## 1 Introduction  
Maritime transportation plays a vital role in the global economy, trade, and cultural exchanges as a basic logistic mode. Container terminals (CTs) were developed as one of the most critical components in logistic transportation systems. CTs consist of three major facilities: quay cranes (QCs), yard cranes (YCs), and automated guided vehicles (AGVs). AGVs cooperate with QC and YC operations. Effective task scheduling of AGVs is the key to reducing the operating cost of CTs. In the terminal's traditional transportation mode, it is essential to ensure the working efficiency of the quay crane. Most terminals use more AGVs to avoid waiting for the quay crane. However, this will increase the waiting time of AGVs, thus reducing their utilization rate. Furthermore, the complex constraints among quay cranes, yard cranes, and AGVs further deteriorate the utilization of AGVs. Many studies focused on the real-time scheduling of AGVs over the past few decades. For example, [PERSON] and [PERSON] proposed a mathematical model composed of a job shop scheduling problem and a conflict-free routing problem [1]. The objective was to minimize AGVs' delay time and achieve multi-AGV integrated scheduling. A two-stage ant colony algorithm was proposed to solve it. [PERSON] and [PERSON] studied job assignments of AGVs in CTs under various uncertainty conditions [2]. A flexible algorithm for real-time AGV scheduling was proposed. [PERSON] et al. designed an algorithm to prevent AGVs from collisions based on labeled Petri nets [3]. [PERSON] et al. studied the combinatorial optimization of two problems in the synchronous loading and unloading operation mode of the automated container terminal. They designed two bi-level optimization algorithms based on a conflict resolution strategy [4]. [PERSON] and [PERSON] investigated the AGV dispatching and routing problem with multiple bidirectional paths to generate conflict-free routes. A mixed-integer programming model was formulated to minimize the completion time of all jobs. A tailored branch-and-bound algorithm was developed to solve the problem [5]. [PERSON] et al. studied a conflict-free AGV path planning problem and rail-mounted gantry crane scheduling problem [6]. In terms of AGVs' offline path planning, [PERSON] et al. showed how timed Petri nets can be efficiently used to solve problems related to resource planning in intermodal freight transport terminals. They considered a real case study and simulated it based on the timed Petri net model of the terminal [7]. [PERSON] and [PERSON] formulated an AGV scheduling problem as a least-cost flow model [8]. They proposed an extended Network Simplex Algorithm (NSA+). [PERSON] et al. analyzed a container-cluster operation and an optimal scheduling strategy of the yard truck [9]. A multi-objective mathematical programming model was established and then solved by a genetic algorithm. [PERSON] et al. proposed a cooperative strategy for scheduling trucks to minimize the total delay time of requests and the travel time of yard trucks [10]. They applied a particle swarm optimization algorithm to solve the problem. The studies, as mentioned above, significantly improved the utilization of AGVs and the operational efficiency of CTs, but they cannot guarantee a full load and ensure the least total waiting time of AGVs during their transportation. Hence, we propose a double-cycling AGV route planning model in this work to maximize the loading rate and minimize the waiting time of AGVs during their transportation. In CTs, AGVs cooperate with the working of QCs and YCs. The work efficiency of QCs directly affects the berth time, which is also an important research topic. Many studies focused on an efficient quay crane scheduling problem (QCSP) under various complex conditions. [PERSON] and [PERSON] proposed a bounded two-level dynamic programming algorithm aiming at a QCSP [11]. A method was proposed to evaluate the practicability of the algorithm. [PERSON] et al. considered optimizing a loading sequence in CTs [12].