## 1 Introduction  
The increasing regionalization of production and the advancement of economic globalization in recent decades have resulted in a flourishing of international trade. Container transportation has been an indispensable cornerstone in international trade since the utilization of standardized containers has greatly facilitated the transportation of goods and materials across vast distances, connecting businesses and consumers worldwide. In a typical container transportation chain, rail and maritime transport predominantly handle long-distance transportation between terminals, while trucks are typically in charge of short-distance transportation from terminals to customers. The short-distance transportation by truck is also known as inland container transportation. Despite accounting for only 5\(\sim\)10% of the total transportation distance, inland container transportation can incur disproportionately high costs, sometimes reaching up to 30 percent of the overall costs. Inland container transportation also has substantial environmental and logistical impacts, such as air contamination, greenhouse gas emissions, and road congestion. This necessitates the proper planning of truck routes for inland container transport. The inland container transportation problem involves the efficient collection and timely delivery of empty and fully laden containers between terminals, depots, and clients. Depots and containers are critical resources in the container transport system and play an extremely important role in the smooth transport of goods. The existing literature on inland container transportation often assumes that depots possess sufficient capacity to store empty containers and that there is an abundant supply of empty containers. However, this assumption does not always align with the real-life situation. In reality, depots are physical facilities. Factors such as land availability, infrastructure limitations, budget constraints, and zoning regulations often impose restrictions on the expansion of depot capacities. Consequently, these limitations inherently restrict the container storage capacity of a designated depot. Besides, the inventory of available empty containers within depots is also constrained in quantity. Admittedly, purchasing and holding containers will incur a significant financial burden. Containers require storage space, and maintenance to ensure they remain in good condition. If the trucking firm does not have a constant need for containers, it might not be cost-effective to keep an extensive stock of empty ones. Overlooking these restrictions when optimizing truck routes might result in sub-optimal and even infeasible routing plans. Therefore, it is crucial to relax these assumptions for better alignment with real-life applications. In the literature, three representative studies, including [PERSON] et al. [1], [PERSON] et al. [2], and [PERSON] et al. [3], rigorously investigate the depot's limited initial inventory of empty containers. However, the representation of empty container inventory, proposed by [PERSON] et al. [1] and [PERSON] et al. [2], is characterized by non-linear constraints. Consequently, the methodology adopted for solving the problem is restricted to heuristic algorithms. The limited depot capacity is neglected by [PERSON] et al. [3]. To our knowledge, we are pioneering in addressing the constraint of limited depot capacity. In view of this, this paper aims to investigate the inland container transportation problem with simultaneous consideration of the depot capacity and limited empty containers, and particularly emphasizes the constraint imposed by depot capacity. The main contributions of our work can be listed as follows:  
1. A graphical model is proposed for the problem, which allows for the constraints related to empty containers and depot capacity to be represented linearly. 2.