## 2 The Theoretical Background
### Optimisation Approaches to the EC Problem  
[PERSON] and [PERSON] [6] categorise approaches to the EC problem into two broad categories: technical solutions and optimisation approaches. The technical solutions include innovations such as collapsible containers and connectors, which aim to reduce the space occupied by ECs during transport, thus improving the efficiency of the logistics system. The optimisation strategies, on the other hand, focus on the use of mathematical models and advanced techniques to minimise costs and improve the operational efficiency. These optimisation techniques often take the form of deterministic or stochastic programming models. [PERSON] et al. [16] focused on cargo routing, while [PERSON] et al. [17] examined multi-scenario policies for fleet sizing. Additionally, [PERSON] and [PERSON] [18] addressed long-haul liner service route design and EC repositioning. These various aspects highlight the complexity of the EC problem, requiring a comprehensive approach that integrates both strategic and operational decision-making. [PERSON] et al. [19] employed a multistage inventory issue approach, using discrete-time Markov decision processes to formulate the optimal inventory policies. Meanwhile, ref. [20] investigated the application of time-expanded networks with recovery, identifying recovery activities that could be solved polynomially. In contrast, [PERSON] et al. [21] approached the EC problem by first framing it as a knapsack problem and then as a network flow problem. The application of agent-based models has gained popularity in EC management research due to their ability to simulate the complex interactions between different logistics actors. [PERSON] and [PERSON] [9] developed a preliminary agent-based model for an intermodal container transport network, focusing on resource sharing to reduce CO\({}_{2}\) emissions and improve the time drivers spend at home. This model, although it was still in its early stages, produced promising results by demonstrating that resource sharing can improve the operational efficiency and reduce the environmental impact in intermodal networks. In addition, [PERSON] et al. [1] proposed a mathematical model to improve the resilience in intermodal transportation networks through the coordination of bulk cargo transportation and EC repositioning. Their model, designed to maximise the individual profits for logistics companies, showed a significant reduction in EC transportation costs. These studies demonstrate that mathematical modelling and simulation approaches are not only effective in optimising the operation of complex logistics networks but also have the potential to reduce environmental impacts if implemented strategically in intermodal systems. Despite previous research that focused on cost and efficiency, our research seeks to quantify the environmental advantages of optimising the container utilisation in this novel logistics architecture. This study presents a novel approach to decreasing emissions and increasing the sustainability in global supply chains by examining ECs in the context of the PI. Our work focuses on discovering environmentally friendly solutions in container transportation and providing new insights into how we can move products more responsibly. [PERSON] et al. [13] also used a noteworthy method to handle the extended transportation problem with substitution flows. This method recognised the possibility of street turns and container substitution by branching and bounding for the substitution sub-problem and generating an LP solution. [PERSON] et al. [22] introduced an agent-based simulation to model the micro-level interactions in rail-road intermodal transport networks, addressing gaps in the Physical Internet (PI)'s deployment for urban rail systems. Their results highlighted that PI-enabled systems reduce empty truck driving, promoting sustainable and efficient logistics operations. Table 1 presents a summary of several key studies on the EC problem.