# Heuristics for a Two-Stage Assembly-Type Flow Shop with Limited Waiting Time Constraints
## 5 Discussion and Conclusions  
This study investigated a scheduling problem in a two-stage assembly-type flow shop with limited waiting time constraints to minimize the makespan. For this problem, a mixed-integer programming formulation was provided to mathematically describe and find an optimal solution using CPLEX. If the limited waiting times are treated as zero time, we can also use the MIP for solving a no-wait constrained problem. This problem is NP-hard, which requires an excessive computational time to solve the problem optimally. Therefore, this study focused on heuristic algorithms to quickly find good solutions. Various heuristic algorithms were provided, including priority rule-based list scheduling methods, a modified NEH algorithm, a genetic algorithm, an iterated greedy algorithm, and a simulated annealing algorithm. In addition, this study suggested equations to calculate the completion times of jobs in a given sequence. Computational experiments demonstrated that IG and SA in small- and large-sized problems, respectively, were effective and efficient within a short computation time.  
This study had some limitations. First, this study considered only two stages and a single machine in the second stage. Next, the suggested problem assumed that all jobs are ready to process at time zero and considered the given and fixed information of jobs. In addition, because finding exact solutions for large problems requires a significant computation time or may be impossible, the solutions obtained by the heuristics were examined by comparing them with each other. These limitations necessitate further studies.  
Future studies may need to consider more general models, such as multiple stages and machines in each stage. In addition, it may be necessary to consider a problem in which jobs arrive at the first stage dynamically or a problem with stochastic processing times. Furthermore, the multi-objective of the throughput of the system and the due date-based measure may also be considered in future studies. Additionally, one may develop an exact algorithm or effective lower bounds to evaluate the solutions obtained by the heuristic algorithms. Finally, optimal solution properties may be necessary to reduce the solution space to be explored.  
Conceptualization, [PERSON] and [PERSON]; methodology, [PERSON]-Y.L.; software, [PERSON]-Y.L.; validation, [PERSON]-[PERSON]; formal analysis, [PERSON]-Y.L.; investigation, [PERSON] and [PERSON]-[PERSON]; resources, J.-Y.L.; data curation, J.-Y.L.; writing--original draft preparation, J.-Y.L.; writing--review and editing, [PERSON] and [PERSON]-[PERSON]; visualization, [PERSON] and [PERSON]; supervision, [PERSON]-Y.L.; project administration, [PERSON]; funding acquisition, [PERSON] and [PERSON]. All authors have read and agreed to the published version of the manuscript.  
This study was supported by the National Research Foundation of Korea (NRF) Grant Funded by the Korean Government (MSIT) (no. NRF-2020R1G1A1006268, NRF-2020R1G1A1099829). The APC was funded by NRF-2020R1G1A1006268.  
Not applicable.  
This data for the computational experiments are generated randomly and the method for generating data is written in Section 4 in the paper.  
The authors declare no conflict of interest.