# A Recursive Optimization Approach for Buffer Allocation in Large Production Lines
## 6 Numerical Experiments
### Discussion  
The above experiments show that the accuracy of the RBD and ROD algorithms is quite good. For production lines with five machines, they always produce the optimal solutions. For production lines with no more than 20 machines, they are as accurate as the gradient algorithm. Although the ROD algorithm has a small margin of error for longer lines (less than 0.1%), the accuracy is acceptable, considering the fact that the machine parameter data available on a factory floor are rarely within 1% of their actual values. In terms of computational efficiency, the RBD and ROD algorithms show a great advantage over the gradient algorithm. For long production lines, the computation time can be reduced by more than 95%. Therefore, the proposed algorithms achieve a good trade-off between solution quality and computation time.  
In general, a gradient-based search algorithm is more efficient than combinatorial approaches (e.g., genetic algorithm, simulated annealing, etc.), since it exploits the concavity of BAPs [7,8]. Theoretical analysis and extensive numerical experiments show this property does hold. See [7,8] for a detailed discussion. For production systems with large buffers, the error of the proposed recursive algorithms can be neglected, but they are more efficient than the gradient algorithm for BAPs in long production lines. However, for just-in-time production systems with small buffer capacities, it is recommended to use the gradient algorithm directly, because the computational complexity is not high when the total buffer capacity is small.  
Another limitation of the RBD and ROD algorithms is that the gradient of the production rate has to be evaluated. In some cases, the gradient is unavailable. For example, if the machine failure time is not exponentially distributed, the production rate cannot be evaluated by the robust decomposition method or the ADDX method. In this case, the proposed algorithms cannot be applied. An alternative solution approach is to combine metaheuristics algorithms with simulation-based evaluation methods [30].  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \(i\) & **Processing Time (min)** & **Availability** & \(MTR_{i}\) **(min)** & **Isolated Production Rate** \\  & \(\tau_{i}=1/s_{i}\) & \(e_{i}\) & & \(\rho_{i}=s_{i}e_{i}\) \\ \hline
14 & 0.488 & 0.988 & 39 & 2.025 \\
15 & 0.482 & 0.975 & 38 & 2.023 \\
16 & 0.43 & 0.989 & 48 & 2.281 \\
17 & 0.473 & 0.981 & 47 & 2.097 \\
18 & 0.5 & 0.992 & 55 & 1.948 \\
19 & 0.467 & 0.974 & 43 & 2.071 \\ \hline \hline \end{tabular}
\end{table}
Table 10: _Cont._  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\begin{array}{c}\textbf{Target Rate}\\ \textbf{$P^{*}$}\end{array}\) & **Algorithm** & \begin{tabular}{c} **Total Capacity** \\ \(\textbf{$N^{*}$}\) \\ \end{tabular} & \
\begin{tabular}{c} **Actual Rate** \\ \(\textbf{$P(b^{*})$}\) \\ \end{tabular} &
\begin{tabular}{c} **Computation Time** \\ **(s)** \\ \end{tabular} \\ \hline
1.80 & \begin{tabular}{c} Gradient \\ RBD \\ ROD \\ \end{tabular} & \begin{tabular}{c} 485 \\ 485 \\ 485 \\ \end{tabular} & \
\begin{tabular}{c} 1.800202 \\ 1.800202 \\ 1.800202 \\ 1.800202 \\ \end{tabular} &
\begin{tabular}{c} 46.1 \\ 9.2 \\ 7.1 \\ \end{tabular} \\ \hline
1.90 & \begin{tabular}{c} Gradient \\ RBD \\ ROD \\ \end{tabular} & \begin{tabular}{c} 1.900065 \\ 1201 \\ 1201 \\ 1201 \\ \end{tabular} & \
\begin{tabular}{c} 1.900065 \\ 1.900064 \\ 1.940010 \\ 1.940009 \\ \end{tabular} &
\begin{tabular}{c} 63.3 \\ 6.8 \\ 4.5 \\ 8.6 \\ 8.7 \\ 3.9 \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table}
Table 11: Optimization results for the real-life production line.