## 2 Materials and Methods
### AGV Double-Cycling
#### 2.1.2 Mathematical Model
The objective function is the shortest waiting time for AGVs:  
\[\min\sum_{1}^{l}T_{g} \tag{1}\]  
where \(T_{g}\) is the waiting time of \(a_{g}\). The constraints are defined as follows:  
\[\sum\chi_{i}^{j}=1,\hskip 14.226378 pt\forall M_{i}\in M+\{m_{e}\} \tag{2}\]  
\[\sum\chi_{i}^{j}=1,\hskip 14.226378 pt\forall M_{i}\in M+\{m_{b}\} \tag{3}\]  
\[\sum\chi_{x\prime}^{x}=0,\hskip 14.226378 pt\forall M_{x},M_{x\prime}\in X \tag{4}\]  
\[\sum\chi_{y\prime}^{y}=0,\hskip 14.226378 pt\forall M_{y},M_{y\prime}\in Y \tag{5}\]  
\[\alpha^{k}=1,\forall\,k\in N_{m} \tag{6}\]  
\[\beta^{k}=1,\forall\,k\in N_{m} \tag{7}\]  
\[C_{1}^{i}=(\mu-2\sigma)+V_{Q}/D_{i}^{q}+\Delta t_{i} \tag{8}\]  
\[C_{2}^{i}=(\mu+2\sigma)+V_{Q}/D_{i}^{q}+\Delta t_{i} \tag{9}\]  
\[\min\sum_{1}^{l}T_{g} \tag{1}\]0  
\[T_{i}^{l}=\left\{ \begin{array}{ll}\frac{D_{(i-1)}^{V_{(i-1)}}}{V_{Y}}+T_{G }&,\ \gamma_{i}^{w}=0\\ T_{G}&,\ \gamma_{i}^{w}=1\end{array} \right. \tag{11}\] \begin{table}
\begin{tabular}{l l} \hline \hline
**Symbol** & **Definition** \\ \hline \(Q\) & particle swarm size; \\ \(T\) & current iteration index; \\ \(t_{max}\) & maximum iteration; \\ \(P\) & particle fitness; \\  & personal best position, a set of the coding schemes in our algorithm. The personal \\ \(pBest\) & best position means the optimal scheduling sequence for each particle in the \\  & algorithm iteration process; \\ \(gBest\) & global best particle. \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The symbols used in the algorithm and their definitions. \begin{table}
\begin{tabular}{l l} \hline \hline \(\chi_{j}^{j}=\left\{\begin{array}{ll}1&,\ \text{an AGV transports task point $j$ after task point $i$}\\ 0&,\ \text{otherwise}\\ \end{array}\right.\) \\ \(\alpha^{g}=\left\{\begin{array}{ll}1&,\ a_{g}\text{ is fully loaded to drive out of the yard}\\ 0&,\ \text{otherwise}\\ \end{array}\right.\) \\ \(\beta^{g}=\left\{\begin{array}{ll}1&,\ a_{g}\text{ is fully loaded to drive out of the quayside}\\ 0&,\ \text{otherwise}\\ \end{array}\right.\) \\ \(\gamma_{i}^{w}=\left\{\begin{array}{ll}0&,\ \text{the YC cannot be prepared in advance at yard $w$}\\ 1&,\ \text{the YC can be prepared in advance at yard $w$}\\ \end{array}\right.\) \\ \(\delta_{i}=\left\{\begin{array}{ll}1&,\ \text{task point $i$ is transported to the quayside}\\ 0&,\ \text{task point $i$ is transported to the yard}\\ 1&,\ \text{container $i$ is loaded at yard $w$}\\ 0&,\ \text{container $i$ is unloaded at yard $w$}\\ \end{array}\right.\) \\ \(\epsilon=\left\{\begin{array}{ll}1&,\ \text{an AGV loads containers at the same yard after unloading}\\ 0&,\ \text{an AGV loads containers at a different yard after unloading}\\ \end{array}\right.\) \\ \hline \end{tabular}
\end{table} 
Table 4: The decision variables and their definitions. \[T_{i}^{ul}=\left\{ \begin{array}{rl}max\bigg{(}\frac{D_{(i-1)i}^{V}}{V_{Y}}, \frac{D_{(i-1)i}^{y}}{V_{y}}\bigg{)}+2\frac{d_{i}}{C_{y}}+\max\bigg{(}\frac{D_{ \text{B}}^{V}}{V_{Y}},\frac{D_{\text{B}}^{y}}{V_{y}}\bigg{)}+T_{\text{G}},&\gamma _{i}^{w}=0\\ T_{\text{G}}&,&\gamma_{i}^{w}=1\end{array} \right. \tag{12}\]  
\[T_{i}=\sum_{0}^{i-2}T_{k}^{q}+C_{2}^{i-1}\,\ \delta_{i}=1 \tag{13}\]  
\[T_{j}^{i}=T_{j}^{q}+\frac{D_{i}^{q}}{V_{\text{Q}}} \tag{14}\]  
\[T_{i}^{q1}=T_{i-1}^{q1}+T_{i-1}^{q}+\frac{D_{i}^{q}}{V_{\text{Q}}} \tag{15}\]  
\[T_{i}^{q2}=T_{i-1}^{q2}+T_{i-1}^{q}+\frac{D_{i}^{q}}{V_{\text{Q}}} \tag{16}\]  
\[T_{i}^{1}=\left\{ \begin{array}{rl}0&,\gamma_{i}^{w}=1\\ \max\big{(}T_{i-1}^{3}+T_{i-1}^{4}-T,0\big{)}&,\varepsilon_{u}^{i-1}=1,\gamma _{i}^{w}=0\\ \max(T_{i-1}^{1}+T_{i-1}^{2}+\frac{D_{i-10}^{Y}}{V_{y}}-T,0),\varepsilon_{u}^ {i-1}=0,\gamma_{i}^{w}=0\end{array} \right. \tag{17}\]  
\[T_{i}^{2}=\left\{ \begin{array}{rl}T_{\text{G}}&,\gamma_{i}^{w}=1\\ T_{\text{G}}&,\varepsilon_{u}^{i-1}=1,\gamma_{i}^{w}=0\\ max\big{(}\frac{D_{(i-10)}^{V}}{V_{Y}},\frac{D_{(i-10)}^{y}}{V_{y}}\big{)}+T_{ \text{G}},\varepsilon_{u}^{i-1}=0,\gamma_{i}^{w}=0\end{array} \right. \tag{18}\]  
\[T_{i}^{3}=\left\{