## 5 Experimental Results  
In this research, 10 small problems were firstly collected from the literature. The corresponding information about the problems is summarized in Table 1. For each problem, the velocities of walking and conveyor (\(VW\) and \(VC\)) are assumed to be 4 and 1 per time unit. So the length of the conveyor is equal to the total processing time. In addition, the widths of the U-shaped conveyors (\(b\)) are set as 10% and 15% of the length of the conveyor, respectively. The proposed integer programming was tested using software LINGO 17 with 6 h maximum computing time. The results of two, three and four workstations are shown in columns 2, 5 and 8 in Tables 2 and 3. As for the SA, it is assumed that total processing time (\(\sum_{i=1}^{l}T_{i}\)) is probably the cycle time of the worst case in which all tasks are assigned to same workstation, and \(AC\) (\(\frac{\sum_{i=1}^{l}T_{i}}{W_{max}}\)) is the cycle time of the best case. This means \(\sum_{i=1}^{l}T_{i}-\frac{\sum_{i=1}^{l}T_{i}}{W_{max}}\) would be the upper bound of difference in performance between \(Z(S_{new})\) and \(Z(S)\). This research set the initial temperature \(E\) as the total processing time. This means when there are two workstations, the acceptance probability for the initial temperature will be \(e^{\frac{-1}{2}}\), which is greater than 60%. The minimum acceptance probability for the initial temperature will reduce as the number of workstations  
Figure 11: Proposed SA procedure. increases. However, the reduction is small. For example, when there are 20 workstations, the acceptance probability for the initial temperature will be \(e^{\frac{-19}{20}}\), which is greater than 38%. For the remaining SA parameters, \(F\), \(H\), \(\alpha\) and \(\beta\), a full factorial experiment is adopted based on Problem 10 with \(b\) = 10% and 4 workstations. For the four parameters, two levels are designed as shown in Table 4. \begin{table}
\begin{tabular}{c c c c c} \hline
**No.** & **Source** & **Number of Tasks** & **Total Processing Time** \\ \hline
1 & [PERSON] and [PERSON] (2009) [11] & Figure 3 & 7 & 23.00 \\
2 & [PERSON] et al. (2013) [12] & Figure 3 & 7 & 34.00 \\
3 & [PERSON] and [PERSON] (2017) [14] & Figure 4 & 7 & 4.07 \\
4 & [PERSON] (1960) [13] & Figure 1 & 8 & 75.00 \\
5 & [PERSON] and [PERSON] (2017) [14] & Figure 3 & 8 & 3.03 \\
6 & [PERSON] et al. (2018) [15] & Figure 1 & 8 & 2.21 \\
7 & [PERSON] et al. (2018) [15] & Figure 2 & 8 & 3.80 \\
8 & [PERSON] and [PERSON] (2009) [11] & Figure 1 & 9 & 37.00 \\
9 & [PERSON] et al. (2017) [16] & Figure 1 & 9 & 55.00 \\
10 & [PERSON] and [PERSON] (2009) [11] & Figure 2 & 10 & 41.00 \\ \hline \end{tabular}
\end{table} 
Table 1: Summary of small problems. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline
**No.** & \multicolumn{6}{c}{**Number of Workstations**} \\ \cline{2-10}  & **LINGO** & **SA** & **Improvement** & **LINGO** & **SA** & **Improvement** & **LINGO** & **SA** & **Improvement** \\ \hline
1 & 12.9451 & 12.9451 & 0.00\% & 9.7334 & 10.0000 & \(-\)2.74\% & 7.5000 & 7.5000 & 0.00\% \\
2 & 19.7955 & 19.7955 & 0.00\% & 13.8362 & 13.8362 & 0.00\% & 12.1336 & 12.1336 & 0.00\% \\
3 & 2.2361 & 2.2361 & 0.00\% & 1.6778 & 1.7820 & \(-\)5.52\% & 1.3466 & 1.3466 & 0.00\% \\
4 & 42.9706 & 42.9706 & 0.00\% & 29.7622 & 29.5053 & 0.00\% & 24.7666 & 24.7666 & 0.00\% \\
5 & 1.7098 & 1.7098 & 0.00\% & 1.2375 & 1.2375 & 0.00\% & 1.0250 & 1.0250 & 0.00\% \\
6 & 1.2270 & 1.2270 & 0.00\% & 0.9132 & 0.9132 & 0.00\% & 0.6875 & 0.6875 & 0.00\% \\
7 & 2.1851 & 2.1863 & \(-\)0.06\% & 1.4465 & 1.4465 & 0.00\% & 1.2424 & 1.2424 & 0.00\% \\
8 & 19.9250 & 19.9250 & 0.00\% & 14.8620 & 15.0000 & \(-\)0.93\% & 11.2500 & 11.2500 & 0.00\% \\
9 & 30.8381 & 30.8381 & 0.00\% & 21.9413 & 21.9412 & 0.00\% & 16.9909 & 16.9909 & 0.00\% \\
10 & 22.1655 & 22.1654 & 0.00\% & 16.0250 & 16.0250 & 0.00\% & 12.2809 & 12.2809 & 0.00\% \\ \hline  & Average & \multicolumn{3}{c}{\(-\)0.01\%} & \multicolumn{3}{c}{\(-\)0.41\% *} & \multicolumn{3}{c}{0.00\%} \\ \end{tabular}
\end{table}