## 2 Materials and Methods
### AGV Double-Cycling
#### 2.1.1 Problem Description  
In a CT, QCs, YCs, and AGVs are, respectively, responsible for loading/unloading containers on ships, loading/unloading containers in the yard, and transporting containers between the quayside and the yard. This work proposes a double-cycling scheduling model. An AGV carries a container at the quayside, drives to the yard to unload it, takes another one from the yard, and returns to the quayside for unloading. Such a process is repeated to finish the handling of containers at CTs. First, we present the following assumptions:  
1. Two QCs are used for loading and unloading containers on ships. One is responsible for loading containers from an AGV to a vessel, and another QC is responsible for unloading containers from a vessel to an AGV. After unloading at the first QC, AGVs run empty to the other QC for loading and finally drive to the yard. After unloading  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Studies** & **Scheduling Problem** & \
\begin{tabular}{c} **Uncertainty** \\ **Considered?** \\ \end{tabular} &
\begin{tabular}{c} **AGV Scheduling Sequence** \\ **Obtained?** \\ \end{tabular} \\ \hline [PERSON] and [PERSON], 2015 [25] & QCSP with YCSP & No & Yes \\ [PERSON], [PERSON], [PERSON], and [PERSON], 2019 [26] & QCSP with CSP & No & No \\ [PERSON] et al., 2018 [27] & Integrated QCSP-YTSP & No & Yes \\ [PERSON] et al., 2015 [28] & QCSP with CSP & No & No \\ [PERSON] and [PERSON], 2015 [29] & QCSP with CSP & No & No \\ [PERSON] and [PERSON], 2010 [30] & YMSP & Yes & No \\ [PERSON], 2010 [31] & QCSP with CSP & No & No \\ Goodchild and Daganzo, 2006 [32] & QCSP & Yes & No \\ [PERSON], [PERSON], [PERSON], [PERSON] & Integrated QCSP-YTSP-YCSP & Yes & No \\ and [PERSON], 2021 [33] & Our Work & Integrated QCSP-YCSP & Yes & Yes \\ \hline \hline \end{tabular}
\end{table}
Table 1: The problems solved by using double-cycling strategies.  
at the yard, AGVs run to the next yard for loading and eventually go to the quayside. Notice that more QCs can be divided into several pairs.
2. Tidal fluctuation and other factors will cause ship instability, resulting in QC operations not being fully automated. At present, most QC operations need manual assistance, resulting in the uncertainty of QC operation time. [PERSON] mentions that the QC operation time roughly conforms to Gaussian distribution [34]. The probability is that the distributed value in (\(\mu-2\sigma\), \(\mu+2\sigma\)) is 0.9544, so we assume that QC completes the transportation of containers within the time (\(t+\mu-2\sigma\), \(t+\mu+2\sigma\)), where \(t\) is the QC operation time.
3. AGVs travel at a uniform speed during their transportation.
4. When transporting a container, the YC and the YC's spreader are required to reach the position of the container. To save time, YC can be prepared before the arrival of an AGV [35], adopted in our work.  
We set up two fixed bays on the ship for storing containers that need to be unloaded and loaded, respectively. Notice that more bays can be divided into several pairs. AGVs are fully loaded in and out of the yard and quayside, thus realizing the double-cycling of AGVs. It is the basic constraint of the model. Figure 1 show the flow chart of this work.  
The second constraint is that no delay of QCs should be guaranteed. QC operational uncertainty leads to an uncertain time when an AGV arrives at the quayside. The latest time required for an AGV to arrive at the quayside (LTQ) is considered. Because the time of QC transporting containers conforms to the Gaussian distribution with parameters (\(\mu\), \(\sigma^{2}\)), LTQ is the moment when the QC finishes the transportation of the previous container.  
Figure 1: A double-cycling AGV scheduling mode.  
To make the algorithm and model more concise, we chose task points to represent the transport of containers. A task point represents a container with its current position and destination information. We give a detailed definition of the task point in Section 2.1.2.