## 2 Materials and Methods
### AGV Double-Cycling
#### 2.1.2 Mathematical Model
\begin{table}
\begin{tabular}{l l} \hline \hline
**Symbol** & **Definition** \\ \hline \multirow{4}{*}{\(M\)} & \(M=\{m_{1},\ m_{2},\ \dots,\ m_{n}\}\) is a set of task points, \(\text{ where }m_{i}=\left\{P_{1}^{i},\ w,P_{2}^{i},T_{i}\right\}\), \(n\) is the number of task points. \\  & For containers to be unloaded from ships, \(P_{1}^{i}\) representsthe position of container ion the ship, \(w\)represents that \\  & container \(i\) will be loaded at yard \(w\), \(\text{ and }P_{2}^{i}\)represents the destination of container \(i\) in the yard. For containers \\  & to be loaded onto ships, \(P_{1}^{i}\)represents the destination of the container inthe ship, \(w\)represents that container \(i\) will be \\  & unloaded at yard \(w\), \(\text{ and }P_{2}^{i}\) represents the position of container \(i\)on the yard. \(T_{i}\) represents the LTQ of container \(i\); \\  & a set of unloading task points that represent unloading the containers in the ship onto the yard; \\  & a set of loading task points that represent loading the containers in the yard onto the ship; \\  & \(l\) & the number of AGSVs; \\  & \(a_{g}\) & the \(g\)-th AGV where \(1\leq g\leq l\); \\  & \(m_{b}\) & a virtual begin task point; \\  & \(m_{e}\) & a virtual end task point; \\  & \(V_{Q}\) & average movement speed of QC; \\  & \(\Delta t_{i}\) & average time of QC spreader grasping container \(i\); \\  & \(D_{i}^{j}\) & the distance that a QC moves to from the position of the AGV to the position of container \(i\); \\  & a time – interval conforming to the Gaussian distribution, including the minimum and maximum time taken by QC to \\  & \(C_{i}\) & transport container \(i\),  where \(Ci_{i}=\left(C_{1}^{i},\ C_{2}^{i}\right)\), \(C_{1}^{i}\) and \(C_{2}^{i}\) are the minimum time and maximum time for QC to transport \\  & container \(i\) transportation, respectively; \\  & moving speed of YC; \\  & the moving distance of the YC from the position of container \(i\) to the position of container \(j\). It represents the distance \\  & from the I/O point to the position of container \(j\) when \(i=0\); it represents the distance of the YC moved from the position \\  & of container \(i\) to the I/O point when \(j=0\); \\  & \(V_{y}\) & movement speed of spreader; \\  & the moving distance of the YC’s spreader from the position of container \(i\) to the position of container \(j\). It represents the distance of the \\  & \(D_{ij}^{y}\) & distance of the spreader moving from the I/O point to the position of container \(j\) if \(i=0\); it represents the distance of the \\  & \(S_{y}\) & the speed at which the YC’s spreader moves up and down; \\  & \(d_{i}\) & the distance of the YC’s spreader moves to grab the container \(i\); \\  & \(T_{i}^{d}\) & time taken by QC to transport container \(i\); \\  & \(T_{y}\) & time required by the spreader to transport a container at the I/O point; \\  & \(T_{i}^{l}\) & time is taken by YC to load the container \(i\); \\  & \(T_{i}^{ul}\) & time is taken by YC to unload the container \(i\); \\  & \(T_{g}\) & =\(\{t_{1},t_{2},\ \dots,t_{1\ |\ 1}\}\), where \(t_{g}\) is the waiting time of \(a_{g}\); \\  & \(H_{ij}\) & the empty running time of an AGV loading task point \(m_{i}\) after unloading task point \(m_{j}\) at the yard; \\  & \(T\) & the time interval between two consecutive AGVs exiting the quayside; \\  & \(T_{i}^{d1}\) & the waiting time before task point \(i\) is unloaded at the quayside; \\  & \(T_{i}^{d2}\) & the waiting time before task point \(i\) is loaded at the quayside; \\  & \(T_{i}^{l}\) & the waiting time before task point \(i\) is unloaded in the yard; \\  & \(T_{i}^{l}\) & the time taken for task point \(i\) to be unloaded in the yard; \\  & \(T_{i}^{s}\) & the waiting time before task point \(i\) is loaded in the yard; \\  & \(T_{i}^{t}\) & the time taken for task point \(i\) to be loaded in the yard. \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The symbols and their definitions. The double-cycling model considers not only the efficiency of the QC but also the waiting time of AGV.