The Influence of Intelligent Guided Vehicle Configuration on Equipment Scheduling in the Railway Yards of Automated Container Terminals

Abstract

The efficiency of collecting and distributing goods has been improved by establishing railway lines that serve new automated container terminals (ACTs) and by constructing central railway stations close to ports. To aid in this process, intelligent guided vehicles (IGVs), which are renowned for their flexibility and for the convenience with which one can adjust their number and speed, have been developed to be used as horizontal transport vehicles that can transport goods between the railway yard and the front of the port. However, they also introduce some difficulties and complexities that affect terminal scheduling. Therefore, we took the automated rail-mounted container gantry crane (ARMG) scheduling problem as our main research object in this study. We established a mixed-integer linear programming (MILP) model to minimize the makespan of ARMGs, designed an adaptive large neighborhood search (ALNS) algorithm, and explored the influence of IGV configuration on ARMG scheduling through a series of experiments applied to a series of large-scale numerical examples. The experimental results show that increasing the number of IGVs can improve the operational efficiency of railway yards, but this strategy reduces the overall time taken for the ARMG to complete various tasks. Increasing or decreasing the speed of the IGVs within a given range has a clear effect on the problem at hand, while increasing the IGV travel speed can effectively reduce the time required for the ARMG to complete various tasks. Operators must properly adjust the IGV speed to meet the requirements of the planned operation.

1. Introduction

As sea–rail intermodal container transportation has developed, an increasing amount of attention has been paid to improving the efficiency of the connection between water transportation modes and railways. In some newly built ACTs, the long-standing physical barrier that has existed between ACTs and central container railway stations has begun to break down, and some of these ACTs have been amalgamated into the same customs supervision areas and operational areas to achieve integrated operations across regions; one such example is the Qinzhou ACT in China. As the time taken to process transfers of goods between railways and ports has decreased, the transportation of goods between railways and port terminals has become easier. Mobile machinery is responsible for the improved connection between railways and port terminals, and it is through this connection that containers are transferred between the railway yards and the front areas of ACTs. Studying the configuration of mobile machinery is of great significance to improve logistics, cost control, and the discharge of pollution in sea–rail intermodal transport.

The mobile machinery used in a typical ACT includes AGVs and IGVs. The “Magnetic rail + AGV” is the “standard configuration” for most ACTs. AGVs are required to bury tens of thousands of magnetic rails in the ground to facilitate automatic driving. However, this is a long process, and laying the magnetic rails incurs high costs. Compared with AGVs, IGVs are more flexible and can operate without the need for any markers. There is no need to make complex changes to the original layout of the railway yard as an IGV can travel without magnetic rails, making their paths flexible and changeable, and allowing them to freely travel around terminals. As IGV routes are flexible and changeable, their schedules can be adjusted according to actual production needs, and the accuracy of their positioning can reach ±1 mm. As autonomous driving and 5G communication technologies continue to develop, the broad prospects of the application of IGVs in the construction of new ACTs and the transformation of traditional container terminals become more apparent [1]. In recent years, IGVs have been introduced at the Yongzhou Terminal of Ningbo Zhoushan Port, the Nansha Phase IV Terminal of Guangzhou Port, and the Qinzhou ACT of Beibuwan Port. AGVs and IGVs are the main types of mobile machinery used to transport containers between the railway yards and front areas of automated container terminals (RYACTs), and this study takes Beibuwan Port’s Qinzhou ACT as an example. As the main horizontal transportation machinery used at the Qinzhou ACT, we chose IGVs as the object of our study and we investigated the influence of IGV configuration on equipment scheduling in the railway yards of automated container terminals. Importantly, research on IGV configurations has practical significance for controlling costs and preventing collisions in the railway yards of ACTs.

In this study, the Qinzhou ACT of Beibuwan Port was taken as a reference, and the coordinated scheduling of ARMGs and IGVs was taken as the main research object. To solve the problem, a MILP model was established, and a random search algorithm (RSA) and ALNS algorithm were designed. Through a series of experiments, the influence of the IGV configuration on ARMG scheduling in central railway depots was explored. The contributions of this paper are as follows: (1) A MILP model was established to solve the problem of coordinating the schedules of ARMGs and IGVs in the RYACTs. (2) An ALNS algorithm was similarly designed to solve this problem. (3) The influence of both the number of IGVs and their speed on ARMG scheduling was also discussed.

2. Literature Review

Several researchers have conducted studies on the mechanical scheduling of berths, storage yards, and RYACTs. Quay cranes (QCs) are positioned at the forefront of ACTs, and the equipment in railway yards generally includes yard cranes (YCs), ARMGs, and double-cantilever rail cranes or rail-mounted gantry cranes. The horizontal transportation vehicles include automated guided vehicles (AGVs), autonomous straddle carriers (ASDCs), and intelligent guided vehicles (IGVs).

Some scholars have studied single-machine scheduling problems from the perspectives of the order of operations, the allocation of storage space within the yard, and energy consumption. Chu et al. [2] studied multi-crane scheduling problems in adjacent container blocks. In a previous study, Luo et al. [3] focused on AGV scheduling and container storage problems. Xiang and Liu [4] built a nested semi-open queueing network model to consider AGV battery management. Cao et al. [5] studied AGV scheduling and bidirectional conflict-free path problems. Considering that the battery-powered AGV scheduling problem is time-intensive, Yang et al. [6] proposed a partitioning method based on depth-first and breadth-first searches to resolve the model. Oladugba et al. [7] designed an algorithm based on the modified Johnson algorithm to resolve the twin-ASC scheduling problem. Gao et al. [8] proposed a digital twin-based approach to optimize the operation of ASCs in terms of energy consumption.

The problem of the joint scheduling of vehicles in the wharf front, yard, and railway yard is also an important branch of mechanical scheduling research. Many authors have carried out research on the problem of the joint scheduling of ship loading/unloading and horizontal transportation [9,10,11,12,13]. Skaf, Lamrous, Hammoudan, and Manier [11] designed a heuristic algorithm and obtained superior results compared to the real results from a port in Tripoli. Liu, Zhu, Wang, and Wang [13] established a bi-level programming model to determine the paths of AGVs and obtain the YC scheduling scheme. Several authors have studied the problem of the joint scheduling of horizontal transportation and yard storage [14,15,16]. Wang, Hu, and Tian [15] studied the problem of scheduling both AGVs and ASCs by considering hybrid, buffer, and direct modes of transporting containers. Zhang, Li, and Sheu [16] researched the problem of the integrated scheduling of AGVs and double YCs in a one-yard block. Numerous authors have studied the problem of the joint scheduling of ship loading/unloading, horizontal transport, and yard storage [17,18,19,20,21,22,23,24,25,26,27]. Luo and Wu [22] established a model to ensure a minimum berthing time. Xu et al. [24] focused on conflict-free AGVs, dual-trolley QCs, and dual-cantilever rail cranes, proposing an integrated scheduling optimization model characterized by a U-shaped layout. Niu et al. [25] studied the problem of scheduling automated DCRCs, QCs, IGVs, and external trucks, aiming to reduce energy consumption. The problem of integrated scheduling for YCs, QCs, and AGVs has also been studied. Taking the ACT layout data of Yangshan Port as an example, Yue, Fan, and Fan [26] studied block allocation and scheduling. To solve the “QC-ASDC-ARMG” complex scheduling problem, Yu et al. [27] designed a particle swarm optimization algorithm based on adaptive tent chaos mapping. In order to reflect the complex interactions between pieces of terminal equipment such as rail gantry cranes (RGCs), IGVs, and double-cantilever rail cranes, Li, Yan, and Xu [28] included safe-distance and non-crossing constraints. To optimize the joint scheduling of ARMGs and external truck arrangements, He et al. [29] designed a study with multiple objectives, including balancing these arrangements across periods of time and among different blocks. To design a new concept based on multi-story frame bridges, Zhen et al. [30] studied how to schedule QCs and bridge cranes that transfer containers between different stories.

In summary, many researchers have studied the mechanical scheduling of key steps in the container transportation process such as ship loading/unloading and container stacking in ACTs from different perspectives, but few studies have been conducted on the combination of mobile machinery configurations and railway yard operations. Yang et al. [31] studied the joint scheduling of ARMGs and AGVs alongside the trains, yard, and ships to minimize the energy consumption within the port, and then obtained a reasonable ratio between ARMGs and AGVs. Yang et al. [32] proposed a strategy to adjust the number of AGVs to accommodate delays in ship arrivals and changes in the types of train containers, but did not discuss the effect of AGV speed on scheduling. Chen and Liu [33] studied the “RYACT–train” cooperative optimization problem. On the basis of this previous research, this paper takes the U-shaped layout at the Qinzhou ACT as a reference, the “IGV-RYACT–train” scheduling problem as the main research object, establishes a MILP model, and discusses the influence of IGV configuration on ARMG scheduling through a series of experiments.

3. Description of the Scheduling Problem

Traditional ACTs with a vertical block layout have ARMGs at both ends (the “land side” and “sea side”), so their civil engineering and equipment requirements are higher. However, the wharf in the Qinzhou Port area is long and deep, making it unsuitable for traditional technology; as such, Qinzhou Port adopted a “U-shaped” layout.

The “U-shaped” layout (Figure 1) facilitates traffic diversion; that is, through the vertical layout of the storage yard and the alternating “U”- and “I”-type roads, the traffic diversion, physical isolation, and mutual non-interference between the external trucks and the IGVs can enable the same storage yard to implement both land-side and sea-side collection and distribution operations, while ensuring the safety of these automated operations.

This paper explores the ARMG–IGV coordinated scheduling problem in an RYACT. ARMGs and IGVs are the main pieces of equipment used for the loading, unloading, and transporting of containers between trains, the RYACT, and the front area of the port. By using the arrival time of the train and the container handling information, the ARMG configured in the RYACT can operate between the inbound trains and the IGVs simultaneously. The ARMG’s operating processes include loading/unloading the trains and IGVs.

The IGV’s docking position within the IGV lane corresponds to the bay in which the container is stored in the RYACT, the main purpose of which is to decrease the movement time of the ARMG and enhance its efficiency. The ARMG’s working points include the location of the train carriage, the position of the container stored in the block, and the IGV docking position. The ARMG does not need to move within this bay when it is handling a container transported by an IGV.

4. Problem Formulation

In this section, a MILP model is formulated to minimize the makespan of ARMGs.

4.1. Assumptions

According to the above descriptions, the assumptions made in this study can be summarized as follows:

(1)

Repositioning or rehandling activities that are included in the ARMG’s operations are not taken into consideration;

(2)

Unloading containers from the train does not affect the subsequent loading operation;

(3)

The speed of the ARMGs is fixed;

(4)

The origin–destination (OD) locations of the containers handled by the ARMGs are known;

(5)

The time taken for the ARMG to grab a container is equal to the time it takes for it to release it;

(6)

To ensure operational safety, the trolley within an ARMG remains stationary while its gantry moves;

(7)

Congestion and IGV breakdowns are not considered.

4.2. Parameters and Decision Variables

The indices, parameters, and variables used in the proposed model are summarized in

Table 1. Indices used in the formulation of the model.

Index	Definition
$S$	Set of ARMGs in the RYACT, indexed by $s$
$V$	Set of IGVs used for transporting containers between yards, indexed by $v$
$I_s^R$	Set of container loading/unloading tasks to/from trains handled by ARMG $s$, indexed by $i$ and $j$
$I_s^V$	Set of container loading/unloading tasks to/from IGVs handled by ARMG $s$, indexed by $i$ and $j$
$I_s$	Set of container loading/unloading tasks handled by ARMG $s$, where $I_s=I_s^R\cup I_s^V$
$I$	Set of all container loading/unloading tasks handled by ARMGs
$I_s^O$	Set of loading and unloading tasks: $I_s^O=I_s\cup \left\{O\right\},$ where $\left\{O\right\}$ is a virtual starting task; $I_s^{VO}$ is defined similarly
$I_s^F$	Set of containers loaded and unloaded: $I_s^F=I_s\cup \left\{F\right\}$, where $\left\{F\right\}$ is a virtual ending task; $I_s^{VF}$ is defined similarly
$C$	Set of pairs of tasks in which two tasks handled by adjacent ARMGs are in conflict, i.e., $\left(i,j\right)\in C$ demonstrates that there is a conflict between the operating processes of two ARMGs that handle tasks $i$ and $j$, respectively

,

Table 2. Parameters used in the formulation of the model.

Parameter	Definition
$A_i$	The ARMG assigned to handle task $i$
$\left({\alpha }_i,{\alpha }_j^{\prime }\right)$	The location of the ARMG at the middle node for tasks $\left(i,j\right)\in C$; if ${\alpha }_i$ or ${\alpha }_j^{\prime }$ is 0, it denotes the middle of the operation in the yard block, and if ${\alpha }_i$ or ${\alpha }_j^{\prime }$ is 1, it denotes the middle of the operation in the train carriage
$\left({\beta }_i,{\beta }_j^{\prime }\right)$	The OD position in the middle of the ARMG’s operation time for tasks $\left(i,j\right)\in C$; if ${\beta }_i$ or ${\beta }_j^{\prime }$ is 0, they refer to the intermediate moment at the starting location of the task, and if ${\beta }_i$ or ${\beta }_j^{\prime }$ is 1, they refer to the intermediate moment at its ending location
$2T_i^B$	The time taken for an ARMG to grab/release containers at the yard block
$2T_i^R$	The time taken for an ARMG to grab/release containers at the train carriage
$T_i^S$	The time taken for task $i$ to be completed by an ARMG
$T_{ij}^{Int}$	The minimum interval between tasks $i$ and $j$ handled by ARMGs
$T_{ij}^{ES}$	The time taken for an ARMG to move from the end location of task $i$ to the starting location of task $j$
$T_i^G$	The moving time of the trolley as an ARMG completes task $i$
$T_i^V$	The operating time of task $i$ for an IGV to transport a container
$T_{ij}^{EV}$	The time taken for the IGV to move from the end location of the transport container task $i$ to the starting location of the transport container task $j$
$M_n$	A large enough positive number, $n\in \left\{2,3,4\right\}$

and

Table 3. Decision variables used in the formulation of the model.

Variable	Definition
$x_{ij}$	If $x_{ij}=1$, it demonstrates that an ARMG begins to operate task $j$ immediately after task $i;$ otherwise, $x_{ij}=0$
$z_{ij}$	If $z_{ij}=1$, it demonstrates that task $i$ is completed before task $j$, $\left(i,j\right)\in C$; otherwise, $z_{ij}=0$
$y_{ij}$	If $y_{ij}=1$, it demonstrates that an IGV starts transport task $j$ immediately after task $i;$ otherwise, $y_{ij}=0$
$t_i^S$	The starting time of task $i$ handled by an ARMG
$t_i^E$	The completing time of task $i$ handled by an ARMG
$t_i^B$	The starting time of task $i$ transported by an IGV
$t_i^D$	The completing time of task $i$ transported by an IGV
$t_i^M$	The middle node in the time taken for an ARMG to grab or release a container in task $i$

, respectively.

4.3. Model Formulation

This study investigated the problem of coordinating the schedules of ARMGs and IGVs, such that an ARMG can load/unload IGVs and trains simultaneously, aiming to improve the production efficiency of ARMGs. The time taken for an ARMG to complete the whole set of tasks should be as short as possible, so the goal of the model is to minimize the makespan of ARMGs. The total task completing time should be calculated according to the ARMG with the longest operation time. Therefore, Formula (1) was used as the objective function, and the model was constructed as follows:

$$[\mathrm{M}1] f^{Min}=\min \left(\underset{i\in I}{\max }{t}_i^E\right)$$

(1)

subject to:

$$t_i^S\ge T_0+T_{Oi}^{ES},\forall i\in I$$

(2)

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

(3)

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

(4)

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

(5)

$$t_j^S\ge t_i^E+T_{ij}^{ES}+M_2\left(x_{ij}-1\right),\forall i,j\in I_s,i\ne j,s\in S$$

(6)

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

(7)

$$t_i^B=t_i^E,\forall i\in I^V$$

(8)

$${\sum }_{i\in I^{VF}}{y}_{O,i}=\left|V\right|$$

(9)

$${\sum }_{i\in I^{VO}}{y}_{i,F}=\left|V\right|$$

(10)

$${\sum }_{i\in I^{VO},i\ne j}{y}_{ij}={\sum }_{i\in I^{VF},i\ne j}{y}_{ji}=1,\forall j\in I^V$$

(11)

$$t_j^B\ge t_i^D+T_{ij}^{EV}+M_3\left(y_{ij}-1\right),\forall i,j\in I^V,i\ne j$$

(12)

$$t_i^D\ge t_i^B+T_i^V,\forall i\in I^V$$

(13)

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

(14)

$$t_i^M=\left\{\begin{array}{l}\left(1-{\alpha }_i\right)\left(t_i^S+T_i^B\right)+{\alpha }_i\left(t_i^E-T_i^R-{0.5\beta }_i{T}_i^G\right), \mathrm{if} i\in I^R\\ {\displaystyle \frac{t_i^S+t_i^E}{2}}, \mathrm{if} i\in I^V\end{array},\forall \left(i,j\right)\in C\right.$$

(15)

$$t_j^M=\left\{\begin{array}{l}\left(1-{\alpha }_j^{\prime }\right)\left(t_j^S+T_j^B\right)+{\alpha }_j^{\prime }\left(t_j^E-T_j^R-{0.5\beta }_j^{\prime }{T}_j^G\right), \mathrm{if} j\in I^R\\ {\displaystyle \frac{t_j^S+t_j^E}{2}}, \mathrm{if} j\in I^V\end{array},\forall \left(i,j\right)\in C\right.$$

(16)

$$t_j^M-t_i^M\ge T_{ij}^{Int}{z}_{ij}+M_4\left(z_{ij}-1\right),\forall \left(i,j\right)\in C$$

(17)

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

(18)

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

(19)

Constraint (2) can li mit the starting time of the initial task handled by the ARMG, where $T_0=0$. Constraints (3) and (4) convey that the ARMG’s activity starts with a virtual start task and terminates with a virtual end task. Constraint (5) can keep the flow of task $j\in I_s$ balanced. When an ARMG is continuously performing tasks $i$ and $j$, Constraint (6) is used to ensure a minimum interval between the ARMG’s completion of task $i$ and the beginning of task $j$. Constraint (7) describes the connection between the ARMG beginning and completing task $i$. Constraints on the IGV transport containers include the following: Constraint (8) ensures that the completion of an ARMG task occurs at the beginning of an IGV transport task. Constraints (9) and (10) are used to determine the number of IGVs used in a joint scheduling problem. Constraint (11) is used to ensure tasks $j\in I^V$ flow balancing for IGVs. When an IGV is continuously performing transport tasks $i$ and $j$, Constraint (12) is used to ensure a minimum interval between the IGV’s completion of task $i$ and the beginning of task $j$. Constraint (13) describes the connection between the IGV beginning and completing transport task $i$. Constraint (14) represents the sequence in which any two tasks, i and j, should be finished by an ARMG. The intermediate nodes within the time taken to complete each task are calculated according to the types of tasks being carried out by the ARMG. Constraints (15)–(17) are used to avoid interference between adjacent ARMGs. Constraints (18) and (19) control the range of variable values.

5. Proposed Algorithm

The ARMG scheduling stage in the ARMG and IGV scheduling problem can be viewed as several traveling salesman problems, and the interference among the different operational cranes should be considered at the same time. The studied problem is NP-hard, and it is difficult to find exact solutions and for optimization solvers such as CPLEX to solve mid- and large-scale examples of this problem in an acceptable amount of time. Many different heuristic algorithms have been used to solve scheduling optimization problems in ACTs with mid- or large-scale examples, including a spatiotemporally greedy genetic algorithm [13], the cuckoo search algorithm [34], and a reinforcement learning-based hyper-heuristic genetic algorithm [24]. Heuristics based on large neighborhood search (LNS) have presented notable success in the fields of scheduling and transportation [35]. Ropke and Pisinger [36] adjusted LNS by defining several destroy-and-repair methods, and they named their algorithm ALNS. ALNS can adaptively select the destroy–repair operators within each iteration, and has been used to solve various routing problems [37,38,39,40,41,42,43], such as transportation planning and heterogeneous container loading problems. Considering the remarkable success of ALNS, we decided to apply the ALNS algorithm to our problem.

5.1. Encoding and Decoding Methods

The ARMG task sequence, that is, the order of the coding, is used to represent the relative order of the tasks handled by ARMGs, where the coded value represents the task’s sequence number and its position represents the task assignment. The ARMG task sub-sequence generation procedures are shown in Figure 2 according to a specific ARMG task sequence. Considering the ARMG task assignment, task types, and the operation mode of loading/unloading synchronization, the ARMG task sub-sequence for each ARMG can be generated. The AMRG’s task sub-sequence corresponding to the encoding sequence (i.e., the ARMG’s task sequence) is generated from the input. Finally, the decoding method for evaluating the ARMG task sequence can be designed according to the ARMG task sub-sequence that has been generated. In this process, the priority of all tasks handled by multiple ARMGs is determined by the ARMG task sequence to avoid any interference between adjacent ARMGs.

In the process of loading/unloading a train, the positions of ARMGs at the beginning of a task are not consistent with those at the end. At this time, each task performed by the ARMG has some key time nodes: the starting time ($t_i^S)$ and the completing time ($t_i^E$). During the loading or unloading of an IGV by an ARMG, the ARMG stays within its bay. However, to fully describe the process of an ARMG carrying out a task between the block and the IGV, the two critical time nodes, the starting time ($t_i^B$) and the completing time ($t_i^D$), need to be recorded. The earliest starting and completing times for the ARMG task, and the earliest starting and completing times for the IGV transportation task, can be calculated according to the descriptions of the problem and the model [M1]. Detailed steps of the proposed decoding method are shown in

Table 4. Decoding steps.

Step 1	Input data: $S,I,I_s,I_s^R,I_s^V,I_s^O,I_s^F,C,A_i,2T_i^B,2T_i^R,T_i^S,T_i^V,T_{ij}^{Int},T_{ij}^{ES},T_{ij}^{EV},T_i^G,Z$ Initialize the intermediate parameters:  No. of task position $n^Z=1$  The completion time of the virtual origin task $t_O^S=t_O^E=0$
Step 2	Generate the task sub-sequence $X_s$ of ARMG $s$ and the task assignment ($I_s^R$ and $I_s^V$) according to $Z$ and the task assignment
Step 3	While $n^Z<\left|I\right|$
Step 3.1	Ensure the insertion task $i$
Step 3.2	Calculate the earliest starting and completing time $\left\{t_i^{ES},t_i^{EE},\forall i\in I\right\}$ of task $i$ handled by an ARMG and the earliest starting and completing time $\left\{t_i^{EB},t_i^{ED},\forall i\in I^V\right\}$ of task $i$ transported by an IGV
Step 3.3	$n^Z\leftarrow n^Z+1$
	End while
Step 4	Ensure the score of the task sequence, $f=\underset{i\in I}{\max }\left\{t_i^{EE}\right\}$, and update the time node $t_i^S\leftarrow t_i^{ES}, t_i^E\leftarrow t_i^{EE}, t_i^B\leftarrow t_i^{EB}$ and $t_i^D\leftarrow t_i^{ED}$
Step 5	Return $f^{Score},t_i^S,t_i^E,t_i^B,t_i^D$

.

5.2. Random Search Algorithm

To verify the feasibility of encoding and decoding, and to search the solution space quickly, we used an RSA to optimize the joint scheduling of ARMGs and IGVs in the RYACT. The process of designing the RSA was similar to that described in our previous paper [33]. Therefore, IGV job sequence scoring was added. Finally, the problem’s target value and the ARMG and IGV operation time nodes were returned. The specific steps in the RSA are presented in

Table 5. RSA steps.

Step 1	Input parameter: $S,I,I_s,I_s^R,I_s^V,I_s^O,I_s^F,C,A_i,2T_i^B,2T_i^R,T_i^S,T_i^V,T_{ij}^{Int},T_{ij}^{ES},T_{ij}^{EV},T_i^G$ Set the maximum iteration number $N^{Max}$
Step 2	Initialize the intermediate parameters:  Set the iteration number $n=1$  The score $f_n$ of the $n-th$ sequence  Initialize the minimum score $f^{best}$ and its optimal sequence $Z^{best}$
Step 3	While $n\le N^{Max}$
Step 3.1	Randomly generate a task sequence $Z$
Step 3.2	Evaluate the task sequence using the method in Table 4 and return the score $f^{Score}$
Step 3.3	$f_n\leftarrow f^{Score}$  $n\leftarrow n+1$
Step 3.4	If $f_n<f^{best}$, then $f^{best}=f_n$ and $Z^{best}=Z$; otherwise, go to Step 3
	End while
Step 4	Calculate $\left\{t_i^S,t_i^E,t_i^B,t_i^D,\forall i\in I\right\}$ according to the optimal sequence $Z^{best}$ $f\leftarrow f^{best}$
Step 5	Return $f,t_i^S,t_i^E,t_i^B,t_i^D$

.

5.3. Adaptive Large Neighborhood Search

In this section, ALNS is applied to optimize the job sequence for the given initial coding sequence. The neighborhood search operator is set to search the neighborhood adaptively to generate a new ARMG task sequence. The ARMG task sequence is optimized iteratively, and the objective values and the schedules of the ARMGs and IGVs are returned.

(1)

Encoding and decoding method:

The encoding and decoding methods of the ALNS are set according to Section 5.1. The order coding shown in Figure 2 is used as the encoding method, and the decoding method is shown in Table 4.

(2)

Generating an initial solution:

An initial solution for this algorithm can be generated by the RSA with a given iteration number.

(3)

The pairs of destroy–repair operators for generating the neighborhood:

Operators include single-point reinsertion, fragment reinsertion, local reverse order, two-point exchange, fore-and-aft interchange, and regeneration [33].

(4)

The stopping criteria:

The algorithm will stop when the iteration number of the algorithm reaches the maximum number of iterations ($N^G$) or when the number of iterations in which the optimal value remains unchanged reaches the maximum number of stall score iterations ($N^{St}$) [33].

(5)

The algorithm’s steps:

The steps involved in ALNS are exhibited in

Table 6. The steps involved in ALNS.

Step 1	Input data: $S,I,I_s,I_s^R,I_s^V,I_s^O,I_s^F,C,A_i,2T_i^B,2T_i^R,T_i^S,T_i^V,T_{ij}^{Int},T_{ij}^{ES},T_{ij}^{EV},T_i^G$ Initialize the ALNS settings:  The number of encodings created per iteration $N^P$  The maximum iteration number $N^G$  The maximum iteration number with stall score $N^{St}$
Step 2	Adapt the RSA to generate an initial solution $Z$, and calculate its score $f_0$
Step 3	Initialize the intermediate parameters:  Set the No. of iterations $n=1$  The score $f_{nm}$ and sequence $Z_{nm}$ of the code $m-th$ encoding the $n-th$ iteration  Save the optimal score $f^{best}$ and its sequence $Z^{best}$
Step 4	While $n\le N^G$
Step 4.1	Ensure the list of destroy–repair operators
Step 4.2	Generate the neighborhood of $Z^{best}$ where it includes $Z_{nm},\forall m\in \left\{1,\dots ,N^P\right\}$
Step 4.3	For $m\in \left\{1,\dots ,N^P\right\}$
Step 4.3.1	Use Table 2 to calculate the $f_{nm}$ score of $Z_{nm}$
	End for
Step 4.4	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 $Z^{best}$
	End while
Step 5	Calculate $\left\{t_i^S,t_i^E,t_i^B,t_i^D\forall i\in I\right\}$ according to the optimal sequence $Z^{best}$ $f\leftarrow f^{best}$
Step 6	Return $f,t_i^S,t_i^E,t_i^B,t_i^D$

.

6. Computational Experiments

The experimental dataset was composed of randomly selected data from the original dataset of wharf operation tasks. The input parameters of the ARMGs and IGVs are listed in

Table 7. The ALNS settings [31].

Parameter	Value
The number of encodings per generation ($N^P$)	24
The maximum number of iterations ($N^G$)	500
The maximum number of stall score iterations ($N^{St}$)	100
The types of operators ($\left|R\right|$)	6
The initial weight values of operators ($w_r$)	50

, and the ALNS settings are presented in

Table 8. The input parameters [31] for ARMGs and IGVs.

Parameter	Value
The gantry movement speed of the ARMG	0.56 m/s
The trolley movement speed of the ARMG	2 m/s
The vertical operating time of the ARMG in the yard ($2T_i^B$)	80 s
The vertical operating time of the ARMG at the train ($2T_i^R$)	100 s
The IGV transport time ($T_i^V$)	$\mathrm{U}\left(180,240\right)$ s
The empty IGV time ($T_{ij}^{EV}$)	$\mathrm{U}\left(180,240\right)$ s

. The algorithm was implemented in MATLAB R2017b, and all the solution procedures were performed on a computer with an 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.

6.1. Comparison of Computing Performance of [M1], RSA, and ALNS

The experimental steps can be summarized as follows: (1) Set the number of tasks as 10, 12, …, 30, 40, …, 160 and create examples according to the experimental dataset. (2) Set the number of RSA iterations to $N^{Max}={10}^3$. (3) Set the ALNS parameters as shown in Table 7, and set other parameters as shown in Table 8. (4) For each set of tasks, solve the model [M1] via the solver CPLEX and obtain the optimal solution, and then run the RSA and ALNS to return the results. The experimental results are presented in

Table 9. Computing performance of [M1], RSA, and ALNS (the units for $f^{Min}$, $f^{Mean}$, and $Cpu$ are seconds).

No. of Tasks	[M1]			RSA			ALNS
	${\mathit{f}}^{\mathit{M}\mathit{i}\mathit{n}}$	$\mathit{G}\mathit{a}\mathit{p}$ a	$\mathit{C}\mathit{p}\mathit{u}$	${\mathit{f}}^{\mathit{M}\mathit{i}\mathit{n}}$	$\mathit{G}\mathit{a}\mathit{p}$ a	$\mathit{C}\mathit{p}\mathit{u}$	${\mathit{f}}^{\mathit{M}\mathit{i}\mathit{n}}$	${\mathit{f}}^{\mathit{M}\mathit{e}\mathit{a}\mathit{n}}$	$\mathit{C}\mathit{p}\mathit{u}$
10	1548	0.00	0.19	1548	0.00	<0.01	1548	1548	0.93
12	1742	0.00	0.31	1799	3.26	<0.01	1742	1754	0.31
14	1996	−0.01	2.46	2168	8.58	<0.01	1997	2019	0.19
16	2102	−1.93	117.97	2344	9.36	<0.01	2144	2227	0.95
18	2275	0.00	89.20	2771	21.82	<0.01	2275	2301	0.40
20	2443	0.36	1200.86	3006	23.45	<0.01	2435	2566	0.58
22	1833	−2.88	36.41	1930	2.25	<0.01	1887	1889	0.42
24	1922	−1.46	34.23	2158	10.63	<0.01	1951	1969	0.55
26	2160	0.00	27.39	2522	16.78	<0.01	2160	2236	0.72
28	2321	−4.24	1212.71	2674	10.35	<0.01	2423	2446	0.36
30	2467	1.33	1210.65	2804	15.16	<0.01	2435	2506	0.86
32	—	—	—	2157	9.81	<0.01	1964	1984	0.91
34	—	—	—	2395	13.94	<0.01	2102	2117	1.39
36	—	—	—	2432	14.61	<0.01	2122	2195	1.50
38	—	—	—	2506	9.74	<0.01	2283	2315	0.40
40	—	—	—	2867	17.40	<0.01	2442	2544	2.23
50	—	—	—	2571	12.52	<0.01	2285	2330	1.39
60	—	—	—	4085	23.47	<0.01	3308	3410	1.36
70	—	—	—	5063	27.55	<0.01	3970	4278	3.03
80	—	—	—	5833	27.96	<0.01	4559	4892	3.01
90	—	—	—	6496	25.98	<0.01	5156	5305	3.26
100	—	—	—	7594	27.20	<0.01	5970	6272	5.00
110	—	—	—	8410	27.95	<0.01	6573	6742	4.64
120	—	—	—	8319	23.24	<0.01	6750	6980	4.11
130	—	—	—	9729	26.81	<0.01	7672	7874	4.59
140	—	—	—	10,433	25.70	<0.01	8300	8576	6.18
150	—	—	—	11,448	26.12	<0.01	9077	9409	5.61
160	—	—	—	12,696	22.66	<0.01	10,351	10,831	5.37

^a This value was calculated as follows: $Gap=\left(f\left(\mathsf{\Theta }\right)-f\left(\mathrm{A}\mathrm{L}\mathrm{N}\mathrm{S}\right)\right)/f\left(ALNS\right)\times 100$, where $\mathsf{\Theta }$ represents model [M1] and RSA, with the unit $\%$. “—” means that no feasible solution could be obtained within the 1200 s limit.

.

As shown in Table 9, model [M1] has an outstanding computational performance in cases with a small number of tasks. For instance, it can obtain results quickly in a limited time when there are no more than 26 tasks. When the number of tasks ranges from 28 to 30, only feasible solutions can be obtained by the model. Appendix A shows the computational times and objective values corresponding to 15 different cases of different small sizes solved by the model [M1]. It can be seen in Appendix A that as the number of tasks increases, the model [M1] cannot obtain the optimal solution to the problem in a limited time, and the range of objective values is small. Secondly, the RSA can quickly solve cases of various sizes, but its optimization is poor. Appendix B shows a comparison of the solution results when the number of iterations increases. Let us set the number of RSA iterations to $N^{Max}=\mathrm{20,000}$ and the number of ALNS iterations to $N^G=800$; as can be seen in Appendix B, when the number of iterations is increased, the optimal value determined by the RSA is still smaller than the optimal value determined by ALNS. Finally, ALNS can resolve cases of various sizes within 9 s, has a good optimality in solving small- and medium-scale examples (no more than 3.8% compared with [M1]), and has a good convergence for multiple solutions.

Other algorithms are often used to solve scheduling optimization problems in ACTs. Appendix C shows the use of a genetic algorithm as an example to compare computing performance with ALNS. In this group of examples, the ALNS algorithm demonstrates a better performance. In future work, ALNS will be applied to solve the problem of the influence of the IGV configuration on equipment scheduling in the RYACT.

6.2. The Effect of the Number of IGVs on the Target Value

The experimental steps can be summarized as follows: (1) Set the number of tasks to 60, 80, …, 140; the corresponding ARMG number is 5. (2) For each combination of task and ARMG number, set the IGV number to 2, 3, …, 8. (3) For each example, run the ALNS algorithm to solve the example 10 times and return the minimum value. The experimental results are exhibited in Figure 3.

As shown in Figure 3, for each example, with an increase in the number of IGVs, the target value first decreases and then tends to converge. According to the experimental results, when the number of IGVs is six, the method can be adapted to most cases. ACT managers can use this method to configure the appropriate number of IGVs, as implementing more IGVs is not always the optimal solution.

6.3. The Effect of IGV Speed on the Target Value

The experimental steps can be summarized as follows: (1) Set the number of tasks to 60, 80, …, 140; the corresponding number of ARMGs is five, and the corresponding number of IGVs is five. (2) Based on the initial IGV speed, set the new IGV speed according to the difference [−50%, −25%, 0, +25%, +50%], and then construct a new calculation. (3) For each example with different numbers of tasks, run ALNS to solve the example 10 times and return the minimum value. The experimental results are exhibited in Figure 4.

According to Figure 4, changing the IGV’s travel speed clearly influences the time taken for the ARMG to complete a series of tasks. Transportation safety, energy consumption, engine power, container’s weight, and other factors can restrict the IGV speed in a container terminal. Considering the safety of the traffic, the speed of IGVs can be adjusted flexibly within the range of 25% in actual operation. From the experimental data, it can be seen that when the speed is increased by 25%, the time taken to complete the tasks is shortened by 14% on average, and when the speed is reduced by 25%, the time taken to complete the tasks is increased by 30% on average. Therefore, increasing the IGV travel speed can effectively reduce the task completion time, while conversely, reducing the IGV travel speed leads to a sharp rise in the task completion time. To take the operation efficiency and energy consumption of the ACT into account, operators must properly adjust the IGV speed to meet the requirements of the expected operation plan. Moreover, to reduce the impact on the operation of the central railway station, the IGVs at the front of the port must actively move out of the way when they intersect with an IGV from the RYACT.

7. Conclusions

This study explored the influence of IGV configurations, numbers, and speed adjustments on ARMG scheduling problems in the RYACT, in which the sequence of ARMGs, and the assignment and sequence of IGVs are fixed. Aiming to minimize the makespan of ARMGs, a MILP model was established, and an ALNS algorithm was designed. Through a series of experiments, the influence of the IGV configuration on ARMG scheduling was discussed in the case of different scales. The experiment revealed that increasing the number of IGVs can improve the efficiency of the railway yard, but the influence of increasing the number to a certain value on the overall task completion time of ARMGs is reduced, and the appropriate number of IGVs should be configured during task scheduling. In the experimental example in this paper, six IGVs were found to be most suitable for completing the tasks. From the experimental data, when the speed is increased by 25%, the time taken to complete a task is shortened by 14% on average, and when the speed is reduced by 25%, the time taken to complete a task is increased by 30% on average. Therefore, speed has a clear influence on task completion, whereby increasing the IGV travel speed can effectively reduce the time it takes ARMGs to complete tasks, while conversely, reducing the IGV travel speed leads to a sharp rise in the task completion time. To reduce the impact on the operation of the central railway station, an IGV from the front yard must actively yield when it intersects with an IGV from the RYACT. This study can help to provide a reference for resource conservation and avoiding collisions at ports. More practical constraints will be considered in future studies, such as constraints on train arrival time windows, restrictions on the number of machines, and environmental requirements. Although the proposed model and algorithm provide a good method and suggestions for IGV configuration, the algorithm designed for handling more loading and unloading tasks still needs to be improved, and the method for optimizing scheduling, considering more freight tasks and faster trains entering and leaving the railway yard, still needs to be studied. ACT managers also need to consider the energy consumption of IGVs, the configuration of other machinery, and the overall efficiency of the port area, all of which are worthy of further study.