## 1 Introduction  
Containerised transport has revolutionised the flow of products by utilising standardised load units and enabling smooth transitions between various modes of transportation, including road, rail, and maritime systems [1]. Mono-modal and multi-modal hubs are the most commonly used forms of cargo transportation. Containers are key elements in the supply chain and are considered one of the most important assets for shipping companies [2]. However, containers often travel half empty or without cargo, with this imbalance a stark example of the associated logistical challenges, with significant financial and environmental consequences [3, 4]. Some research has indicated that around 20% of the containers around the world are empty [5]. One of the causes of this inefficiency is the locations where commodities are produced and consumed differing [6]. As a result, containers are frequently left empty at ports where they are not needed, waiting to be returned or repositioned to places where cargo is ready to be exported [1]. The issue of empty containers (ECs) presents significant challenges to the supply chain, affecting environmental, social, and performance aspects [2]. Moreover, these ECs do not bring any added value to the system. From an environmental standpoint, the transportation of these ECs is a significant source of CO\({}_{2}\) emissions [6]. Particularly within truck-based logistics, ECs often account for a significant portion of the overall trips, leading to unnecessary fuel consumption and contributing to CO\({}_{2}\) emissions [7]. This inefficiency underscores the need for innovative solutions to improve container sharing and capacity utilisation within logistics networks. To address modern problems within supply chains, the Physical Internet (PI) introduces a novel vision for logistics and supply chain management. The PI concept proposesan open and interconnected logistics system where physical products are transported with the efficiency and scalability of digital data across the Internet [8]. The PI introduces key concepts such as standardised modular containers, open networks for logistics collaboration, and real-time information sharing, which can significantly enhance the utilisation of transportation assets, including trucks. By applying PI principles, logistics systems can dynamically share the container space between different actors, reducing the number of ECs and optimising the transport capacity [9]. Despite the growing interest in the PI as a solution to modern logistics challenges, very limited research has focused on how the PI can directly address the issue of EC movements in barge-road hubs. This research is significant, as it addresses the EC problem through the lens of the PI, emphasising environmental and social indicators. This paper has two main contributions:  
* The problem is formulated and solved using a two-stage optimisation (2 Stage-Opt) model that optimises the container allocation by minimising the empty space in the containers (ECs). * A multi-agent system simulation (MASS) model is developed to test the validity of the solutions provided by the 2 Stage-Opt model by simulating the real-world dynamics of container sharing. The MASS will also allow us to evaluate two indicators: CO\({}_{2}\) emissions and truck utilisation. The remainder of this article is structured as follows: Section 2 offers an analysis of the existing literature on the EC problem in supply chains, the PI perspectives, and an analysis of the gaps. Section 3 presents a description of the problem and assumptions about the barge-road hub terminal. In Section 4, a mathematical formulation of the problem as a form of two-stage optimisation (2 Stage-Opt) and the components of the MASS developed are presented. Section 5 presents the implementation of the 2 Stage-Opt and MASS models and the experimental results, with a statistical analysis. Section 6 presents several industrial insights for the application of the proposed approach.