## 5 Numerical Experiments
### Comparison with Previous Work  
In this section, we test the performance of the AG algorithm on two existing benchmark problems. The first problem investigates a production line consisting of ten machines, which was originally proposed by [PERSON] et al. [23]. A total of 270 buffer units are allocated to this production line. The processing rate \(\mu=1\) is the same on all machines. Failure and repair times are exponentially distributed. Table 3 shows the MTTFs and MTTRs of the machines. [PERSON] et al. [23] solved this problem using a degraded ceiling (DC) algorithm and a simulated annealing (SA) algorithm and the results are given in Table 3. It should be noted that they use the DDX method, proposed by [PERSON] et al. [36], to evaluate the line throughput. The DDX method is an approximate analysis technique for evaluating the performance of continuous flow lines. Compared to simulation-based evaluation methods, the DDX method may introduce inherent errors that cannot be eliminated by increasing the \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{1}{c}{
\begin{tabular}{c} **Problem Set** \\ **(K, N)** \\ \end{tabular} } & **Algorithm** & **Avg. Percent Error** & **Avg. CPU Time (s)** \\ \hline \multirow{2}{*}{(10, 500)} & FG & 0.022\% & 186.9 \\  & AG & 0.029\% & 16.7 \\ \hline \multirow{2}{*}{(10, 1000)} & FG & 0.013\% & 368.6 \\  & AG & 0.021\% & 18.4 \\ \hline \multirow{2}{*}{(20, 1000)} & FG & 0.017\% & 1952.7 \\  & AG & 0.046\% & 189.1 \\ \hline \multirow{2}{*}{(20, 2000)} & FG & 0.022\% & 3717.1 \\  & AG & 0.034\% & 203.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Performance of FG and AG algorithms for randomly generated BAPs. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multicolumn{1}{c}{
\begin{tabular}{c} **Machine** \\ \end{tabular} } & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **10** \\ \hline \(MTTF_{i}\) & 20 & 30 & 22 & 22 & 25 & 40 & 23 & 30 & 45 & 20 \\ \(MTTR_{i}\) & 7 & 7 & 5 & 10 & 9 & 14 & 5 & 8 & 10 & 10 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: A ten-machine production line. computational time. As shown in Table 4, the throughput obtained by the DDX method differs by a certain amount from the corresponding simulation result. According to the simulated throughputs, the FG and AG algorithms obtain better solutions than the DC and SA algorithms. We compare the computational efficiency of these algorithms based on the number of simulation runs. It is clear that the AG algorithm is much more efficient than the other algorithms. It needs only 34 evaluations to obtain a good quality solution. In comparison, the DC and SA algorithms take more than 10,000 evaluations. In the second example, we study a large-scale BAP in a production line with 40 machines and 39 buffers. This problem is originally proposed by [PERSON] et al. [31]. The line consists of two duplicated blocks of 20 machines. Each block is described in Table 5. The machine processing rates are set to one, and the failure and repair times are exponentially distributed. There are a total of 1600 buffer units to allocate. [PERSON] et al. [31] solved the problem using a local search algorithm and a degraded ceiling algorithm. As in the previous example, they also used the DDX method [36] for continuous flow production lines to evaluate throughput. Table 6 compares the results obtained by different algorithms. In this example, the DDX method leads to a relatively large error: the evaluated throughput differs greatly from the corresponding simulated throughput. The throughput of the AG algorithm is very close to that of the FG algorithm, with a negligible error of 0.03%, and is better than the result in [PERSON] et al. [31]. Figure 4 shows the buffer capacity distributions obtained by different algorithms. Note that the distribution curves of the FG and AG algorithms almost overlap. In terms of computational efficiency, the AG algorithm takes a tenth of the time of the FG algorithm.