## 2 The Theoretical Background
### The Physical Internet (PI) Paradigm
[28] studied truck scheduling in rail-road PI-hubs, aiming to minimise the delays and container travel distances while considering uncertainty in the truck arrival times. They proposed using a Fuzzy Multi-Objective Mixed-Integer Programming model (FMO-MIP) to generate a robust Pareto-optimal solution. This study reviewed the existing research on truck scheduling in cross-docking facilities and outlined a description of the problem and its mathematical formulation. It developed a methodology combining an \(\epsilon\) constraint and chance-constrained programming and conducted implementation experiments to demonstrate its efficiency and robustness. Overall, this study contributed to addressing uncertain multi-objective truck scheduling in PI-hubs. Other studies, such as those by [PERSON] et al. [29], have focused on rail-road hubs from a sustainability perspective by minimising both the number of trucks used and the energy consumption of the PI-sorters. The PI concept of encapsulation is a key innovation, which refers to standardised packaging of goods into modular containers known as PI-containers [8]. These containers are compatible with different handling systems and methods of transportation because they are universally standardised in terms of their size, design, and functionality [30]. Because of their modular design, smaller containers can be combined to form bigger units, maximising the use of space and accommodating various storage and transportation needs [26; 31]. Numerous studies have looked into various facets of using PI-containers, with an emphasis on how they improve the sustainability and efficiency of logistical operations. Several works in the literature have studied the \"activeness\" of PI-containers, improving their role in PI management and the container loading efficiency. [PERSON] et al. [32] introduced the idea of the \"activeness\" of PI-containers actively participating in PI management. After outlining the requirements and limitations of existing smart container solutions, this paper proposed a framework for PI-container activeness and demonstrated its application in a grouping scenario within a PI-hub through simulation experiments. It suggested future research directions, including refining the framework, conducting pilot tests with instrumented containers, and exploring the roles of active PI-containers in hub control and optimisation. In another study, [PERSON] et al. [33] addressed the challenge of container loading in dynamic conditions. The authors presented an online packing heuristic for the three-dimensional container loading problem (3 DCLP) in the context of the PI. The results showed a significant reduction in the packing time and improved space efficiency. Other researchers have explored PI-containers in road-rail hubs, focusing on optimising the truck scheduling and container grouping for greater sustainability. For instance, [PERSON] et al. [31] investigated the utilisation of PI-containers in road-rail PI-hubs, with an emphasis on sustainable truck scheduling and container grouping. Their research developed a MAS integrated with a Mixed-Integer Linear Programming (MILP) paradigm to handle the interruptions and inefficiencies that are frequent in such settings. Their major goal was to optimise the truck scheduling and container grouping at these multi-modal hubs. By using PI-container grouping algorithms, the model was able to reduce both the number of vehicles and the distance travelled by the containers. This study found that grouping PI-containers into optimum clusters could result in fewer cars on the road, shorter distances, and a more sustainable logistics system. The MAS method was successful in enhancing the overall efficiency and sustainability. Finally, routing solutions for PI-containers are another key area of focus in the literature. [PERSON] et al. [34] presented a routing solution for PI-containers that aimed to improve the scalability and efficiency of container mobility inside the PI.