## 4 Case Study--Numerical Experiments
### Inter-Facility Material Transfer Process
#### 4.1.2 Optimal Buffer Allocation  
Table 4 presents the optimal buffer allocations obtained for each scenario investigated in the study. The allocations differ across scenarios, indicating the significance of tailoring buffer allocation strategies to specific operational conditions. A uniform truck allocation was employed for all material types in the initial three scenarios. Furthermore, an analysis of the results reveals that an increase in the number of trucks allocated to each material type and the total number of trucks corresponded to an increase in throughput, quantified as the total tons of materials transferred. However, it is worth noting that this increase in throughput was accompanied by a potential decrease in the average throughput per truck. In scenarios 4 and 5, different truck allocations were implemented for each material type while maintaining a constant total of 15 trucks. Consequently, distinct optimal buffer allocations were obtained for these scenarios, resulting in disparate throughput values. These findings underscore the influence of the number and allocation of trucks on buffer optimisation and subsequent material transfer efficiency in inter-facility operations. Furthermore, in scenario 4, we observed a near-uniform allocation of buffers across the service stations, except for a single station. This finding underscores the significance of conducting a comprehensive analysis to determine each case's optimal buffer allocation strategy. Additionally, noteworthy observations can be drawn from scenarios 1 and 2. Despite an increase in the total number of trucks employed (from 9 to 12), the total number of buffers utilised decreased from 22 to 21. This suggests that an increase in truck quantity does not necessarily correlate with a proportional increase in buffer requirements. Similarly, in scenario 4, where the total number of buffers used was 18, an equivalent number of trucks in scenario 5 necessitated 25 buffers to achieve the optimal throughput. This discrepancy highlights the varying impact of truck allocations and compositions on buffer requirements, \begin{table}
\begin{tabular}{l c c c} \hline \hline \multirow{2}{*}{**Scenario \#**} & **Optimal Buffer Allocation** & \
\begin{tabular}{c} **Throughput (Tons of** \\ **Materials** \\ **Transferred)** \\ \end{tabular} &
\begin{tabular}{c} **Total \# of Buffers** \\ **Used** \\ \end{tabular} \\ \hline Scenario 1 & 4,1,4,3,1,3,2,2,1,1 & 3101 & 22 \\ \hline Scenario 2 & 1,1,3,2,2,4,3,2,2,1 & 3656 & 21 \\ \hline Scenario 3 & 1,3,2,4,1,1,2,7,2,1 & 3677 & 24 \\ \hline Scenario 4 & 1,1,1,1,1,10,1,1,1,1 & 3628 & 18 \\ \hline Scenario 5 & 2,1,6,2,1,5,1,1,3,3 & 3794 & 25 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Optimal buffer allocations in each scenario. Figure 7: DES model for SM inter-facility material transfer operations. considering that certain service stations are exclusively utilised by trucks carrying a single product type. In contrast, others are shared among multiple product types. The decision to set the maximum iterations to 5000 was carefully considered to strike a balance between ensuring convergence towards optimal results and computational efficiency. The figures presented in Appendix A (Figures A1-A4) demonstrate that convergence typically occurs well before reaching the maximum of 5000 iterations, indicating the efficiency of the proposed method in achieving optimal solutions in most experimental examples. However, to account for potential problem size and complexity variations, we adopted a conservative approach by setting a maximum of 5000 iterations. For larger-scale problems, the convergence might occur more slowly than in the experimented examples, making the maximum iteration limit crucial in capturing longer convergence trends. This choice ensures the robustness of our approach, enabling effective handling of a broader range of problem sizes and complexities.