## 3 Problem Statement and Assumptions  
In this paper, the focus is on the scheduling and container assignment and the grouping problem within a barge-road PI-hub. Figure 1 illustrates a comparison between a traditional cross-docking hub and a PI-hub. In a typical cross-docking hub, the containers move directly from the receiving area to outbound trucks with limited sorting, as seen in the simple flow shown in Figure 1A. In contrast, a PI-hub, as depicted in Figure 1B, incorporates a structured sorting process within a central grid where the modular PI-container is divided in the receiving zone. In this setup, the barge consists of multiple EC slots, which are treated as blocks. These blocks are loaded with PI-containers that arrive by truck. In the sort section, the modular PI-containers are processed through the PI-sorter and then loaded onto outgoing trucks.  
The primary assumptions for the problem studied in this paper are as follows:  
* All incoming containers are assumed to be 20 ft or 40 ft, representing the traditional logistics format. Upon their arrival at the PI-hub, these containers are deconsolidated into smaller, modular PI-containers with a standard \(H\) size. This separation allows for efficient assignment based on their shared destination features and compatibility with the truck capacities, as illustrated in Figure 1B.
* All the trucks in the fleet are considered to have uniform capacity and operational characteristics and to be capable of carrying multiple modular PI-containers. Depending on the size of the containers, trucks are loaded accordingly to maximise capacity utilisation.
* The number of containers arriving at the hub and the number of trucks available for dispatch are predetermined for each planning period.
* Each container's destination is determined upon its arrival at the hub, guiding the deconsolidation process into modular PI-containers based on destination-specific grouping.
* Containers are assumed to be in perfect condition upon their arrival and throughout their handling at the hub.
* Containers are grouped into modular PI-containers based on their destinations, which facilitates efficient routing and transport assignment.
* Upon their arrival, containers are immediately available for processing and assignment without any delay. This assumption helps to simplify the logistical considerations within the hub, focusing on the assignment and dispatch processes.  
To build the model, we take into consideration the container set \(I\). Containers are offloaded from the ships and added to the hub's logistical system. An occupancy, or the quantifiable quantity of space taken, is associated with each container in set \(I\) and a destination in set \(D\). The destinations within set \(D\) reflect the closest cities to which the products need to be transported. This assignment is dependent on the system's internal demand for items.  
Moreover, the notion of PI-containers, represented by set \(P\), adds the PI component to generate modularity and standardisation inside the hub. Based on the shared destination features from set \(D\), containers from set \(I\) are classified into PI-containers from set \(P\). The PI-containers model a cooperative transport approach that strives for environmental sustainability while also increasing efficiency. For every PI-container, the next stage of the expedition is assigned by the truck set \(J\). The PI-containers are planned to be delivered to trucks from set \(J\) according to their availability and capacity. The method of assignment takes into account the size of the containerssss as well as the capacity restrictions of the trucks.