## 5 Discussion  
The benefit of using U-shaped production lines has been demonstrated both in industry and in research. This research deals with the problem of U-shaped production line design and aims to minimize cycle times when given the number of operators. An integer programming formulation was developed to optimize the assignment of tasks to locations and locations to workstations. Unlike other related studies, the proposed integer programming formulation takes both precedence relationships of tasks and walking times between locations into consideration. The proposed integer programming formulation was then validated on 13 problems that have been reported in the literature. As the information about walking times was not provided in these problems, it was assumed that the walking times were a proportion of the average production times. The experimental results confirm that U-shaped production lines can reduce cycle times when compared with traditional straight production lines. The results also show that when walking times are taken into consideration for balancing production lines, the advantages of U-shaped production lines are more obvious, especially when walking times are longer.  
Computation time increases when the number of workstations or tasks increases. According to the experimental results, the computation times increase seriously for problems with more than 10 tasks and 4 workstations. When the demand has a small volume and a large variety, new products are always being requested by customers. Then, the proposed integer programming formulation needs to be optimized frequently. Therefore, an efficient heuristic, such as genetic algorithms, tabu search or simulated annealing, might be appropriate, and this may be an opportunity for future research. Then, more results can be compared and analyzed.  
The proposed integer programming formulation was validated on 13 problems in the literature. The 13 problems were not all originally proposed for U-shaped production lines. Therefore, only precedence relationships of tasks and task times were used for validation. In this regard, the information concerning layout and walking times was designed for this research. Therefore, it is difficult to compare the results with similar problems in the literature. Applying the proposed integer programming formulation to a real case and analyzing the results of a practical application would be future work of this research.  
Figure 10: Computation time.  
**Author Contributions:** Conceptualization, Y.K. and T.Y.; methodology, Y.K.; software, Y.K.; validation, Y.K., T.Y. and T.-L.H.; formal analysis, Y.K. and T.-L.H.; investigation, Y.K.; resources, Y.K.; data curation, Y.K. and T.-L.H.; writing--original draft preparation, Y.K. and T.Y; writing--review and editing, Y.K. and T.Y.; visualization, Y.K.; supervision, Y.K. and T.Y.; project administration, Y.K.; funding acquisition, Y.K. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.  
This work was supported in part by the Ministry of Science and Technology of Taiwan, Republic of China, under grant MOST-109-2221-E-006 -152 -MY3.  
The authors declare no conflict of interest.