## 3 Problem Statement and Formulations
### Graphical Formulation
#### 3.2.2 Nodes for Pickup Requests/Depots  
IF requests and the depot as the suppliers of empty containers are represented by nodes. With respect to IF requests, each of them is denoted by one node, signifying the ongoing sequence of loading a fully laden container from the terminal (Origin), transporting it to the receiver (Destination), and subsequently unpacking it. Such a node possesses three key attributes: (1) the service time \(T_{i}\) that is required to complete the activities; (2) the time window \([L_{i},U_{i}]\) during which its corresponding activities should begin; and (3) the travel cost \(C_{i}\) accrued from truck travel. Specifically, it is known that each IF request is \begin{table}
\begin{tabular}{l l} \hline \(C_{IF}\) & The quantity of the IF requests \\ \(Q\) & Depot capacity: the maximum capacity of the depot is limited to \(Q\) empty containers \\ \(H\) & Planning horizon \\ \([L_{i}^{0},U_{i}^{0}]\) & The time window at the origin of node \(i\in N_{IF}\cup N_{OF}\) \\ \([L_{i}^{1},U_{i}^{10}]\) & The time window at the destination of node \(i\in N_{IF}\cup N_{OF}\) \\ \([L_{i},U_{i}^{1}]\) & The consolidated time window of node \(i\) \\ \(T_{i}\) & Service duration of node \(i\) \\ \(C_{i}\) & Travel cost of node \(i\) \\ \(C_{ij}\) & The cost of traveling from node \(i\) to node \(j\) \\ \(T_{ij}\) & The time taken to travel from node \(i\) to node \(j\) \\ \(P_{i}\) & Packing/unpacking time associated with IF/OF request node \(i\) \\ \(\alpha_{i}\) & Change in the quantity of empty containers maintained at the depot when visiting \\  & node \(i\), \(\alpha_{i}\in\{-1,+1\}\), \(i\in N_{E}^{P}\cup N_{IF}^{P}\cup N_{IF}^{P}\) \\ \(\overline{M}\) & A adequately large number \\ \hline \end{tabular}
\end{table} 
Table 2: _Cont._  
Figure 2: Graphical representation \(G\). subject to two distinct time windows, namely the origin time window \([L_{i}^{O},U_{i}^{O}]\) imposed by the terminal and the destination time window \([L_{i}^{D},U_{i}^{D}]\) imposed by the receiver. These time windows should accordingly be consolidated into a single one \([L_{i},U_{i}]\) for each node. Additionally, it is further required to compute the service time \(T_{i}\), which is determined by the duration of its activities and unavoidable waiting time. Besides, we should also calculate the travel cost \(C_{i}\) consumed by each node, which is assumed to proportionally correlate with travel time from the terminal to the receiver's site. Assume that \(t_{od}\) is the time required to travel from the origin to the destination and \(\rho_{1}\) is the cost incurred by each unit of travelling time. Then, computation of three attributes can be formally conducted in the following manner (refer to [37] for a thorough explanation):  
\[L_{i}=min\{max\{L_{i}^{O},L_{i}^{D}-t_{od}\},U_{i}^{O}\} \tag{1}\]  
\[U_{i}=min\{U_{i}^{D}-t_{od},U_{i}^{O}\} \tag{2}\]  
\[T_{i}=max\{L_{i}^{D}-U_{i}^{O},t_{od}\}+P_{i} \tag{3}\]  
\[C_{i}=\rho_{1}t_{od} \tag{4}\]  
The depot provides empty containers sourced from the initial inventory or from returned containers that have been emptied after fulfilling IF requests. Thus, we introduce \(e\) nodes, each representing an available empty container initially stocked at the depot. In other words, visiting such a node indicates the _pickup_ of one empty container initially kept at the depot. For ease of presentation, the set of \(e\) nodes is denoted by \(N_{E}^{P}\). We can analogously define the node set \(N_{IF}^{P}\) to represent \(q_{IF}\) depot duplicates, where \(C_{IF}\) is the count of the IF requests. Each duplicate within this set corresponds to an empty container released by a specific IF request. Obviously, visiting such a node signifies that an empty container released from a specific IF request has been already returned to the depot and is now being _picked up_ for usage again. These nodes have the following attributes attached to them. Firstly, the time window for node \(i\in N_{E}^{P}\cup N_{IF}^{P}\) is defined as the interval of \([0,H]\).