## 4 Conclusions  
To improve the utilization rate of AGVs in the process of CTs, this work proposes a double-cycling AGV scheduling model, which specifies that AGVs should enter and exit the yard and quayside with the full load during transportation. It reduces the empty running and minimizes the total waiting time of AGVs. The uncertainty of QC's operation time is also considered. The container scheduling sequence is obtained by arranging the loading and unloading of containers based on the HPSO algorithm with a penalty function. The double-cycling mode can be adopted whenever containers need to be loaded and unloaded simultaneously. If the numbers of loading task points and unloading task points are not equal, we can choose some of the task points and use the double-cycling mode. We can adopt a multi-QC transportation mode for other complicated cases, and the QCs are divided into groups by the divide-and-conquer strategy. Each group has two quayside cranes for loading and unloading, respectively. The proposed model and algorithm are still key to realizing double cycling. More heuristic algorithms [37; 38; 39; 40; 41; 42; 43; 44; 45; 46; 47] will be used to solve the proposed problem in our future work.  
Conceptualization, H.Z. and L.Q.; methodology, H.Z.; software, H.Z.; validation, H.Z., L.Q. and W.L.; formal analysis, H.Z.; investigation, H.Z.; resources, W.L.; data curation, H.Z.; writing--original draft preparation, H.Z.; writing--review and editing, L.Q.; visualization, H.Z.; supervision, L.Q.; project administration, H.M.; funding acquisition, L.Q. All authors have read and agreed to the published version of the manuscript.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Number of Task Points** & **GA** & **HPSO** & **SA** \\ \hline
12 & 90.19\% & 98.41\% & 88.23\% \\
16 & 86.27\% & 98.33\% & 80.39\% \\
20 & 84.31\% & 98.27\% & 75.55\% \\
24 & 76.47\% & 96.07\% & 68.62\% \\ \hline \hline \end{tabular}
\end{table}
Table 13: The success rate of the algorithm under different number of task points.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**(\(\mu\), \(\sigma\))** & **GA** & **HPSO** & **SA** \\ \hline
(4, 4) & 88.46\% & 98.38\% & 87.23\% \\ (4, 8) & 86.27\% & 98.50\% & 85.10\% \\ (8, 8) & 82.35\% & 98.30\% & 82.22\% \\ (8, 16) & 78.43\% & 98.14\% & 74.5\% \\ \hline \hline \end{tabular}
\end{table}
Table 14: The success rate of the algorithm under different \(\mu\) and \(\sigma\).  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Number of Task Points** & **GA** & **HPSO** & **SA** \\ \hline
12 & 25.96 s & 12.85 s & 32.76 s \\
16 & 28.87 s & 14.82 s & 41.13 s \\
20 & 42.65 s & 17.12 s & 69.50 s \\
24 & 65.42 s & 18.24 s & 113.38 s \\ \hline \hline \end{tabular}
\end{table}
Table 15: The average execution time of the algorithm under different number of task points.  
**Funding:** This work was supported in part by the National Natural Science Foundation of China under Grant 61903229, in part by the Natural Science Foundation of Shandong Province under Grants ZR2019 BF004 and ZR2019 BF0041, and in part by Jingying Program of Shandong University of Science and Technology under Grant skr19-D-049.  
**Institutional Review Board Statement**: Not applicable.  
**Informed Consent Statement**: Not applicable.  
**Data Availability Statement**: Not applicable.  
**Conflicts of Interest**: The authors declare no conflict of interest.