## 4 Numerical Experiments and Results
### The Change in the Number of Yard Blocks and Time Windows  
As shown in Table 1, we experimented with three cases of yard blocks and time windows based on two transport companies (forwarders). The input parameters for each case reflect random values generated according to the uniform distribution (as shown in Table 2). Table 3 and Figure 5 show the experimental results for the three mathematical models.  
According to the experimental results, in all cases, the objective value when the transport companies (forwarders) and the terminal operator cooperated was lower than when the transport company (forwarder) and the terminal operator decided independently. As the number of yard blocks and time windows increased, the gap in the objective value gradually increased.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Cases**} & **No. of Transport Companies** & \multirow{2}{*}{**No. of Yard Blocks**} & \multirow{2}{*}{**No. of Time Windows**} \\  & **(Forwarders)** & & \\ \hline Case 1 & 2 & 2 & 2 \\ Case 2 & 2 & 3 & 3 \\ Case 3 & 2 & 4 & 4 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Cases for the change in the number of yard blocks and time windows.  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Cases**} & **The Objective** & **The Objective** & **The Objective** & **Gap (\%)** \\  & **Value of Model 1** & **Value of Model 2** & **Value of Model 3** & **(A+B)** \(-\) \\  & **(A)** & **(B)** & **(C)** & **(C))/(C** \\ \hline Case 1 & 210.19 & 16.56 & 225.19 & 0.69 \\ Case 2 & 229.41 & 19.40 & 244.79 & 1.64 \\ Case 3 & 251.78 & 21.59 & 266.80 & 2.46 \\ \hline \hline \end{tabular}
\end{table}
Table 3: The experimental results for the change in the number of yard blocks and time windows.  
Figure 5: Comparison of experimental results for the change in the number of yard blocks and time windows.  
According to the experimental results, in all cases, the objective value when the transport companies (forwarders) and the terminal operator cooperated was lower than when the transport company (forwarder) and the terminal operator decided independently. Also, as the number of transport companies (forwarders) increased, the gap in the objective value increased rapidly.  
Analysis of the Results from the Perspective of Each Transport Company (Forwarder) and Terminal Operator  
Based on the results of the sensitivity analysis performed in Sections 4.1 and 4.2, we analyzed how the changes in the number of yard blocks, time windows, and transport companies (forwarders) affected each transport company (forwarder) and terminal operator. Figure 7 shows the experimental results of the impact of the three factors on the transport company (forwarder). Figure 8 shows the experimental results of the effect of these changes on the terminal operator.  
Figure 8: The relationship between the terminal operator and the change in the number of yard blocks, time windows, and transport companies (forwarders).  
Figure 7: The relationship between the transport company (forwarder) and the change in the number of yard blocks, time windows, and transport companies (forwarders).  
Both the number of yard blocks and time windows and the number of transport companies (forwarders) have a positive effect on the transport company (forwarder). In other words, as the number of yard blocks and time windows and the number of transport companies (forwarders) increased, the objective value of the transport company (forwarder) increased.  
The terminal operator has a positive relationship with the number of yard blocks and time windows. However, it has the opposite relationship to the number of transport companies (forwarders). In other words, as the number of yard blocks and time windows increased, the terminal operator's objective value increased. On the other hand, as the number of transport companies (forwarders) increased, the terminal operator's value decreased.