## 6 Conclusions  
In this study, we tackle the inland container transportation problem, focusing on the challenge posed by the limited depot capacity. A novel graph model is defined to characterize the depot-capacity-related constraints in a more efficient way, based on which the considered problem can be directly formulated as a mixed-integer linear programming model. A tailored branch-and-price-and-cut algorithm is proposed, which operates without requiring time horizon discretization and incorporates enhancement strategies to achieve optimal solutions. Rigorous numerical experiments are undertaken to validate the proposed algorithm. The results clarify that the proposed algorithm is significantly superior to CPLEX, GA, TS, LNS, and SA. The algorithm is capable of identifying the optimal solutions for instances involving up to 75 requests within 1 h. Comparative results also show the effectiveness of enhancement strategies.  
Sensitivity analysis brings insightful findings. Firstly, increasing the initial stock of empty containers at the depot helps in reducing the total cost. However, it will encounter the law of diminishing marginal returns, with each additional container contributing less to the overall cost saving. Secondly, expanding the depot capacity offers benefits by cutting down the overall cost and the number of dispatched trucks at the expense of a modest increase in the travel cost. Thirdly, lengthening the time window could benefit inland container transportation, providing operators with more flexibility and cost-effective solutions.  
The findings of this study hold valuable implications for practitioners in the logistics and transportation industries, especially those managing containerized cargo at inland depots. Decision makers, including logistics managers, fleet operators, and urban planners, can use these insights to optimize their operations, improve cost efficiency, and better  
Figure 9: Impact of time window length.  
manage capacity constraints. Furthermore, policymakers may find the study useful in shaping infrastructure development decisions, particularly those related to depot expansion or the management of empty container flows.  
This work also opens several avenues for future research. While the proposed model and algorithm show promising results in theory, validating these findings in real-world settings is of great importance. Therefore, future studies will aim to test the algorithm in real operational environments, taking into account factors such as road congestion, traffic conditions, weather fluctuations, and the variability of infrastructure. This will allow us to assess the algorithm's practical effectiveness and refine its assumptions to better match the complexities of real-world container logistics. Another area for exploration involves the use of heterogeneous trucks with varying capacities to handle multi-size containers, which would better reflect the diverse fleet configurations found in real-world logistics operations. Additionally, further research could focus on developing more efficient algorithms, particularly those capable of handling larger, more complex instances within a shorter time.  
Conceptualization, Y.Z. and Y.S.; methodology, Y.S.; software, Y.S.; formal analysis, Y.S. and Y.Z.; writing Y.S. and Y.Z.; visualization, Y.Z.; supervision, Y.S.; funding acquisition, Y.S. All authors have read and agreed to the published version of the manuscript.  
This study is supported by the National Natural Science Foundation of China (No. 72101152).  
Not applicable.  
The data for this study can be requested from the corresponding author.  
The authors declare no conflicts of interest.