&\quad z_{ij} \geq (T_{ij}+T_{0i}+s_j)x_{ij} \quad \forall i\in I\setminus{\{0\}},j\in I, i\neq j\tag9 \\&\quad z_{0i} = T_{0i}x_{0i} \quad \forall i\in I\setminus{\{0\}}\tag{10} \\&\quad x_{ij} \in \{0,1\}, z_{ij} \in \mathbb{R}_{\geq 0}, m\in \mathbb{Z}_{\geq 0} \end{align}
Next, I want to constrain the number of packages each truck can deliver. Say, we have a capacity of $K$ packages for each truck, and $P_j$ is the number of packages to deliver at each node. How can I incorporate these constraints?
Reference
[1] Almoustafa, S., Hanafi, S., Mladenović, N. (2013). New exact method for large asymmetric distance-constrained vehicle routing problem. European Journal of Operational Research. 226(3):386-394.