# A Network-Based Model for Optimization of Container Location Assignment at Inland Terminals
## 6 Validation of the Mathematical Model
### Simulating Container Storage Using the Mathematical Model  
The simulation suggested that the criteria of container owner, dimension, and final destination are decisive in determining the container's storage location (Table 5).  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Occupancy** & \(\mathbf{w}\) & \(\mathbf{X}\) \\ \hline Filled & 0.500 & 0.000 \\ \hline Empty & 0.500 & 0.000 \\ \hline \hline \end{tabular}
\end{table}
Table 10: Ranking of container occupancy according to retention time.  
Figure 4: Layout of the inland terminal in the FlexSim CT simulation to validate the mathematical model for inbound container placement.  
The random numbers in Table 12 were adapted from statistics given in Table 2. The random numbers in Tables 13 and 14 were adapted from statistical analysis of the 11 European inland terminals.  
Table 15 shows the storage tier locations based on tier ranking for 32 inbound containers arriving daily in the inland terminal. The aim of defining the storage tier was to shorten the number of unproductive manipulations at the inland terminal and to lower overall terminal costs related to container storage.  
\begin{table}
\begin{tabular}{c c c} \hline
**Distance (km)** & **Random Number** & **Retention (Days)** \\ \hline
0–50 & 0–0.25 & 4.25 \\ \hline
50–100 & 0.25–0.50 & 5.5 \\ \hline
100–150 & 0.50–0.75 & 4.25 \\ \hline
150–\(\infty\) & 0.75–1 & 4.25 \\ \hline \end{tabular}
\end{table}
Table 14: Simulation of container retention according to the distance to the container’s final destination.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Dimension** & **TeU** & **Random Number** & **Retention (Days)** \\ \hline
20 & NA 1 & 0–0.32 & 9.75 \\ \hline
30 & 1.5 & 0.32–0.42 & 5.5 \\ \hline
40 & 2 & 0.42–0.79 & 8 \\ \hline
45 & 2.25 & 0.79–1 & 4.75 \\ \hline \end{tabular}
\end{table}
Table 12: Simulation of container retention according to container dimension.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Dimension** & **Retention** \\
**(Feet)** & **TeU** & **Random Number** & **Retention (Days)** \\ \hline
20 & NA 1 & 0–0.32 & 9.75 \\ \hline
30 & 1.5 & 0.32–0.42 & 5.5 \\ \hline
40 & 2 & 0.42–0.79 & 8 \\ \hline
45 & 2.25 & 0.79–1 & 4.75 \\ \hline \end{tabular}
\end{table}
Table 13: Simulation of container retention according to container owner.  
\begin{table}
\begin{tabular}{c c c} \hline
**Owner** & **Random Number** & **Retention (Days)** \\ \hline Railway operator & 0–0.24 & 4.25 \\ \hline Road operator & 0.24–0.36 & 5.5 \\ \hline Client & 0.36–0.52 & 5 \\ \hline Shipping company & 0.52–1 & 4.25 \\ \hline \end{tabular}
\end{table}
Table 13: Simulation of container retention according to container owner.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline
**Container ID** & **Criterion** & **TEU** & **Retention Time** & **Average** & **Storage Tier** \\ \hline  & Dimension & Owner &
\begin{tabular}{c} Final destination \\ \end{tabular} & \(t_{1}\) & \(t_{2}\) & \(t_{3}\) & \(\tilde{t}\) \\ \hline
1 & 0.938 & 0.774 & 0.201 & 2.25 & 4.75 & 4.25 & 4 & 3 \\ \hline
2 & 0.966 & 0.625 & 0.790 & 2.25 & 4.75 & 4.25 & 4.25 & 4 & 3 \\ \hline
3 & 0.884 & 0.785 & 0.999 & 2.25 & 4.75 & 4.25 & 4.25 & 4 & 3 \\ \hline
4 & 0.894 & 0.560 & 0.717 & 2.25 & 4.75 & 4.25 & 4.25 & 4 & 3 \\ \hline
5 & 0.839 & 0.936 & 0.967 & 2.25 & 4.75 & 4.25 & 4.25 & 4 & 3 \\ \hline
6 & 0.991 & 0.122 & 0.625 & 2.25 & 4.75 & 4.25 & 4.25 & 4 & 3 \\ \hline
7 & 0.343 & 0.748 & 0.109 & 1.5 & 5.5 & 4.25 & 4.25 & 5 & 3 or 2 \\ \hline
8 & 0.356 & 0.917 & 0.684 & 1.5 & 5.5 & 4.25 & 4.25 & 5 & 3 or 2 \\ \hline \end{tabular}
\end{table}
Table 15: Modeling the retention time and storage tier of inbound containers at the simulated inland terminal.  
The proposed model suggested a different ranking of criteria for container storage at the simulated inland terminal than what was empirically observed at the inland terminal whose data were used in the simulation (Figure 5). In this way, the model was validated by a simulation of container storage processes at an inland terminal representative of many European inland terminals.