## 1 Introduction
A model was established to integrate the loading sequence and a rehandling strategy under parallel operations of multi-quay cranes. An improved genetic algorithm was proposed to solve the model. [PERSON] and [PERSON] attempted to minimize the operational cycle time of a QC for unloading and loading containers in a ship bay [13]. They formulated a QCSP as a mixed-integer programming model and solved it using a hybrid heuristic approach. [PERSON] and [PERSON] compared various pooling strategies for constructing the pool [14]. Heuristic algorithms were evaluated in terms of the total delay time of QC operations and the total travel distance of AGVs. [PERSON] et al. studied an integrated optimization QCSP and yard truck scheduling problem (YTSP) in CTs [15]. A mixed-integer programming model was formulated and solved by a particle swarm optimization-based method. [PERSON] et al. extended the QCSP by considering the stability of containers and proposed a QCSP with stability constraints [16]. A bi-criteria evolutionary algorithm was proposed to solve it. Most of the above QC scheduling methods fail to consider the impact of uncertain factors, especially the tide fluctuation and manual operation, on QC transportation tasks, with a few exceptions ([PERSON], 2014 [17]; [PERSON] and [PERSON], 2018 [18]). Uncertainties can be good or bad for a transportation system ([PERSON] and [PERSON], 2021 [19]), and we analyze uncertainties in our model. The dispatch of YCs is another crucial part of CTs. Most researchers combine the scheduling of YCs and AGVs to formulate the model closer to reality. [PERSON] et al. investigated single YC scheduling to minimize the total tardiness of tasks and focus on the case with uncertain release times of retrieval tasks [20]. A two-stage stochastic programming model was proposed, and a genetic algorithm and a rule-based heuristic were developed to solve it, especially for large-scale instances. [PERSON] and [PERSON] studied the scheduling of a YC to perform a given set of loading and unloading tasks with different ready times, which minimizes the sum of the task waiting time [21]. A branch and bound algorithm was proposed to solve it. Some studies focus on a yard crane scheduling problem (YCSP) under various conditions. [PERSON] et al. studied a YCSP among different blocks in CTs to minimize cranes' total travel time and the total delayed workload [22]. It was formulatedas a mixed-integer programming model and solved by a branch-bound method. [PERSON] et al. addressed a YCSP with the uncertainty about the vessels' arrival time and volumes [23]. It optimized the total delay to the estimated ending time of all task groups based on a genetic algorithm-based framework combined with a three-stage algorithm. In CTs, a double-cycling scheduling model of QCs and YCs requires that a crane transports a container, unloads it, and returns by taking another container. Meanwhile, AGVs transport a container to its destination and bring another one back. At present, double-cycling strategies are used in QCSP, YCSP, YMSP, and sequencing the loading of the container (i.e., Container Sequencing Problem, CSP, which refers to sequencing the loading/unloading of the container in the yard [24]), as shown in Table 1. Most of these studies considering a double-cycling scheduling mode do not specifically consider the uncertainty of QC operation. Most of these studies do not obtain specific AGV scheduling, including which containers should be transported by each AGV and the sequence of transporting these containers by an AGV. In this work, a double-cycling model is proposed to realize the full load of AGVs by scheduling loading/unloading containers to minimize the waiting time of the AGVs. A hybrid particle swarm optimization (HPSO) algorithm with a penalty function is used to solve this issue. The rest of this paper is organized as follows. Section 2 describes a double-cycling model and proposes a container scheduling sequence based on an HPSO algorithm. Section 3 shows experimental results.