## 2 Literature Review
### The Inland Container Transportation Problem with Limited Empty Containers
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Paper**} & \multicolumn{4}{c}{**Feature**} & \multicolumn{4}{c}{**Solution Method**} & \multirow{2}{*}{**Objective**} & \multirow{2}{*}{**Model**} \\ \cline{2-2} \cline{5-8}  & **Time.** & **Size** & **Empty.** & **Cap.** & **Solver** & **Heur.** & **Exact** \\ \hline [PERSON] et al. [1] & ✓ & ✓ & & ✓ & & Min Time & Non-linear \\ [PERSON] et al. [2] & ✓ & ✓ & & ✓ & & Min Time/Trucks & Non-linear \\ [PERSON] et al. [3] & ✓ & ✓ & & & ✓ & Min Cost & Linear \\ [PERSON] et al. [14] & ✓ & & & ✓ & ✓ & Min Cost & Non-linear \\ [PERSON] et al. [15] & & & & ✓ & & Min Cost & Linear \\ \hline [PERSON] et al. [16] & ✓ & & & & ✓ & Min Distance/Trucks & Non-linear \\ [PERSON] et al. [17] & & & & ✓ & & Min Time & Linear \\ [PERSON] and [PERSON] [18] & ✓ & & & ✓ & & Min Time & Linear \\ [PERSON] et al. [19] & ✓ & & ✓ & & & Min Cost & Non-linear \\ [PERSON] et al. [20] & ✓ & & ✓ & & & Min Cost & Linear \\ [PERSON] et al. [21] & & & ✓ & & & Min Cost & Non-linear \\ [PERSON] et al. [22] & & ✓ & & ✓ & & Min Distance/Trucks & Linear \\ [PERSON] et al. [23] & ✓ & ✓ & & ✓ & & Min Cost & Linear \\ [PERSON] and [PERSON] [24] & ✓ & ✓ & ✓ & & Min Distance/Time & Linear \\ [PERSON] et al. [25] & ✓ & ✓ & & ✓ & & Min Cost & Linear \\ Bustos-Coral and Costa [26] & ✓ & ✓ & & ✓ & Min Cost & Linear \\ [PERSON] et al. [27] & ✓ & & & ✓ & & Min Cost & Linear \\ [PERSON] et al. [28] & & & & ✓ & & Min Cost & Linear \\ [PERSON] et al. [29] & ✓ & & ✓ & & Min Time/Requests & Linear \\ [PERSON] et al. [30] & & & & ✓ & & Min Cost/Requests & Non-linear \\ [PERSON] et al. [31] & ✓ & & & ✓ & & Max Profit & Linear \\ [PERSON] et al. [32] & ✓ & & & ✓ & & Min Cost & Non-linear \\ [PERSON] et al. [33] & ✓ & & & ✓ & Min Cost & Linear \\ [PERSON] et al. [34] & ✓ & & & ✓ & Min Cost & Non-linear \\ [PERSON] et al. [35] & ✓ & & & ✓ & Min Cost & Non-linear \\ [PERSON] et al. [36] & ✓ & & & ✓ & Min Cost & Linear \\ This study & ✓ & ✓ & ✓ & & ✓ & Min Cost & Linear \\ \hline \hline \end{tabular}  
* “Time.”: time window; “Size”: multi-size containers; “Empty.”: limited empty containers; “Cap.”: limited depot capacity; “Heur.”: heuristic.  
\end{table} 
Table 1: Comparative summary of related studies. By making comparisons with the existing literature, we can observe that these studies are the most similar to our research. However, our study is different from them primarily in three aspects:  
1. Existing studies often ignore the constraint of limited depot capacity, which is a critical factor in real-world container logistics. To address this, we introduce an innovative algorithmic framework that explicitly integrates depot capacity constraints, which allows for a more accurate and realistic representation of container storage. This integration requires novel algorithms to handle both the capacity and the restricted flow of empty containers, which have not been effectively addressed by previous approaches. 2. Unlike the methods in previous studies, we propose a graphical model designed specifically to capture the characteristics of the limited depot capacity and the limited empty containers at the depot. This model enables a linear formulation of both depot capacity and empty container constraints. In contrast to previous works that use more complex, non-linear models or require extensive pre-processing, our MILP formulation is simpler. 3. While most existing studies rely on heuristic algorithms (e.g., genetic algorithms, tabu search), our approach proposes an exact algorithm based on the branch-and-price-and-cut methodology. Compared to heuristic algorithms that often provide approximate solutions, our proposed algorithm ensures accuracy and optimality, offering a clear advantage in terms of solution quality.