## 6 Conclusions  
This research dealt with a U-shaped assembly line balancing problem, in which the assembled items are fixed on the conveyor. Operators have to follow the conveyor while executing the assembly tasks assigned to them. After completing the assembly jobs on one item, the operators then walk to another point to execute the assembly jobs on other items. Therefore, both assembly times and walking times are included in the cycle time.  
For the proposed U-shaped assembly line balancing problem, a mixed-integer programming model was first developed. The developed model can be optimized by commercial software LINGO 17. However, the computation time increases rapidly when the number of tasks and workstations increases. Simulated annealing (SA) was then proposed for optimizing the proposed U-shaped assembly line balancing problem. As there are many parameters in the proposed SA algorithm, a full factor experiment was adopted to find the optimal parameters setting. According to the experimental results from small problems, the proposed SA and LINGO 17 find similar solutions, but the computing times required by the proposed SA are much shorter. In order to show the efficiency of the proposed methodology, six larger problems are also tested. The results show that the efficiency of all problems is higher than 94%. Moreover, the computation times are approximately 20 min when the number of tasks is 40. Therefore, the proposed SA is also efficient for problems with more tasks or workstations.  
Moreover, some research deals with assembly line balancing problems with stochastic assembly times. Although the variance in assembly time can be reduced and ignored when operators receive enough training for specific assembly tasks, the assembly times may vary even for well-trained operators and should be considered when optimizing the assignment of tasks to workstations. Moreover, interference between operators always reduces performance and should be avoided in any production line if possible. Checking all the results that were optimized by LINGO 17, interference was found in one case. This means that avoiding interference is not taken into account by the proposed mixed-integer programming model. Adding the influence of interference between operators into the proposed model and SA procedure could make it more applicable to real cases. This could suggest a future direction for work based on the present study.  
In addition, in the proposed SA, only one strategy was proposed for searching for new solutions. Adopting various strategies for searching for new solutions may enrich the search directions and increase the chances of finding better solutions. Developing more search strategies for the proposed SA algorithm could also be an opportunity for future research.  
Figure 14: The efficiency of the results of larger problems.  
**Author Contributions:** Conceptualization, Y.K. and W.-C.H.; methodology, Y.K, S.-H.C. and. T.Y.; software, S.-H.C. and W.-C.H.; validation, Y.K. and T.Y.; formal analysis, Y.K.; data curation, Y.K. and W.-C.H.; writing--original draft preparation, Y.K. and W.-C.H.; writing--review and editing, Y.K. and T.Y.; visualization, Y.K.; supervision, Y.K.; project administration, Y.K. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research received no external funding.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** The data that support the findings of this study are available from the corresponding author, upon reasonable request.  
**Conflicts of Interest:** The authors declare no conflict of interest.