## 4 Experimental Results
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \# & **Problems** & **0\%** & **5\%** & **10\%** & **15\%** & **20\%** \\ \hline \multirow{8}{*}{2} & 1 & - & 2.44 & 4.47 & 6.2 & 7.68 \\  & 2 & - & 1.54 & 2.86 & 4 & 5 \\  & 3 & - & 0.78 & - & 1.02 & 2.62 \\  & 4 & - & 2.36 & 2.98 & 4.19 & 9.89 \\  & 5 & - & - & - & - & 2.1 \\  & 6 & - & 1.48 & 2.76 & 3.08 & 4.94 \\  & 7 & - & 3.32 & 4.51 & 4.21 & 3.95 \\  & 8 & - & 0.71 & 1.32 & 2.78 & 4.21 \\  & 9 & - & 1.09 & 1.56 & 2.17 & 3.82 \\  & 10 & 0.62 & 2.34 & 2.81 & 2.64 & 2.49 \\  & 11 & - & 1 & 2.17 & 3.29 & 3.65 \\  & 12 & 17.73 & - & 0.2 & 0.28 & 0.35 \\  & 13 & 1.04 & 4.13 & 5.84 & 7.3 & 8.21 \\  & average & 1.49 & 1.63 & 2.42 & 3.17 & 4.53 \\ \hline \multirow{8}{*}{3} & 1 & 7.14 & 6.91 & 3.23 & - & - \\  & 2 & - & 4.54 & 6.34 & 5.2 & 4.18 \\  & 3 & - & 0.87 & - & - & - \\  & 4 & - & 0.82 & 2.49 & 2.21 & 1.2 \\  & 5 & - & 3.8 & 3.45 & - & - \\  & 6 & - & 3.13 & 2.61 & 2.43 & 3.04 \\  & 7 & - & - & - & - & - \\  & 8 & 7.69 & 4.78 & 2.17 & 1.47 & 3.37 \\  & 9 & - & 1.12 & 1.59 & - & - \\  & 10 & - & - & 0.01 & 0.42 & 0.77 \\  & 11 & - & - & 0 & - & - \\  & 12 & 2.5 & 0.43 & 0.81 & 0.29 & 5.93 \\  & 13 & - & 0.44 & 0.83 & 1.17 & 1.48 \\  & average & 1.33 & 2.07 & 1.81 & 1.02 & 1.54 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Improvement in U-shaped production lines (percentage). Figure 9: An illustration of task and location sequences for traditional straight assembly lines. In Table 3, the % indicates the ratio of the average processing time for one unit. Higher ratios indicate longer walking times between locations. In Table 3, from a total of 195 (13 \(\times\) 5 \(\times\) 3) cases, the U-shaped production line results in shorter cycle times in 131 cases. For different numbers of workstations and walking times, the average improvements range between 1.02% and 4.53%. Therefore, the comparison results confirm some of the arguments for the advantage of U-shaped production lines found in the literature. Moreover, the results of 0% for both traditional straight and U-shaped production lines are based on walking times that are 0% of the average processing time. This means that these results correspond to calculations ignoring walking time. Therefore, if the walking time is added onto these results, the real performance without taking walking time into consideration can be evaluated. Table 4 shows the improvements when taking walking time into consideration. In Table 4, there are 156 (13 \(\times\) 4 \(\times\) 3) cases that take walking time into consideration for both traditional straight and U-shaped production lines. For traditional straight production lines, there are 55 cases where performance improves when walking time is considered. For U-shaped production lines, 121 cases result in a shorter cycle time when walking time is considered. This means that the probability of improvement produced by taking walking time into account for U-shaped production lines is higher than that for traditional straight production lines. Moreover, longer walking times give more improvement on average for both traditional straight and U-shaped production lines. For individual cases, some unexpected results were found. For example, when there were two workstations, the improvements in problems 1, 5, 10 and 12 over a traditional straight production line were very obvious. The improvement is even better than for a U-shaped production line. Similar results can also be found in problem 13 with three workstations and problem 10 with four workstations. These results should encourage the managers of traditional straight production lines to try to balance the line by considering walking times if it is difficult to transfer the production lines into U-shaped production lines temporarily. Moreover, for problem 8 with four workstations, the improvements in U-shaped production lines when considering walking times are most obvious when the walking time of one unit is 5% of the average processing time.