## 6 Conclusions  
To address the challenge that existing heuristic algorithms and meta-heuristic algorithms cannot fundamentally solve the three-stage distributed assembly permutation flow shop optimization problem, this paper proposes a hybrid bat algorithm optimization algorithm based on variable neighborhood structure and two learning strategies to minimize the completion time of this problem. The algorithm designs a search-based and capture-based two populations to solve the difficult trade-off between convergence and diversity of the bat algorithm in solving the scheduling optimization problem. Moreover, by fully mining the population information, a new selection mechanism and a new velocity and location update strategy are designed to solve the difficult trade-off between exploration and exploitation when the bat algorithm solves the scheduling optimization problem. The Gaussian learning strategy and elite learning strategy are used to assist the whole population to jump out of the local optimal frontier. Simulation results show that IHBA can solve the optimization problem of three-stage distributed assembly permutation flow shop scheduling well, and performs better than existing algorithms in the literature.  
In future research, the algorithm is extended to more complex multi-objective optimization problems to further improve the efficiency of iterative search, and the proposed algorithm is applied to other scheduling problems.  
Conceptualization, [PERSON]; methodology, [PERSON]; software, Y.W.; validation, [PERSON] and J.Z.; formal analysis, [PERSON]; investigation, Y.W.; resources, [PERSON] and J.Z.; data curation, Y.W.; writing--original draft preparation, [PERSON]; visualization, [PERSON] and J.Z.; supervision, [PERSON] and J.Z.; project administration, J.Z.; funding acquisition, J.Z. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the Fundamental Research Funds for the Central Universities under Grant No. CUSF-DH-D-2018050 (18D310804).  
Not applicable.  
The benchmark of instances for the permutation flowshop scheduling problem can be found here: [[http://soa.iti.es/problem-instances](http://soa.iti.es/problem-instances)]([http://soa.iti.es/problem-instances](http://soa.iti.es/problem-instances)), accessed on 1 September 2021.  
We gratefully acknowledge the anonymous reviewers for their insightful comments on the manuscript.  
The authors declare no conflict of interest.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**CPU**} & \multicolumn{6}{c}{**Algorithms**} \\ \cline{2-7}  & & **VNDH12** & **VNDH22** & **VNDH32** & **VND3 BA** & **VND2 BA** \\ \hline \multirow{3}{*}{\(n\)} & 100 & 0.459 & 0.508 & 0.525 & 0.004 & 0.009 \\  & 200 & 1.836 & 1.786 & 1.798 & 0.007 & 0.027 \\  & 500 & 13.424 & 13.885 & 13.417 & 0.016 & 0.135 \\ \hline \multirow{3}{*}{\(m\)} & 5 & 4.765 & 4.591 & 4.326 & 0.008 & 0.028 \\  & 10 & 5.001 & 5.313 & 5.114 & 0.010 & 0.051 \\  & 20 & 5.953 & 6.274 & 6.300 & 0.011 & 0.091 \\ \hline \multirow{3}{*}{\(f\)} & 4 & 3.662 & 3.522 & 3.668 & 0.012 & 0.083 \\  & 6 & 4.993 & 5.610 & 5.615 & 0.008 & 0.050 \\  & 8 & 7.065 & 7.045 & 6.457 & 0.008 & 0.037 \\ \hline \multirow{3}{*}{\(s\)} & 30 & 5.321 & 5.836 & 5.763 & 0.007 & 0.049 \\  & 40 & 5.224 & 5.354 & 5.084 & 0.009 & 0.058 \\  & 50 & 5.173 & 4.989 & 4.894 & 0.012 & 0.064 \\ \hline \multicolumn{7}{c}{average} & 5.240 & 5.393 & 5.247 & 0.009 & 0.057 \\ \hline \hline \end{tabular}
\end{table}
Table 14: CPU time of VND method.