Research on Train Loading and Unloading Mode and Scheduling Optimization in Automated Container Terminals

Abstract

In some automated container terminals, railway lines have been implemented into the port, saving container transfer time. However, the equipment scheduling level of the railway yard needs to be improved for managers. In the equipment scheduling of loading and unloading containers for railway trains, the operation modes “full unloading and full loading” and “synchronous loading and unloading” are often adopted. Due to the long length of the railway yard and the line of one train, there are two ways to arrange loading and unloading tasks for automated rail-mounted gantry cranes (ARMGs): one is to pre-assign tasks for ARMGs, and the other is to not pre-assign tasks for ARMGs. To investigate the efficacy of these different operation modes and methods of assigning tasks, this study formulated three mixed-integer linear programming (MILP) models with the goal of minimizing the ARMG task completion time. An adaptive large neighborhood search algorithm was used to tackle the scheduling problem. The scheduling effects of different operation modes and methods for assignment tasks were compared in terms of their calculation time and the completion time of ARMG tasks. Notably, the findings reveal that, with an increase in the number of tasks, the “pre-assign” task arrangement had a limited effect on the completion time of the ARMG tasks, made the calculation time shorter, and reduced the complexity of the problem. From the perspective of the completion time of ARMG tasks, the time under the “synchronous loading and unloading” operation mode was less than that of the “full unloading and full loading” operation mode. Therefore, it is recommended that the managers of the railway yard in an automated container terminal adopt the “synchronous loading and unloading” operation mode but determine the task assignment method according to decision time requirements. In addition, when the number of tasks is large, to decrease the time to complete ARMG tasks, the manager can adopt the “non-pre-assign” task distribution method.

1. Introduction

With the evolving global economic landscape, there is a notable increase in the proportion of container freight. However, a persistent challenge lies in the disparity between the growing capacity of rapidly developing ports and the infrastructure available to facilitate seamless transportation to and from inland hinterlands. Therefore, there is a pressing need for further optimization and enhancement of container transport systems linking ports with inland regions.

In recent years, China has witnessed a notable trend toward the automation of container terminals, with ports like Xiamen, Tianjin, and Shanghai undergoing transformations or new constructions to adopt automated systems [1]. The investment in automated container terminals (ACTs) has significantly improved the operational efficiency of ports. However, other logistic aspects of the port also need improvement, such as the port collection and distribution links. Road and railway transportation are the main methods of port collection and distribution. With the advantages of large cargo volume, fast transport speed, and independence from climate conditions, railway transportation can effectively improve the efficiency of port cargo collection and distribution. The railway yard is used to connect railway transport with shipping. Some ports have even implemented railway lines directly into ports, elevating the efficiency of sea–rail combined transport. Consequently, it is necessary to improve railway yard operations to increase the operational efficiency of container terminals. Nevertheless, there remains room for improvement in combined sea–land container transport infrastructure, transportation organization modes, and scheduling optimization levels.

In the daily equipment scheduling of loading and unloading containers by railway trains, after a train arrives at the RYACT, each train wagon can only hold one container. A train line is long, and multiple train wagons need to be loaded or unloaded at the same time. The operation mode of “full unloading and full loading” or “synchronous loading and unloading” is often adopted. Due to the long length of the railway yard and the line of one train, there are two ways to arrange loading and unloading tasks for ARMGs: one is to pre-assign tasks for ARMGs, and the other is not to pre-assign tasks for ARMGs. “Full unloading and full loading” means that ARMGs first operate multiple unloading tasks and then operate multiple loading tasks. “Synchronous loading and unloading” means that loading and unloading are carried out at the same time. However, because only one container can be placed in the same carriage, unloading before loading is adopted for carriages with both loading and unloading. The “pre-assign” task arrangement method means that tasks are assigned to each ARMG of the RYACT in advance according to the number of tasks. The “non-pre-assign” task arrangement method means that tasks are not assigned in advance but are adjusted in real time according to the position of the task and the ARMG position.

This study focused on investigating the effects of these different operation modes and methods of assigning tasks using the sea–rail intermodal storage yard layout of the Qinzhou U-shaped ACT. The MILP model was established with the goal of minimizing the ARMG task completion time for “full unloading and full loading” and “synchronous loading and unloading” with the “pre-assign” task arrangement method, as well as the “non-pre-assign” task arrangement method under the “synchronous loading and unloading” operation mode, respectively. This study finally provides scheduling suggestions to the managers of the RYACT by comparing the scheduling effects of different operation modes and methods of assigning tasks. A conclusion drawn is that the time spent in the “synchronous loading and unloading” operation mode is less than the time spent in the “full unloading and full loading” operation mode from the perspective of the completion time for ARMG tasks. With an increase in the number of tasks, adopting the “pre-assign” task distribution method can shorten the computation time and reduce the complexity of the problem, but the ARMG’s maximum task completion time is longer. Synthesizing the findings of this study, we provide optimization strategies for the operation of the RYACT: managers can pre-assign tasks to ARMGs if there is not enough decision time and adopt the “synchronous loading and unloading” operation mode to enhance the overall efficiency and productivity of the railway yard. In addition, when the number of tasks is large, to decrease the time to complete ARMG tasks, the manager should adopt the “non-pre-assign” task distribution method. The main contributions of this research are described below:

• In this study, the “full unloading and full loading” and “synchronous loading and unloading” operation modes in RYACT train loading and unloading were researched. To fill a key research gap in the existing literature, an MILP model was established with the objective of minimizing the maximum completion time of ARMG tasks, and the scheduling effects of the two modes were compared.
• An MILP model was established with the objective of minimizing the maximum completion time for ARMG tasks in the case of “non-pre-assign” task arrangement under the “synchronous loading and unloading” mode. It was then compared with “pre-assign” task arrangement in terms of the calculation and completion times.
• The models, solution method, and experimental results presented in this paper have good reference value for managers of the RYACT, helping them to make decisions on equipment scheduling in the RYACT.

The remainder of this paper is organized as follows: Section 2 provides a literature review. The problem is described and modeled in Section 3. The solution approach is introduced in Section 4. In Section 5, the different operation modes and task distribution methods are compared using experiments. Section 6 provides a discussion, and conclusions are drawn in Section 7.

2. Literature Review

In the Section 1, the situation of a railway yard being built into an automated container terminal and the operation mode and task assignment methods in the RYACT were introduced. In this section, a survey on the methods of improving the operational efficiency and optimization of berths, port yards, and railway yards in ACTs found in the literature is presented. With container throughput increasing year by year, many scholars and port managers have researched various aspects of container terminals, especially ACTs. These research branches include equipment scheduling problems, terminal layout design, and battery charging strategies. There is a wide variety of machinery involved: ship handling can use a quay crane (QC) to load and unload containers. In practice, automated guided vehicles (AGVs), intelligent guided vehicles, and autonomous straddle carriers are used for road horizontal transport. Meanwhile, automatic stacking cranes (ASCs), ARMGs, double cantilever rail cranes, and rail gantry cranes (RGCs) are responsible for yard stacking and picking up.

The logistic process of ACTs includes ship loading and unloading in the berth, road horizontal transportation, and yard stacking and picking up. In the berth section, Vacca et al. [2] presented a mathematical optimization model, along with a branch-and-price algorithm, to solve the berth assignment and QC scheduling problem. Zhen et al. [3] investigated crane assignment problems for a type of ACT system founded on multi-story frame bridges. Iris and Lam [4] used an adaptive large neighborhood-based heuristic framework to address the QC assignment problem for recoverable robustness in weekly berth schedules. Tan et al. [5] analyzed the automated QC scheduling problem by examining the connection between energy consumption and operation efficiency. In these studies, the QC carried out the loading task after completing the unloading operation of the ship, which is also the actual operation of the mechanical scheduling in the berth section.

In the horizontal transport link, Cai et al. [6] focused on exploring replanning schemes for large-scale autonomous straddle carrier problems under the situation of new task arrivals at random times. Choe et al. [7] presented an online preference learning algorithm that can dynamically adjust the strategy for scheduling AGVs in response to environmental changes. Luo et al. [8] studied container storage and AGV scheduling problems. To solve the problem of path optimization and obstacle avoidance of AGVs, Wu et al. [9] established an object-oriented timed colored stochastic Petri net algorithm. Xiang and Liu [10] researched the battery replacement strategy for AGVs and established a queueing network model to minimize the annual cost. Li et al. [11] designed a simulation-based ant colony optimization algorithm and established a two-stage stochastic programming model for the joint scheduling of battery replacement and random task operations. Drungilas et al. [12] studied the energy-saving problem of AGVs and proposed a deep reinforcement learning method to dynamically control the AGV speed according to path conditions.

In terms of yard equipment scheduling, Gharehgozli et al. [13], Yang et al. [14], and Vallada et al. [15] studied yard cranes (YCs) and yard resource scheduling problems. Hu et al. [16], Lu and Wang [17], and Oladugba et al. [18] solved the sequencing and scheduling problem of twin ASCs. Chu et al. [19] concentrated on the scheduling of three YCs in two blocks. Han et al. [20] and Feng et al. [21] studied the storage space allocation problem for containers arriving in the future and built a nonlinear mathematical model. To synchronize the production of ASCs, Gao et al. [22] used digital twin technology and established a model aimed at optimizing the production of ASCs with the goal of minimizing total energy consumption. In these studies, the YC of the yard is operated according to the principle of sequential operation because the type of YC operation is related to the transportation needs of external vehicles.

For multi-mechanical systems, Skaf et al. [23] and Kong et al. [24] conducted joint scheduling studies on ship loading, unloading, and horizontal handling. Considering the factors of traffic congestion and conflicts, Kong et al. [24] studied the joint scheduling problem of AGVs and tandem QCs and solved the model using a multi-start local search algorithm. Luo and Wu [25], Chen et al. [26], and Liu et al. [27] studied the integrated scheduling of YCs and AGVs. The joint scheduling of ship loading and unloading, horizontal transportation, and yard operation machinery were also studied [28,29,30]. Most of these studies take the total operation time or energy consumption of the machine as the goal, and this study also takes the task completion time of ARMGs as the goal for comparison.

From the above literature, it is evident that mechanical scheduling, path planning, charging management, and other aspects of ACTs have been widely studied. However, research on the scheduling of railway central stations in ACTs is not comprehensive enough. Anghinolfi et al. [31] optimized the operation of loading containers into wagon slots to maximize train loading. Based on the U-shaped yard layout in a sea–rail intermodal container terminal, Liu et al. [27] introduced a bi-level programming model with the aim of minimizing the makespan and total waiting time of trucks and AGVs. Chang et al. [32] researched the joint scheduling of equipment between the railway yard and dockyard in ACTs. Yang et al. [33] addressed the cooperative scheduling challenges among RGCs, AGVs, and YCs under the loading and unloading modes in sea–rail ACTs. Chen and Liu [34] introduced an adaptive large neighborhood search (ALNS) algorithm to tackle the “RYACT–train” cooperative optimization problem and verified its efficiency. Li et al. [1] considered safe distance constraints and non-crossing constraints to reflect intricate interactions among RGCs, double cantilever rail cranes, and intelligent guided vehicles (IGVs). Each train wagon can only hold one or two containers after a train arrives at the RYACT. A China–Europe train is equipped with about 45 wagons at a time. The loading and unloading lines in RYACTs are long, and ARMGs need to load and unload multiple wagons in a short period of time. The operation modes and methods for assigning ARMG tasks generated by these operational characteristics have been less studied.

The existing key literature on the different links in ACTs is summarized in

Table 1. Key publications in the literature on the different links in ACTs.

Literature	Solved Problem	Berth	Horizontal Transport	Container Yard	Railway Yard
Vacca et al. [2]	QC scheduling problem	√
Iris and Lam [4]	QC assignment problem	√
Luo et al. [8]	Container storage and AGV scheduling		√
Xiang and Liu [10]	Battery replacement strategy of AGVs		√
Drungilas et al. [12]	Energy-saving problem of AGVs		√
Gharehgozli et al. [13]	YCs and yard resource scheduling			√
Oladugba et al. [18]	Twin ASC sequencing and scheduling			√
Skaf et al. [23]	Joint scheduling		√	√
D. Anghinolfi et al. [31]	Optimization of the operation of loading containers to wagon				√
Chang et al. [32]	Joint scheduling of equipment		√	√	√
Yang et al. [33]	Cooperative scheduling		√	√	√

. As can be seen from Table 1, many aspects of ACTs have been widely studied to date, such as container storage [8] and the energy-saving problem of AGVs [12]; however, there has been very little discussion about the operation modes and methods for assigning ARMG tasks in the RYACT, which represents a research gap in this area. This study established MILP models for “full unloading and full loading” and “synchronous loading and unloading” with the “pre-assign” and “non-pre-assign” task arrangement methods. Experiments were conducted to compare the scheduling effects of different operation modes and methods for task assignment. Finally, this study provides scheduling suggestions to the managers of the RYACT based on the results. A comparative study of the operation mode and task distribution method is conducive to improving the equipment scheduling efficiency of the RYACT.

3. Problem Definition and Formulations

This section establishes models for “full unloading and full loading” with pre-assigned tasks, “synchronous loading and unloading” with pre-assigned tasks, and “synchronous loading and unloading” without pre-assigned tasks, respectively.

3.1. Problem Description

The innovative U-shaped ACT layout was implemented at Qinzhou Port and designed specifically for efficient sea–rail combined transport operations. The terminal, depicted in Figure 1, represents a novel approach to terminal design, emphasizing seamless container transfer between ships and trains [1]. Within the U-shaped ACT, the railway entry facilitates uninterrupted container handling, optimizing the interface between maritime and rail logistics. Notably, the storage yard adjacent to the container terminal aligns strategically with the railway central station, enhancing operational efficiency. The container area along the waterfront is organized vertically, with U-shaped lanes designated for IGVs and external trucks on either side. ARMGs are deployed for container retrieval and stacking within the railway yard.

The research described in this paper mainly focuses on a railway central depot. Due to the train operation line, the storage yard of the railway central station has a low stacking height and a long length, so ARMGs usually need to run a long distance, which causes interference between ARMGs. In task allocation, there are “pre-assign” and “non-pre-assign” methods. While the “full unloading and full loading” mode requires the sequential completion of all unloading tasks before loading commences, leading to potential underutilization of resources and increased idle times, the “synchronous loading and unloading” mode, although more efficient, introduces operational complexity due to the need for precise coordination of simultaneous tasks. This simultaneous coordination demands advanced planning and real-time adjustment capabilities to avoid conflicts and ensure smooth transitions between loading and unloading operations. Additionally, the dynamic prioritization of tasks in the “synchronous loading and unloading” mode complicates the scheduling algorithms, as it requires continuous reassessment of task urgency and resource availability.

3.2. Assumptions

To facilitate the modeling process in this study, the following key assumptions are made without hindering the applicability of the methodology to real-world situations:

• The loading and unloading of containers by the ARMG system does not include shifting activities.
• Containers that have been unloaded from the train will not impact subsequent loading operations.
• The operational speed of the ARMG system remains consistent throughout the modeling process.
• The precise starting and ending positions of container tasks, along with their corresponding coordinate values, are known and accounted for.
• The time required for the ARMG system to both grab and release containers is assumed to be equal, with the initial position of the ARMG system being predetermined.
• To ensure the safety of container handling operations, the trolley will remain stationary during the movement of the ARMG system’s gantry.

3.3. Parameter Description

The relevant sets, parameters, and decision variables of the model proposed in this study are shown in

Table 2. Sets for the model formulation.

Set	Definition
$S$	Set of ARMGs indexed by $s$.
$I_s^L$	Set of ARMG tasks for loading containers indexed by $i$ and $j$.
$I_s^U$	Set of ARMG tasks for unloading containers indexed by $i$ and $j$.
$I_s$	Set of ARMG tasks for loading and unloading containers that verify $I_s=I_s^L\cup I_s^U$.
$I_s^O$	Set of tasks for loading and unloading containers $I_s^O=I_s\cup \left\{O\right\}$, where $\left\{O\right\}$ represents a virtual start task; similarly, we can define sets $I_s^{LO}$ and $I_s^{UO}$.
$I_s^F$	Set of tasks for loading and unloading containers verifying $I_s^F=I_s\cup \left\{F\right\}$, where $\left\{F\right\}$ represents a virtual start task; similarly, we can define $I_s^{LF}$ and $I_s^{UF}$.
$C$	Set of tasks that may interfere with one another when adjacent ARMGs operate tasks, where $\left(i,j\right)\in C$.

,

Table 3. Parameters for the model formulation.

Parameter	Definition
$A_i$	The ARMG assigned to task $i$.
$\left({\alpha }_i,{\alpha }_j^{\prime }\right)$	A 0–1 integer parameter that represents the conflict between the nodes selected in the middle of the moment of tasks $i$ and $j$. When its value is 0, it is the middle time of ARMG operation in the RYACT, and when it is 1, it is the middle time of ARMG operation in the train carriage.
$\left({\beta }_i,{\beta }_j^{\prime }\right)$	A 0–1 integer parameter that represents the middle of the moment to avoid conflict between tasks $i$ and $j$. When its value is 0, it refers to the middle moment of the starting position, and when it is 1, it refers to the middle moment of the ending position.
$2T_i^B$	The operation time of the ARMG for grabbing/releasing containers in the RYACT.
$2T_i^R$	The operation time of the ARMG for grabbing/releasing containers on the train.
$T_i^S$	Operation task $i$ time of the ARMG for loading/unloading containers.
$T_{ij}^{Int}$	The minimum time interval between ARMG container tasks $i$ and $j$.
$T_{ij}^{ET}$	The time the ARMG moves from the end position of container task $i$ to the start position of container task $j$.
$T_i^G$	The moving time of the trolley during ARMG operation to complete task $i$.
$M_n$	A positive number that is large enough, where $n\in \left\{\mathrm{4,5},\dots ,10\right\}$.

and

Table 4. Decision variables for the model formulation.

Variable	Definition
$x_{ij}$	A binary decision variable that takes the value of $1$ if the ARMG starts container task $j$ immediately after completing container task $i$; otherwise, it takes the value of 0.
$z_{ij}$	A binary decision variable that takes the value of $1$ if task $i$ is completed before task $j$; otherwise, it takes the value of 0, $\left(i,j\right)\in C$.
$t_i^S$	A real variable representing the start time of task $i$.
$t_i^E$	A real variable representing the end time of task $i$.
$t_i^M$	A real variable representing the midpoint of the moment to grab or release the container for the ARMG operating conflict task $i$.

, respectively.

3.4. Model Formulation

In this section, the ARMG scheduling problem of the RYACT is modeled in three cases: “full unloading and full loading” with the “pre-assign” task arrangement method (M1), “synchronous loading and unloading” with the “pre-assign” task arrangement method (M2), and the “non-pre-assign” task arrangement method under the “synchronous loading and unloading” operation mode (M3).

Based on the above problem description and assumptions, the known input parameters of the problem include the number of ARMGs in a block, the number of unloading and loading container tasks, the origin and destination (OD) positions of these tasks, the moving speeds of the ARMGs and spreaders, and the container task assignment for each ARMG. Figure 2 shows the unloading, loading, and moving processes of the ARMG operating tasks, which correspond to three types of operation times: unloading time, loading time, and transferring time. The ARMG moving time between different block bays is the main component of these operation times. The time parameters can be calculated in advance [16].

Because the OD positions of the tasks are generally different, the ARMG needs to perform vertical operations at two block bays when handling a task. According to the OD position of the task and the task assignment for the ARMG, the set of interference tasks of the adjacent ARMG can be determined. Figure 3 shows the interference between the ARMGs for handling two tasks, and the position (origin position or destination position) of the interference can also be determined. Referring to the computing method proposed in the literature [16], the minimum interval time for avoiding interference can be computed in advance.

3.4.1. “Full Unloading and Full Loading” Mode

In this section, it is assumed that the ARMG for handling the inbound train is scheduled according to the “full unloading and full loading” mode; that is, all unloading tasks are handled first before the loading tasks. The model M1 is formulated as follows:

$$f^{Min}=\min \left(\underset{i\in I}{\max }{t}_i^E\right)$$

(1)

subject to

$$t_i^S\ge T_0+T_{Oi}^{ET},\forall i\in I_s^U,s\in S$$

(2)

$${\sum }_{i\in I_s^{UF}}{x}_{O,i}=1,\forall s\in S$$

(3)

$${\sum }_{i\in I_s^{UO}}{x}_{i,F}=1,\forall s\in S$$

(4)

$${\sum }_{i\in I_s^{UO},i\ne j}{x}_{ij}={\sum }_{i\in I_s^{UF},i\ne j}{x}_{ji}=1,\forall j\in I_s^U,s\in S$$

(5)

$${\sum }_{i\in I_s^U}{\sum }_{j\in I_s^L}{x}_{ij}=1,\forall i\in I_s^U,j\in I_s^L,s\in S$$

(6)

$$t_j^S\ge t_i^E+T_{ij}^{ET},\forall i\in I_s^U,j\in I_s^L,s\in S$$

(7)

$${\sum }_{i\in I_s^{LO}}{x}_{O,i}=1,\forall s\in S$$

(8)

$${\sum }_{i\in I_s^{LF}}{x}_{i,F}=1,\forall s\in S$$

(9)

$${\sum }_{i\in I^{LO},i\ne j}{x}_{ij}={\sum }_{i\in I^{LF},i\ne j}{x}_{ji}=1,\forall j\in I_s^L,s\in S$$

(10)

$$t_j^S\ge t_i^E+T_{ij}^{ET}+M_4\left(x_{ij}-1\right),\forall i,j\in I_s,i\ne j,s\in S$$

(11)

$$t_i^E\ge t_i^S+T_i^S,\forall i\in I_s,s\in S$$

(12)

$$z_{ij}+z_{ji}=1,\forall \left(i,j\right)\in C$$

(13)

$$\left\{\begin{array}{l}{t}_i^M=\left(1-{\alpha }_i\right)\left(t_i^S+T_i^B\right)+{\alpha }_i\left(t_i^E-T_i^R-{0.5\beta }_i{T}_i^G\right)\\ t_j^M=\left(1-{\alpha }_j^{\prime }\right)\left(t_j^S+T_j^B\right)+{\alpha }_j^{\prime }\left(t_j^E-T_j^R-{0.5\beta }_j^{\prime }{T}_j^G\right)\\ t_j^M-t_i^M\ge T_{ij}^{Int}{z}_{ij}+M_5\left(z_{ij}-1\right)\end{array},\forall \left(i,j\right)\in C\right.$$

(14)

$$x_{ij},z_{ij}\in \left\{\mathrm{0,1}\right\},\forall i,j\in I$$

(15)

$$t_i^S,t_i^E,t_i^M\ge 0,\forall i\in I$$

(16)

In order to improve the operational efficiency of the equipment in the RYACT, the objective of the problem is to minimize the makespan (the maximum completion time of all tasks handled by ARMGs). The objective function, outlined in Formula (1), serves as the guiding principle for achieving this goal, aiming to minimize the maximum latest completion time of all tasks. The ARMG unloading task phase is governed by a set of constraints, including Formula (2), which sets the start time of the initial task to $T_0=0$. Formula (2) dictates that the task’s start time is at least equal to the initial start time plus the operation time. In addition, the commencement and end of the unloading operation are established with virtual start and end tasks, respectively. Formula (3) dictates that the ARMG unloading operation begins with a virtual start task. Formula (4) dictates that the ARMG unloading operation ends with a virtual termination task. To ensure smooth operation, Formula (5) maintains flow balancing for ARMG tasks $j\in I_s^U$ in the unloading phase, while Formula (6) dictates the sequence, prioritizing unloading tasks over loading tasks.

Moving to the loading task phase, Formula (7) dictates that the start time of the initial loading task $i$ is equal to the end time of unloading task $i$ plus the interval of the ARMG from the end position of task $i$ to the start position of task $j$, while Formulas (8) and (9) introduce virtual start and end tasks to mark the beginning and end of the loading operation. Formula (8) dictates that the ARMG loading operation begins with a virtual start task. Formula (9) dictates that the ARMG loading operation ends with a virtual termination task. Similar to the unloading phase, Formula (10) enforces flow balancing for ARMG tasks $j\in I_s^L$ in the loading phase.

Further constraints, such as Formula (11), establish the relationship between consecutive tasks, ensuring continuous operation of the ARMG. The start time of task $j$ is equal to the end time of task $i$ plus the interval of the ARMG from the end position of task $i$ to the start position of task $j$ if $x_{ij}$ is equal to 1. Formula (12) dictates that the end time of task $i$ is equal to the start time of task $i$ plus the operation time of task $i$, while Formula (13) ensures that only one of $z_{ij}$ or $z_{ji}$ is 1. To prevent interference between tasks, Formula (14) is introduced, where the middle of the operating time $t_i^M$ for task $i$ is calculated based on the value of ${\alpha }_i$ and the different time periods $t_i^S, T_i^B$, $t_i^E$, and $T_i^R$, as well as ${\beta }_i{T}_i^G$. The time $t_j^M$ for task j is calculated based on the value of ${\alpha }_j^{\prime }$ and the different time periods $t_j^S, T_j^B$, $t_j^E$, and $T_j^R$, as well as ${\beta }_j{T}_j^G$; this constraint ensures a minimum interval time $T_{ij}^{Int}$ between tasks $i$ and $j$. Formulas (15) and (16) constrain the variable ranges to maintain feasibility.

3.4.2. “Synchronous Loading and Unloading” Mode

In this section, it is assumed that the ARMG for handling the inbound train is scheduled according to the “synchronous loading and unloading” mode; that is, all unloading and loading tasks are handled synchronously, with priority given to unloading for carriages with both tasks. Each train wagon can only hold one container; when a train wagon undergoes both unloading and loading, the ARMG should carry out the unloading task before the loading task. The model M2 is formulated as follows:

$$f^{Min}=\min \left(\underset{i\in I}{\max } t_i^E\right)$$

(1)

subject to

$$t_i^S\ge T_0+T_{Oi}^{ET},\forall i\in I_s,s\in S$$

(17)

$${\sum }_{i\in I_s^F}{x}_{O,i}=1,\forall s\in S$$

(18)

$${\sum }_{i\in I_s^O}{x}_{i,F}=1,\forall s\in S$$

(19)

$${\sum }_{i\in I_s^O,i\ne j}{x}_{ij}={\sum }_{i\in I_s^F,i\ne j}{x}_{ji}=1,\forall j\in I_s,s\in S$$

(20)

$$t_j^S\ge t_i^E+T_{ij}^{ET}+M_6\left(x_{ij}-1\right),\forall i,j\in I_s,i\ne j,s\in S$$

(11)

$$t_i^E\ge t_i^S+T_i^S,\forall i\in I_s,s\in S$$

(12)

$$z_{ij}+z_{ji}=1,\forall \left(i,j\right)\in C$$

(13)

$$\left\{\begin{array}{l}{t}_i^M=\left(1-{\alpha }_i\right)\left(t_i^S+T_i^B\right)+{\alpha }_i\left(t_i^E-T_i^R-{0.5\beta }_i{T}_i^G\right)\\ t_j^M=\left(1-{\alpha }_j^{\prime }\right)\left(t_j^S+T_j^B\right)+{\alpha }_j^{\prime }\left(t_j^E-T_j^R-{0.5\beta }_j^{\prime }{T}_j^G\right)\\ t_j^M-t_i^M\ge T_{ij}^{Int}{z}_{ij}+M_7\left(z_{ij}-1\right)\end{array},\forall \left(i,j\right)\in C\right.$$

(14)

$$x_{ij},z_{ij}\in \left\{\mathrm{0,1}\right\},\forall i,j\in I$$

(15)

$$t_i^S,t_i^E,t_i^M\ge 0,\forall i\in I$$

(16)

The objective functions of models M2 and M1 are the same. Constraints on the ARMG loading/unloading trains include constraint (17), which is used to limit the start time of the initial task of ARMG operation, and constraints (18) and (19), which indicate that the ARMG activity starts with a virtual start task and ends with a virtual end task, respectively. Formula (20) is used to limit task $j\in I_s$ ARMG flow balancing for tasks. The remaining constraints are the same as constraints (11)–(16) in model M1.

3.4.3. Considering “Non-Pre-Assignment” of Tasks

The assignment of ARMG tasks needs to be decided in the ARMG scheduling problem when considering the tasks that are not pre-assigned to ARMGs. At this point, a new interference set $C^{\prime }$ is calculated according to the OD position of the tasks, and then, the parameters $\left({\alpha }_i,{\alpha }_j^{\prime }\right)$, $\left({\beta }_i,{\beta }_j^{\prime }\right)$, and $T_{ij}^{Int}$ are updated. The new parameters for the model are as follows:

-

$C^{\prime }$ indicates the set of tasks that may conflict among ARMG tasks, where $\left(i,j\right)\in C^{\prime }$ indicates that if the start and end positions of tasks i and j overlap, they may interfere with each other when assigned to two ARMGs.

-

$r_i$ indicates the position coordinates of task i in the direction of the railway yard.

At the same time, the following decision variables are added to the expansion problem model:

-

$x_i^s=1$ indicates that container task $i$ is assigned to ARMG $s$ for operation; otherwise, $x_i^s=0$.

-

$x_{ij}^s=1$ indicates that ARMG $s$ starts task $j$ immediately after completing task $i$; otherwise, $x_{ij}^s=0$.

Based on model M2, adding ARMG task assignment constraints, an extended model of the equipment scheduling optimization problem of the RYACT under the “synchronous loading and unloading” mode was constructed, as shown in model M3.

$$f^{Min}=\min \left(\underset{i\in I}{\max }{t}_i^E\right)$$

(1)

subject to

$$t_j^S\ge t_i^E+T_{ij}^{ET},\forall i\in I_s^U,j\in I_s^L,s\in S$$

(7)

$${\sum }_{i\in I_s^{LO}}{x}_{O,i}=1,\forall s\in S$$

(8)

$$t_i^S,t_i^E,t_i^M\ge 0,\forall i\in I$$

(16)

$$t_i^S\ge T_0+T_{Oi}^{ET},\forall i\in I_s,s\in S$$

(21)

$${\sum }_{i\in I^O}{x}_{O,i}^s=1,\forall s\in S$$

(22)

$${\sum }_{i\in I^F}{x}_{i,F}^s=1,\forall s\in S$$

(23)

$${\sum }_{i\in I^O,i\ne j}{x}_{ij}^s={\sum }_{i\in I_s^F,i\ne j}{x}_{ji}^s=x_i^s,\forall j\in I,s\in S$$

(24)

$${\sum }_{s\in S}{x}_i^s=1,\forall i\in I$$

(25)

$$s{x}_i^s\le {kx}_j^k+M_8\left(x_i^s+x_j^k-2\right),\forall s,k\in S,i,j\in I,r_i<r_j$$

(26)

$$t_j^S\ge t_i^E+T_{ij}^{ET}+M_9\left({\sum }_{s\in S}{x}_{ij}^s-1\right),\forall i,j\in I,i\ne j$$

(27)

$$\left\{\begin{array}{l}{t}_i^M=\left(1-{\alpha }_i\right)\left(t_i^S+T_i^B\right)+{\alpha }_i\left(t_i^E-T_i^R-{0.5\beta }_i{T}_i^G\right)\\ t_j^M=\left(1-{\alpha }_j^{\prime }\right)\left(t_j^S+T_j^B\right)+{\alpha }_j^{\prime }\left(t_j^E-T_j^R-{0.5\beta }_j^{\prime }{T}_j^G\right)\\ t_j^M-t_i^M\ge T_{ij}^{Int}{z}_{ij}+M_{10}\left(z_{ij}-1\right)\end{array},\forall \left(i,j\right)\in C^{\prime }\right.$$

(28)

$$x_i^s,x_{ij}^s,z_{ij}\in \left\{\mathrm{0,1}\right\},\forall i,j\in I$$

(29)

The objective function of models M3 and M1 is the same. Constraint (21) is used to limit the start time of the initial task of the ARMG. Constraints (22) and (23) indicate that the activity of each ARMG starts with a virtual start task and ends with a virtual end task, respectively. Constraint (24) is used to restrict the ARMG flow balance for task $j\in I$. Constraint (25) is used to ensure that ARMGs are scheduled for each task. Constraint (26) is used to assign ARMG tasks to form individual operating blocks, ensuring that there is no overlap between blocks. Constraint (27) is used to determine the relationship between the ARMG completion time for $i$ and the ARMG job start time for $j$ when the ARMG is continuously working on tasks $i$ and $j$. Constraint (28) is used to prevent interference between adjacent ARMG operations. Constraint (29) limits the range of values of variables.

4. Solution Approach

Small-scale examples of MILP models can be solved using the solver CPLEX. To improve efficiency, intelligent algorithms are needed to solve the medium- and large-scale examples. The adaptive large neighborhood search (ALNS) algorithm was first proposed by Ropke and Pisinger in 2006 to solve the pickup and delivery problem with time windows [35] and has since been successfully applied to VRP and its variants [36,37,38]. Zhang et al. [39] and Zhang et al. [40] applied ALNS to solve preference-based multi-objective optimization and planning problems for synchromodal transport. Wang et al. [41] compared CPLEX and ALNS solution results for a tugboat scheduling problem. Chen and Liu [34] demonstrated the feasibility and effectiveness of ALNS in tackling medium- to large-scale instances of the “RYACT–train” cooperative optimization problem. This study also applied an ALNS algorithm to solve the problem. The technical parameters of ARMGs and the initial parameter settings of ALNS are the same as in [34]. The steps of the ALNS algorithm are outlined in Algorithm 1. A flowchart of the proposed ALNS algorithm is shown in Figure 4.

Algorithm 1 Adaptive large neighborhood algorithm
1:	Input data: $S,I,I_s, I_s^O, I_s^F, C, A_i, 2T_i^B, 2T_i^R, T_i^S, T_{ij}^{Int}, T_{ij}^{ET}, T_i^G$;
2:	Set algorithm parameters:
3:	The number of solutions generated per iteration: $N^P$;
4:	Maximum number of iterations: $N^G$;
5:	The maximum number of iterations with the same optimal value: $N^{St}$;
6:	The initial solution $Z$ is generated by RSA and its score $f_0$ is calculated;
7:	Initialize the data:
8:	Set iteration number $n=1$;
9:	The score $f_{nm}$ and sequence $Z_{nm}$ of the m code of the n iteration;
10:	Optimal score $f^{best}$ and corresponding sequence $Z^{best}$.
11:	While $n\le N^G$:
12:	Determine the list of domain search operators;
13:	Search the neighborhood of $Z^{best}$ to generate $N^P$ new encodings; $Z_{nm},\forall m\in \left\{1,\dots ,N^P\right\}$.
14:	For $m\in \left\{1,\dots ,N^P\right\}$:
15:	Use Algorithm 1 to calculate the score $f_{nm}$ of $Z_{nm}$;
16:	End for;
17:	Determine the optimal score $f^{best}\leftarrow \min \left\{f^{best},\underset{m\in \left\{1,\dots ,N^P\right\}}{\mathit{min}}\left\{f_{nm}\right\}\right\}$ and corresponding sequence $Z^{best}$;
18:	End while.
19:	Calculate $\left\{t_i^S,t_i^E,\forall i\in I\right\}$ according to the optimal sequence $Z^{best}$; $f\leftarrow f^{best}$.
20:	Return $f,t_i^S,t_i^E$.

Above, “synchronous loading and unloading” combined with the “pre-assign” task arrangement method (M2) was taken as an example to determine the number of ALNS iterations, which can be resolved according to the following experimental steps: (1) set the number of tasks to 40, 60, and 80 and the corresponding ARMG number to 4, 5, and 5, respectively; (2) set the number of iterations in the ALNS algorithm to 300, 400, …, 800; (3) for the combined calculation example of the number of tasks and the number of iterations, run the ALNS algorithm to solve the calculation example 10 times and return the minimum and average values of the 10 results. The minimum and average values of multiple iterations are shown in Figure 5. The experimental results show that when the scale of the example was 40, that is, when the scale of the example was small, changing the number of iterations basically did not affect the target value. When the scale of the example was 60 and 80, that is, with the number of iterations increasing, the minimum and average values decreased slightly. In the experiments, to compare the scheduling effect, the number of ALNS iterations was set to the same number, 600.

5. Numerical Experiments and Results

In this section, we specifically compare the operation mode and task distribution method in the RYACT. The experiments and algorithms were implemented in MATLAB R2017b software, and all the solutions were found using a computer with an Intel(R) Core(TM) i5-7200U CPU @ 2.50 GHz (Intel, Santa Clara, CA, USA), 8 GB RAM, and the Windows 10 operating system.

5.1. Comparison of “Full Unloading and Full Loading” and “Synchronous Loading and Unloading”

In this experiment, the operation efficiency of ARMGs under the two operating modes of “full unloading and full loading” (M1) and “synchronous loading and unloading” (M2) was compared in terms of the computational performance and objective function value. ALNS was used to solve the problem under different numbers of medium- and large-scale tasks. The specific steps of this experiment are as follows:

• Set the number of tasks to values ranging between 20 and 120, and generate corresponding examples.
• Keep the parameter settings of the ALNS algorithm the same as those in [34].
• For each numerical example of the number of tasks, solve the scenario 10 times and return the minimum value obtained.

The experimental results are shown in

Table 5. Experimental results under different operation modes (units of $f^{Min}$, $f^{Mean}$ and $Cpu$: seconds).

	Mode	Full Unloading and Full Loading (M1)			Synchronous Loading and Unloading (M2)
Number of Tasks		${\mathit{f}}^{\mathit{M}\mathit{i}\mathit{n}}$	${\mathit{f}}^{\mathit{M}\mathit{e}\mathit{a}\mathit{n}}$	$\mathit{C}\mathit{p}\mathit{u}$	${\mathit{f}}^{\mathit{M}\mathit{i}\mathit{n}}$	${\mathit{f}}^{\mathit{M}\mathit{e}\mathit{a}\mathit{n}}$	$\mathit{C}\mathit{p}\mathit{u}$
20		2830.07	2867.49	2.68	2654.36	2753.84	2.31
40		2615.44	2661.64	4.57	2518.30	2566.37	4.70
60		3557.49	3690.65	4.89	3417.90	3535.45	4.57
80		4750.69	4993.32	6.27	4661.40	4910.53	5.48
100		6009.37	6291.13	5.68	5949.71	6190.91	5.18
120		7581.89	7724.49	8.25	7374.37	7593.14	7.08
140		8620.06	8999.28	7.61	8473.79	8946.89	7.33
160		9906.03	10,384.06	7.96	9736.71	9943.30	7.58

.

From the experimental results in Algorithm 1, ARMG operation was more efficient with less time spent in the “synchronous loading and unloading” operation mode. From the perspective of the completion time of ARMG tasks, the completion time under the “synchronous loading and unloading” operation mode was less than that of the “full unloading and full loading” operation mode. The calculation time of the ALNS algorithm corresponding to the two operation modes was almost equal. Therefore, it is recommended for the managers of the RYACT to adopt the “synchronous loading and unloading” operation mode for ARMG scheduling.

5.2. Comparison of the Task Distribution Methods

In this experiment, the scheduling efficiency of ARMGs under the two distribution methods of “pre-assign” (M2) and “non-pre-assign” (M3) task arrangement was compared. To verify the validity of the models and solve them, the solver CPLEX was used to figure out the models using small-scale examples. To improve efficiency, an ALNS algorithm is used to solve medium- and large-scale examples. The specific steps of this experiment are as follows:

-

Set the number of tasks from 10 to 20 and generate corresponding examples.

-

Limit the model solution time to 1200 s.

-

Solve the models M2 and M3 using the solver CPLEX and return the experimental results.

-

Set the number of tasks to 40, 60, 80, …, 160, and generate corresponding examples.

-

Solve the models M2 and M3 using an ALNS algorithm with the same number of iterations (600) and return the experimental results.

-

Compare the experimental results between the two models. The experimental results are shown in

Table 6. Experimental results under different task distribution methods (units of $f^{Min}$ and $Cpu$: seconds).

Number of Tasks	“Pre-Assign” Method (M2)		“Non-Pre-Assign” Method (M3)
	${\mathit{f}}^{\mathit{M}\mathit{i}\mathit{n}}$	$\mathit{C}\mathit{p}\mathit{u}$	${\mathit{f}}^{\mathit{M}\mathit{i}\mathit{n}}$	$\mathit{G}\mathit{a}{\mathit{p}\left({\mathit{f}}^{\mathit{M}\mathit{i}\mathit{n}}\right)}^{ \mathit{a}}$	$\mathit{C}\mathit{p}\mathit{u}$	$\mathit{G}\mathit{a}{\mathit{p}\left(\mathit{C}\mathit{p}\mathit{u}\right)}^{ \mathit{a}}$
10	1623.18	0.11	1512.67	−6.80	3.00	2627.27
11	1607.07	0.20	1510.14	−6.03	35.89	17,845
12	1683.74	0.29	1683.74	0	157.65	54,262.06
13	1766.60	0.06	1766.60	0	1201.23	2,001,950
14	1850.19	0.72	1850.19	0	1201.40	166,761.11
15	1960.76	7.79	1960.76	0	1233.00	15,727.98
16	2094.00	1.74	2094.00	0	1202.82	69,027.59
17	2125.84	27.44	—	—	>1200	—
18	2253.09	59.28	—	—	>1200	—
19	2334.37	84.63	—	—	>1200	—
20	2446.46	115.10	—	—	>1200	—
40	2508.77	4.79	2508.77	0	10.58	120.88
60	3514.99	4.27	3506.47	−0.24	9.66	126.23
80	4655.61	5.75	4601.04	−1.17	12.50	117.39
100	5949.71	5.40	5946.44	−0.05	11.80	118.52
120	7371.59	7.25	7286.47	−1.15	15.50	113.79
140	8477	7.37	8448.33	−0.34	15.74	113.57
160	9730.31	7.81	9380.31	−3.60	17.62	125.61

^a This value is calculated as follows: $Gap=\left(f\left(\mathrm{M}3\right)-f\left(\mathrm{M}2\right)\right)/f\left(\mathrm{M}2\right)\ast 100$; units: %.

.

As can be seen from Table 6, when the number of tasks was less than 12, the “non-pre-assign” task distribution method was slightly ahead of the “pre-assign” task distribution method in terms of the ARMG’s maximum task completion time. However, within the finite time, model M3 could only obtain feasible solutions for a number of tasks from 13 to 20 because its calculation time exceeded the limit of 1200 s. Based on the ALNS solution results, when the number of tasks was between 40 and 160, the “non-pre-assign” task distribution method was slightly ahead of the “pre-assign” task distribution method in terms of the ARMG’s maximum task completion time, but the computation time was multiplied. In addition, when the number of tasks was 80, 120, 140, and 160, the difference in the ARMG’s maximum task completion time between the two methods was much larger than the difference in the required computation time. The results show that, with an increase in the number of tasks, adopting the “pre-assign” task distribution method can shorten the computation time and reduce the complexity of the problem, but the ARMG’s maximum task completion time is longer.

6. Discussion

In this study, the operation mode and task arrangement method of ARMG scheduling in the RYACT were discussed, and a scheduling optimization model was established with the goal of minimizing the task completion time of ARMGs. The different links in ACTs have been widely studied; so far, however, there has been very little discussion about operation modes and methods for the assignment of ARMG tasks in the RYACT. This research provides a useful supplement to the scheduling optimization literature on ACTs. At the same time, it also provides a good reference for RYACT policymakers and managers when making scheduling decisions. After a train arrives at an ACT, multiple train wagons undertake loading and unloading tasks at the same time. These train wagon loading and unloading tasks are distributed along the train side in a long line. Thus, the ARMG operation line is long, which brings complexity to train loading and unloading task arrangement. The research results of this study suggest that managers adopt the “synchronous loading and unloading” operation mode to obtain higher operational efficiency and accelerate the completion of loading and unloading tasks. In addition, when the number of tasks is low, such as less than 11 tasks, the scheduling manager can adopt the “non-pre-assignment” task arrangement method to arrange tasks to obtain a lower total task completion time. When the number of tasks is from 11 to 20, as shown in this paper, the scheduling manager can adopt the “pre-assign” task arrangement method to arrange tasks to make scheduling decisions with a shorter computation time. When the number of tasks exceeds 20, it is recommended that the managers of the RYACT adopt the “non-pre-assign” task distribution method to reduce the ARMG’s maximum task completion time if there is sufficient decision time. If there is not enough decision time the “pre-assign” task distribution method can be adopted to reduce the complexity of the problem. When the number of tasks is large, for example, more than 120, it is recommended that the managers of the RYACT adopt the “non-pre-assign” task distribution method because the difference in the ARMG’s maximum task completion time is much larger than the difference in the required computation time. This study provides a new theoretical perspective and actionable decision-making method for the operational management of RYACTs.

Although the methods proposed in this study provide valuable insights into the selection of operation modes and task arrangement methods for ARMGs in RYACTs, in some respects, if an ARMG operates other vehicles and trains in the port at the same time, the time window constraint of trains entering and leaving the RYACT is still beyond the scope of this study. These factors provide valuable directions for further research. Future work will benefit from incorporating more logistic actions and operational constraints into the RYACT to optimize the entire logistics process. In addition, improvements in intelligent algorithms for improving ARMG scheduling operations have also helped reduce the required computation time and improve decision-making efficiency.

7. Conclusions

We studied the operation modes of “full unloading and full loading” and “synchronous loading and unloading” and two methods for ARMG task arrangement in the RYACT. By establishing MILP models with the objective of minimizing the maximum completion time for ARMG tasks, the scheduling effects of different operation modes and methods for task arrangement were compared. The experimental results showed that with an increase in the number of ARMG tasks, the “pre-assign” task distribution method has a limited effect on the completion time of ARMG tasks but shortens the calculation time and reduces the complexity of the problem. From the perspective of the completion time of ARMG tasks, the completion time under the “synchronous loading and unloading” operation mode was less than that of the “full unloading and full loading” operation mode. Therefore, it is recommended that the managers of an RYACT adopt the “synchronous loading and unloading” operation mode and pre-assign tasks to ARMGs if there is not enough decision time. In addition, when the number of tasks is large, managers should adopt the “non-pre-assign” task distribution method. The problems studied herein are common in RYACTs, and the proposed methods and results can be used to guide the operation of ARMGs in RYACTs. In the future, we will further study container transportation scheduling optimization under the “synchronous loading and unloading” mode and consider other transportation links in container sea–rail combined transportation to optimize the whole logistics system. Although the proposed method can be applied to the scheduling optimization of railway yards, container transfer between railway yards and terminals and the direct loading of containers between trains and AGVs were beyond the scope of this study. These factors provide valuable directions for future research. In addition, the energy consumption management of automated container terminal systems, large models for greater freight volumes, and intelligent algorithms for improving ARMG scheduling operations are directions for future research.