## 4 Experimental Results  
The proposed integer programming formulation was validated on 13 problems in the literature. The corresponding information about the problems is summarized in Table 2. Because the walking times and layouts were ignored in previous studies, the present research adopted the ratios of average processing time as the walking time of one unit. All U-shaped assembly lines are laid out in a grid system, as illustrated in Figure 8a. The walking time between any adjacent locations is one unit, and the minimal walking time between both sides of a cell is two units. In this research, the walking time of one unit is the ratio of the average processing time, ranging between 0% and 20%. For example, in problem 1, proposed by [PERSON] [34], there are eight tasks, and the average processing time is 9.38. If one unit of walking time is 5% of the average processing time, the walking time between any adjacent locations is 0.469 (9.38 \(\times\) 5%). To ensure that the walking time between adjacent locations is one unit as in traditional straight production lines, when the number of tasks is even, the right-hand side of the cell is laid out as in Figure 8b. As all tasks are laid out in a grid system, the walking times between all pairs of locations can be easily calculated using Pythagoras' theorem. The purpose of layout design is to systematically generate problems that can be tested. A smaller ratio of the average processing time indicates a shorter walking time between locations. When the ratio of the average processing time is equal to zero, it means that walking times are not taken into consideration. Any real U-shaped production layout can be tested if the position of all locations (location sequence) and all walking times between any pairs of locations (\(d_{pq}\)) are given. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**No.** & **Source** & & **Number of Tasks** & **Average Processing Time** \\ \hline
1 & [PERSON] [34] & Figure 1 & 8 & 9.38 \\
2 & [PERSON] and [PERSON] [35] & Figure 1 & 12 & 5 \\
3 & [PERSON] and [PERSON] [26] & Figure 1 & 9 & 4.11 \\
4 & [PERSON] and [PERSON] [26] & Figure 2 & 10 & 4.1 \\
5 & [PERSON] and [PERSON] [26] & Figure 3 & 7 & 3.29 \\
6 & [PERSON] and [PERSON] [26] & Figure 4 & 11 & 4.18 \\
7 & [PERSON] et al. [2] & Figure 3 & 7 & 5.29 \\
8 & [PERSON] et al. [3] & Figure 1 & 10 & 3.6 \\
9 & [PERSON] et al. [23] & Figure 1 & 9 & 6.11 \\
10 & [PERSON] and [PERSON] [36] & Figure 3 & 8 & 0.38 \\
11 & [PERSON] and [PERSON] [36] & Figure 4 & 7 & 0.58 \\
12 & [PERSON] et al. [37] & Figure 1 & 8 & 0.28 \\
13 & [PERSON] et al. [37] & Figure 2 & 8 & 0.48 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Summary of experimental problems. Figure 8: Layout of the U-shaped assembly cells. (**a**): Number of tasks is odd; (**b**): Number of tasks is even. For the 13 problems, two, three and four workstations are tested using the software LINGO on an Intel Core i7 2.40 GHz notebook computer. Fortunately, the proposed integer programming formulation can be adopted to balance traditional straight production lines, because the position of all locations (location sequence) and all walking times between any pairs of locations are known. The location sequence and task sequence are illustrated in Figure 9. Therefore, this research also adopts the proposed integer programming formulation to balance traditional straight production lines with the consideration of walking times. A comparison between traditional straight production lines and U-shaped production lines is shown in Table 3.