AGV Scheduling and Energy Consumption Optimization in Automated Container Terminals Based on Variable Neighborhood Search Algorithm

Abstract

Automated Guided Vehicles (AGVs) for automated container terminals are mainly used for horizontal transportation at the forefront of the terminal. They shoulder the responsibility of container transportation between the quay cranes and yard cranes. Optimizing their scheduling can not only improve operational efficiency, but also help reduce energy consumption and promote green development of the port. This article first constructs a mathematical model with the goal of minimizing the total energy consumption of AGVs, considering the impact of different states of AGVs on energy consumption during operation. Secondly, by using the variable neighborhood search algorithm, the AGV allocation for container operation tasks is optimized, and the operation sequence is adjusted to reduce energy consumption. The algorithm introduces five types of operators and a random operator usage order to expand the search range and avoid local optima. Finally, the influence of the number and speed of AGVs on the total energy consumption is discussed, and the optimization performance of the variable neighborhood search algorithm and genetic algorithm is compared through computational experiments. The research results show that the model and variable neighborhood search algorithm proposed in this paper have a significant effect on reducing the total energy consumption of AGVs and show good stability and practical application potential.

1. Introduction

In recent years, with the continuous increase in global port container throughput, automated container terminals have gradually replaced traditional terminals with their high efficiency, realizing the automated control of key segments such as quay crane operations, horizontal transportation, and yard crane operations. Among them, the horizontal transportation of the terminal, as a bridge connecting the berth and the yard, is one of the key factors that affects the overall operating efficiency of the automated container terminal. Currently, most automated container terminals use Automated Guided Vehicles (AGVs) to complete horizontal transportation tasks. Their reasonable scheduling and efficient operation can not only significantly improve the loading and unloading efficiency of the terminal, but also effectively reduce the operating costs of the terminal, thereby greatly improving the overall competitiveness of the terminal.

At the same time, energy conservation and emission reduction, a global hot topic throughout all fields, occupies a pivotal position in the port field. As an inevitable trend in the development of modern ports, automated container terminals also show great potential in energy conservation and emission reduction. According to statistics and calculations in the environmental field, 1 kilowatt-hour of electricity saving can reduce carbon dioxide emissions by approximately 0.997 kg. Promoting the construction of green automated container terminals will not only help ports achieve the low-carbon development goal, but also lay a solid foundation for their sustainable development.

Based on this background, combining AGV scheduling with energy consumption optimization has become a hot topic in current research. How to formulate a comprehensive scheduling strategy that effectively reduces AGV energy consumption while improving AGV operating efficiency has become a significant topic in the current research field of automated container terminals. This is not only related to the further improvement of terminal operating efficiency, but also provides theoretical support and practical methods for the development of green and intelligent ports, which has important academic value and practical significance.

2. Literature Review

At present, research on the AGV scheduling optimization problem has made remarkable progress. The relevant literature can be divided into three parts: collaborative scheduling of AGVs and other equipment, AGV scheduling considering charging and battery-swapping, and AGV scheduling considering energy consumption optimization.

In terms of collaborative scheduling of AGVs and other equipment, Yue [1] considered the joint scheduling of the quay crane and AGV, including the completion time, safety distance between the ship and quay crane, and the operating surface and buffer capacity of the quay crane. Wang et al. [2] considered the problems of the AGV catching-up conflict, node conflict, and road occupancy conflict; constructed a two-layer scheduling model for the cooperative scheduling of loading and unloading equipment and conflict-free path planning of AGVs; and designed a two-layer genetic algorithm based on the characteristics of the model for solving it. Bae J W et al. [3] investigated the joint scheduling problem of multiple AGVs and multiple dual-trolley quay cranes, and proposed the Pooled Dispatching Strategy, which means that an AGV does not serve one quay crane alone but operates between multiple quay cranes. Zhao et al. [4] established a cooperative scheduling model for Automated Quay Cranes (AQCs) and AGVs, which considered the capacity limitation of the transfer platform on the AQC and took the minimum total energy consumption of AQC and AGV as the objective function. Yang et al. [5] simultaneously studied the AGV path planning and the integrated scheduling problem involving quay cranes, yard cranes, and AGVs, fully considering the impact of congestion, establishing a two-layer scheduling model, and designing a two-layer genetic algorithm based on the congestion prevention mechanism to solve the problem. Choe et al. [6] proposed an online preference learning method incorporating deep neural networks to dynamically adapt dispatching of AGVs with the objective of minimizing the AQC operating time and AGV unloading travel distance.

In terms of AGV scheduling considering charging and battery swapping, Zhao et al. [7] address the AGV scheduling optimization problem and the battery-swapping problem by constructing an AGV scheduling model that incorporates constraints on AGV power consumption differences and battery-swapping thresholds, with the objective of minimizing the completion time of all tasks. Zeng et al. [8] constructed a spatio-temporal network graph to portray the AGV operation and charging process, established a model for task assignment and charging timing selection aimed at minimizing the operational cost of the AGV system, and solved it using a branch-and-price algorithm. Xu et al. [9] proposed a two-layer scheduling model that optimizes the total task completion time and total battery-swapping time for AGVs, reasonably planned the number of battery packs in the battery-swapping station, and effectively improved the utilization rate of AGVs while reducing the number of battery packs in the battery-swapping station. Ma et al. [10] proposed a mixed-integer programming model with the goal of minimizing the completion time to address congestion and charging issues in AGV operations, and the model was optimized in two stages using the simulated annealing algorithm and Dijkstra-based congestion-aware charging strategy.

In terms of AGV scheduling considering energy consumption optimization, Drungilas et al. [11] constructed a transportation model that integrates the AGV operating time, energy consumption, and task priority, and used a deep reinforcement learning algorithm to dynamically adjust the speed of AGVs to achieve a balance between energy consumption and operating time. Yang et al. [12] constructed a multilevel AGV scheduling model for the container loading and unloading process and used a heuristic algorithm combining breadth-first search and depth-first search to solve the problem, which significantly improved the AGV scheduling efficiency and effectively reduced the energy consumption. Wang et al. [13] proposed a two-layer scheduling model based on the consideration of the AGV task allocation and operating process, and designed a two-layer heuristic method combining the genetic algorithm and ant colony optimization algorithm to solve the model efficiently. Zhou et al. [14] addressed the problem of AGV dynamic energy consumption and the battery-swapping strategy by constructing a multi-objective optimization model aimed at minimizing the AGV completion time and energy system cost, and proposed a discrete multi-objective whale optimization algorithm to solve the problem. Yang et al. [15] coordinated the scheduling of the yard crane, quay crane, and AGV, and established a mixed-integer programming model with the objective of minimizing the energy consumption of AGVs, considering the scheduling optimization of AGVs under changing and constant speed. Xing et al. [16] constructed a mixed-integer programming model with the objective of minimizing the total operating time of yard cranes and quay cranes and the energy consumption of AGVs, and proposed a three-phase solution framework combining the greedy algorithm, the branch-and-bound algorithm, and speed optimization to solve the model. Chu et al. [17] used a variable scheduling strategy algorithm based on the deep Q-learning network algorithm to solve a mathematical model with the objective of maximizing the utilization rate of AGVs and minimizing the energy consumption of AGVs, which effectively optimizes them. Zhao [18] investigated the enhancement of overall operational efficiency in automated container terminals through the configuration of varying amounts of operational equipment, focusing on green energy saving and using Yangshan Port Phase IV as a case study. Fan et al. [19] studied the joint configuration and scheduling optimization problem of dual-trolley quay cranes and AGVs from a low-carbon and green perspective, and constructed a two-stage optimization model with the minimization of the total energy consumption. Ai et al. [20] proposed a multi-objective mixed-integer programming model with the objectives of the minimum total operating time of the operating equipment and the total energy consumption to achieve the balance between time efficiency and energy savings.

Through a comprehensive review and analysis of the relevant literature, it can be observed that research on AGV intelligent scheduling optimization mainly focuses on multi-equipment coordination scheduling at terminals, the AGV charging and battery-swapping strategy, and AGV energy consumption. Genetic algorithms are primarily used to solve the models, with less attention given to the energy consumption under different AGV states and the impact of AGV operating speeds on energy consumption.

This paper focuses on the energy consumption of AGVs in three operational states: loading, unloading, and waiting. By comprehensively considering both energy consumption and scheduling as key factors, it establishes a mathematical optimization model aimed at minimizing total energy consumption. Additionally, the Variable Neighborhood Search (VNS) algorithm is proposed to solve this problem, which can reduce the energy consumption of AGV operation as much as possible and improve the operation efficiency of AGVs in automated container terminals, ultimately contributing to the realization of green and intelligent terminal objectives.

3. Problem Description

3.1. Horizontal Transportation Equipment Operation System

The automated container terminal is mainly composed of berths, the forefront of the terminal, container yards, the gate, and other facilities. Among them, the operation scope of AGVs is mainly concentrated in the forefront of the terminal, which is responsible for connecting the berths and container yards to realize the horizontal transportation operation of container imports and exports. The layout is shown in Figure 1. In the quay crane operation area, the quay crane is responsible for loading and unloading containers on container ships. In the yard crane operation area, the yard crane is responsible for loading and unloading containers in the container yards. Quay cranes and yard cranes, respectively, complete the handover of containers with horizontal transportation equipment in their exchange area. In the horizontal transportation equipment operation area, the horizontal transportation equipment undertakes the horizontal transportation of containers between the quay crane and the yard.

In the automated container terminal, the operation of horizontal transportation equipment on containers is mainly divided into two categories: import operation and export operation (as shown in Figure 2).

Import operation: When the container ship berths, the quay crane discharges the containers from the ship in accordance with the preset operation sequence and loads them onto the horizontal transport equipment waiting in the quay crane exchange area. Subsequently, the horizontal transport equipment delivers the containers to the yard crane exchange area of the designated container yards. The yard crane stores the containers at the designated location of the container yard.

Export operation: When the container ship berths, the yard crane takes out the containers from the yard in accordance with the preset operation sequence and loads them onto the horizontal transport equipment waiting in the yard crane exchange area. Subsequently, the horizontal transport equipment delivers the containers to the designated quay crane exchange area. The quay crane loads the containers at the designated location of the container ship.

3.2. Horizontal Transportation Equipment

The Automated Guided Vehicle (AGV) is the most widely used horizontal transportation equipment in automated container terminals. It is responsible for the transfer of containers between the quay crane and the yard. The AGV is usually equipped with electromagnetic or optical guidance devices to ensure that it can travel along the specified path, and has the functions of self-protection and container fixing and transfer. Currently, batteries are the mainstream power supply method for AGVs, which are used to provide energy for the vehicle body and auxiliary devices.

3.3. Energy Consumption and Scheduling of AGV

In terms of energy consumption, energy consumption is different in different AGV operational states. Therefore, it is necessary to consider three operational states—loading, unloading, and waiting—when calculating the energy consumption generated by an AGV to complete a container task. The operating time of the AGV in the loading and unloading state during the container transportation process can be calculated by the distance between the quay crane and the yard crane, and is multiplied by the energy consumption per unit time in the corresponding state to obtain the value of energy consumption. Since this paper mainly focuses on AGV scheduling, the energy consumption of the quay crane and yard crane in the operation process is not considered.

In terms of scheduling, the various automated equipment in the terminal need to work in close coordination. Based on the Pooled Dispatching Strategy for AGVs proposed in the literature [3], AGVs are not dedicated to a specific container crane but shared among multiple cranes. This may lead to a queuing situation where multiple AGVs arrive at the same yard or quay crane at the same time. Therefore, it is necessary to comprehensively consider the operation process of the yard and quay cranes, and coordinate the possible conflicts between different equipment when scheduling AGVs. The process of an AGV accomplishing a container transportation task can be divided into the following five stages:

• Unloading to pick-up point: the AGV drives from the charging station (if the current task is the first task of the AGV) or the last task drop point to the current task pick-up point in the unloading state.
• Arrive at the pick-up point: After the AGV arrives at the designated yard or quay crane, it will judge whether it needs to wait by comparing the time axis of the AGV with the time axis of the yard or quay crane. If it needs to wait, the AGV will generate the waiting cost until the yard or quay crane completes the last container operation; if it does not need to wait, it will directly proceed to the next operation.
• AGV operation: The AGV waits for the yard or quay crane to load the container onto it (the waiting time depends on the loading operation time). Then, the AGV goes to the drop point of the current container in the loading state.
• Arrive at the drop point: After the AGV arrives at the designated quay or yard crane, it will judge whether it needs to wait again by comparing the time axis of the AGV with the time axis of the quay or yard crane. If it needs to wait, the AGV will generate the waiting cost until the quay or yard crane completes the last container operation; if it does not need to wait, it will directly proceed to the next operation.
• Unloading operation: The AGV waits for the quay or yard crane to unload the container from it. After that, the AGV can directly go to the next pick-up point without waiting for the quay or yard crane to store the current container in the container yard.

In the above process, it is necessary to compare the time axis of the AGV with the time axis of the quay and yard crane many times to accurately calculate the energy consumption of the AGV in the waiting state. Through the unified time axis to describe the operation process of terminal operation equipment, the operation sequence and time constraint between different equipment can be intuitively displayed.

4. Modeling

4.1. Model Assumptions

(1)

An AGV has sufficient power during the entire operation process, and the charging process is not considered.

(2)

Traffic problems such as congestion and AGV path planning are not considered.

(3)

An AGV only transports one container at a time.

(4)

An AGV drives at a constant speed throughout the entire operation process.

4.2. Model Parameter Settings

• $J: \mathrm{The} \mathrm{set} \mathrm{of} \mathrm{AGVs}, j=1, 2, \dots , J;$
• $K: \mathrm{The} \mathrm{set} \mathrm{of} \mathrm{yard} \mathrm{cranes}, k=1, 2, \dots , K;$
• $L: \mathrm{The} \mathrm{set} \mathrm{of} \mathrm{quay} \mathrm{cranes}, l=1, 2, \dots , L;$
• $N: \mathrm{The} \mathrm{set} \mathrm{of} \mathrm{containers} \mathrm{to} \mathrm{be} \mathrm{operated}, i=1, 2, \dots , N;$
• $N^{in}: \mathrm{The} \mathrm{set} \mathrm{of} \mathrm{import} \mathrm{containers} \mathrm{to} \mathrm{be} \mathrm{operated}, N^{in}\in N;$
• $N^{out}: \mathrm{The} \mathrm{set} \mathrm{of} \mathrm{export} \mathrm{containers} \mathrm{to} \mathrm{be} \mathrm{operated}, N^{out}\in N;$
• $T_{\mathit{ji}}: \mathrm{The} \mathrm{time} \mathrm{when} \mathrm{AGV} j \mathrm{transports} \mathrm{container} i \mathrm{to} \mathrm{yard}/\mathrm{quay} \mathrm{crane};$
• $T_{\mathit{jii}\prime }: \mathrm{The} \mathrm{time} \mathrm{when} \mathrm{AGV} j \mathrm{completes} \mathrm{the} \mathrm{last} \mathrm{container} i \mathrm{and} \mathrm{starts} \mathrm{transporting} \mathrm{container} i\prime ;$
• ${\mathit{ET}}_{\mathit{ki}}: \mathrm{The} \mathrm{time} \mathrm{when} \mathrm{yard} \mathrm{crane} k \mathrm{starts} \mathrm{operating} \mathrm{container} i;$
• ${\mathit{LT}}_{\mathit{ki}}: \mathrm{The} \mathrm{time} \mathrm{when} \mathrm{yard} \mathrm{crane} k \mathrm{completes} \mathrm{operate} \mathrm{container} i;$
• ${\mathit{ET}}_{\mathit{li}}: \mathrm{The} \mathrm{time} \mathrm{when} \mathrm{quay} \mathrm{crane} l \mathrm{starts} \mathrm{operating} \mathrm{container} i;$
• ${\mathit{LT}}_{\mathit{li}}: \mathrm{The} \mathrm{time} \mathrm{when} \mathrm{quay} \mathrm{crane} l \mathrm{completes} \mathrm{operating} \mathrm{container} i;$
• $t_{\mathit{ji}}: \mathrm{The} \mathrm{loading} \mathrm{time} \mathrm{of} \mathrm{AGV} j \mathrm{transporting} \mathrm{container} i;$
• $t_{\mathit{jii}\prime }: \mathrm{The} \mathrm{unloading} \mathrm{time} \mathrm{of} \mathrm{AGV} j \mathrm{between} \mathrm{container} i \mathrm{and} \mathrm{container} i\prime ;$
• $d_{\mathit{jki}}: \mathrm{The} \mathrm{waiting} \mathrm{time} \mathrm{of} \mathrm{AGV} j \mathrm{transporting} \mathrm{container} i \mathrm{under} \mathrm{yard} \mathrm{crane} k;$
• $d_{\mathit{jli}}: \mathrm{The} \mathrm{waiting} \mathrm{time} \mathrm{of} \mathrm{AGV} j \mathrm{transporting} \mathrm{container} i \mathrm{under} \mathrm{quay} \mathrm{crane} l;$
• $C^f: \mathrm{The} \mathrm{loading} \mathrm{energy} \mathrm{consumption} \mathrm{of} \mathrm{AGV} \mathrm{per} \sec \mathrm{ond};$
• $C^e: \mathrm{The} \mathrm{unloading} \mathrm{energy} \mathrm{consumption} \mathrm{of} \mathrm{AGV} \mathrm{per} \sec \mathrm{ond};$
• $C^w: \mathrm{The} \mathrm{waiting} \mathrm{energy} \mathrm{consumption} \mathrm{of} \mathrm{AGV} \mathrm{per} \sec \mathrm{ond};$
• $t^k: \mathrm{The} \mathrm{time} \mathrm{of} \mathrm{yard} \mathrm{crane} \mathrm{operating} \mathrm{one} \mathrm{container};$
• $t^l: \mathrm{The} \mathrm{time} \mathrm{of} \mathrm{quay} \mathrm{crane} \mathrm{operating} \mathrm{one} \mathrm{container};$
• $v^f: \mathrm{The} \mathrm{loading} \mathrm{speed} \mathrm{of} \mathrm{AGV};$
• $v^e: \mathrm{The} \mathrm{unloading} \mathrm{speed} \mathrm{of} \mathrm{AGV};$

4.3. Decision Variables

• $x_{\mathit{ji}}: \mathrm{if} \mathrm{AGV} j \mathrm{operates} \mathrm{container} i, \mathrm{it} \mathrm{equals} \mathrm{to} 1; \mathrm{otherwise}, \mathrm{it} \mathrm{equals} \mathrm{to} 0.$
• $y_{\mathit{jii}\prime }: \mathrm{if} \mathrm{AGV} j \mathrm{operates} \mathrm{container} i\prime \mathrm{after} \mathrm{completing} \mathrm{container} i, \mathrm{it} \mathrm{equals} \mathrm{to} 1; \mathrm{otherwise}, \mathrm{it} \mathrm{equals} \mathrm{to} 0.$

4.4. Model Establishment

Objective function:

$$minF=C^f{\displaystyle \sum }_{i=1}^N{\displaystyle \sum }_{j=1}^J{x}_{ji}{t}_{ji}+C^e{\displaystyle \sum }_{i=1}^N{\displaystyle \sum }_{i^{\prime }=1}^N{\displaystyle \sum }_{j=1}^J{y}_{jii\prime }{t}_{jii\prime }+C^w({\displaystyle \sum }_{i=1}^N{\displaystyle \sum }_{j=1}^J{\displaystyle \sum }_{k=1}^K{d}_{jki}+{\displaystyle \sum }_{i=1}^N{\displaystyle \sum }_{j=1}^J{\displaystyle \sum }_{l=1}^L{d}_{jli})$$

(1)

Constraints:

$${\displaystyle \sum }_{i=1}^N{x}_{ji}=1, \forall j\in J;$$

(2)

$${\displaystyle \sum }_{j=1}^J{x}_{ji}=1, \forall i\in N;$$

(3)

$${\displaystyle \sum }_{j=1}^J{y}_{jii\prime }=1, \forall i\in N, \forall i^{\prime }\in N;$$

(4)

$$T_{jii\prime }=T_{ji}+d_{jli}+t_{ji}+d_{jki}+t_{jii\prime } , \forall i\in N^{in}, \forall i^{\prime }\in N, \forall j\in J, \forall k\in K, \forall l\in L;$$

(5)

$$T_{jii\prime }=T_{ji}+d_{jki}+t_{ji}+d_{jli}+t_{jii\prime } , \forall i\in N^{out}, \forall i^{\prime }\in N, \forall j\in J, \forall k\in K, \forall l\in L;$$

(6)

$$L{T}_{ki}=E{T}_{ki}+t^k, \forall i\in N, \forall k\in K;$$

(7)

$$L{T}_{li}=E{T}_{li}+t^l, \forall i\in N, \forall l\in L;$$

(8)

$$d_{jki}=\max \left\{E{T}_{ki}-T_{ji}, 0\right\}, \forall i\in N^{out}, \forall j\in J, \forall k\in K;$$

(9)

$$d_{jki}=\max \left\{E{T}_{ki}-T_{ji}-d_{jli}-t_{ji}, 0\right\}, \forall i\in N^{in}, \forall j\in J, \forall k\in K;$$

(10)

$$d_{jli}=\max \left\{E{T}_{li}-T_{ji}, 0\right\}, \forall i\in N^{in}, \forall j\in J, \forall l\in L;$$

(11)

$$d_{jli}=\max \left\{E{T}_{li}-T_{ji}-d_{jli}-t_{ji}, 0\right\}, \forall i\in N^{out}, \forall j\in J, \forall l\in L;$$

(12)

Equation (1) is the objective function with the optimization goal of minimizing the total energy consumption of AGV operation. Equation (2) constrains an AGV to only transporting one container at a time. Equations (3) and (4) constrain a container to only being transported by one AGV. Equation (5) constrains the time when AGV j completes the last import container i and starts transporting container i’. Equation (6) constrains the time when AGV j completes the last export container i and starts transporting container i’. Equation (7) constrains the time of yard crane k operating one container. Equation (8) constrains the time of quay crane l operating one container. Equation (9) constrains the waiting time of AGV j transporting an export container i under yard crane k. Equation (10) constrains the waiting time of AGV j transporting an import container i under yard crane k. Equation (11) constrains the waiting time of AGV j transporting an import container i under quay crane l. Equation (12) constrains the waiting time of AGV j transporting an export container i under quay crane l.

5. Variable Neighborhood Search Algorithm

5.1. The Introduction of Variable Neighborhood Search

The Variable Neighborhood Search (VNS) algorithm belongs to a kind of local search algorithm, which uses operators to transform the neighborhood structure during local search to realize the search for the optimal solution. Traditional local search algorithms are prone to fall into local optima in the optimization process. In contrast, the VNS algorithm, by switching between different neighborhood structures and combining the neighborhood shaking mechanism, can dynamically adjust the search strategy during the search process, thus effectively avoiding falling into the local optimal solution. This kind of adaptivity and flexibility makes the VNS algorithm show strong robustness in the face of complex and variable optimization problems, and it is often applied to the study of path planning and scheduling problems.

5.2. Algorithm Design

5.2.1. Coding

In this paper, a matrix coding method is used to design the chromosome. The length of chromosomes is the number of containers to be operated assigned to AGV. As shown in Figure 3, the three chromosomes represent three AGVs, and the number in the chromosome represents the container number that the AGV needs to operate. The AGV operates from left to right, according to the container number in the corresponding chromosome. Taking 16 container operation tasks and three AGVs as an example, assuming that the list of operation tasks is (23, 19, 16, 44, 30, 26, 31, 29, 13, 02, 17, 05, 35, 11, 41, 27) and the calculated scheduling scheme is agv1 operates (23, 30, 26, 13, 11, 41), agv2 operates (19, 31, 02, 35, 27), and agv3 operates (16, 44, 29, 17, 05). The three chromosomes form a complete solution.

5.2.2. Operators

In this paper, five operators are used to generate neighborhood structure, including Insert Operator, Swap Operator, Reverse Operator, Block Swap Operator, and Double Task Swap Operator.

(1)

Insert Operator

Select any chromosome and randomly choose a container task within this chromosome to insert after another container task within the same chromosome, as shown in Figure 4.

(2)

Swap Operator

Select any chromosome and randomly choose two container tasks within this chromosome to swap them, as shown in Figure 5.

(3)

Reverse Operator

Select any chromosome and randomly choose two container tasks within this chromosome to reverse all tasks between the two container tasks including the two selected tasks, as shown in Figure 6.

(4)

Block Swap Operator

Randomly choose two blocks of container task with the same length from two different chromosomes and swap the two whole blocks, as shown in Figure 7.

(5)

Double Task Swap Operator

Select any two chromosomes and randomly choose two container tasks within each of the two chromosomes to swap the four selected container tasks two by two, as shown in Figure 8.

5.2.3. Initial Solution

The local search algorithm has relatively high requirements for the quality of the initial solution. A good initial solution can reduce the time of iterative calculation and improve the solution efficiency. In order to find a better initial feasible solution, the energy consumption is used as the evaluation index. The details of the algorithm are shown in Figure 9.

The above pseudo code first generates empty chromosomes with the equal number of AGVs, then traverses all containers in the initial container task list, calculates and compares the total energy consumption of all AGV operations when the container task inserts each chromosome, and inserts the container into the chromosome with the lowest total energy consumption. By analogy, all container tasks in the initial container task list are assigned to each AGV to generate the initial solution.

5.2.4. Calculation of Objective Function

Input the initial solution and the randomly generated pick-up point, drop point, and yard crane and quay crane operation time of each container operation. Initialize the total energy consumption and the AGV operation time to 0. Traverse each operation task in each AGV and calculate the unloading time, loading time, waiting time, and corresponding energy consumption for completing this container task. By analogy, calculate the total energy consumption and total operation time of the whole tasks (shown in Figure 10).

5.3. Process of Algorithmic Solution

The VNS algorithm takes the initial solution s₀ as the starting point and sets the maximum number of unoptimized iterations α_max and the maximum neighborhood range k_max. In the shaking phase and the local search phase, two neighborhood structures, S and L, are generated, respectively. The algorithm initializes the current solution s as the initial solution s₀ and calculates its value of the objective function F(s). Then, the algorithm gradually explores different neighborhoods through circulation: first, k neighborhood structures are randomly selected to shake the current solution and generate the neighborhood solution s’. Then the local optimal solution s” and its energy consumption F(s”) are obtained in the local search phase. If the energy consumption F(s”) of the new solution is better than that of the current solution F(s), the current solution is updated; otherwise, the neighborhood range k is enlarged, or the number of unoptimized iterations α is increased. The algorithm stops after the maximum number of unoptimized iterations α_max is reached and outputs the globally optimal solution and the globally optimal energy consumption. The overall process of the algorithm is shown in Figure 11.

The key of the algorithm is to avoid falling into the local optima through shaking and local search, and to improve the global search capability by using the strategy of gradually expanding the neighborhood.

6. Numerical Experiments

6.1. Example Design

All the calculations in this paper are based on a Windows 10 system with 12th Gen Inter(R) Core(TM) i5-12600KF CPU@3.70 GHz(Santa Clara, CA, USA), 16 GB. In the small-scale example, the AGV needs to operate 100 container tasks: 1 to 50 are export containers, and 51 to 100 are import containers. The large-scale example requires the AGV to operate 500 container tasks, with 1 to 250 being export containers and 251 to 500 being import containers. The operation time of each container at the yard crane and quay crane is randomly generated according to the normal distribution N (75 s,15 s). The parameters of the AGV speed and energy consumption per second in different states used in this paper are quoted from the literature [21]. The specific parameters are shown in

Table 1. AGV parameters.

Parameter	Description	Value
vf	Loading speed of AGV/(m/s)	3.5
ve	Unloading speed of AGV/(m/s)	6
Cf	Energy consumption of AGV in loading state/(kWh/s)	0.006
Ce	Energy consumption of AGV in unloading state/(kWh/s)	0.004
Cw	Energy consumption of AGV in waiting state/(kWh/s)	0.0025

.

6.2. Small-Scale Example

6.2.1. Comparison of the Number of AGVs in Small-Scale

Each group of experiments randomly generates 100 container pick-up and drop points. Under this condition, all container tasks are operated by two, three, four, and five AGVs, respectively. The total energy consumption is calculated for each number of AGVs. The parameters of AGV speed and energy consumption in three states are taken according to Table 1. A total of 10 experimental groups are conducted. The average total energy consumption and operating time for each number of AGVs is shown in

Table 2. Results on comparison of the number of AGVs in small-scale.

Number of AGVs	Pre-Optimization		Post-Optimization		Decrease
	EC (kWh)	OT (s)	EC (kWh)	OT (s)	EC	OT
2	27.85	6406.64	20.57	5568.61	26.14%	13.09%
3	27.98	4358.53	20.73	3845.59	25.91%	11.77%
4	27.99	3372.59	20.85	2999.17	25.51%	11.07%
5	28.23	2621.11	20.99	2294.20	25.65%	12.47%

Notes: Pre-optimization is the result of the initial solution presented in Figure 9; Post-optimization is the result of VNS.

. (EC indicates the energy consumption, and OT indicates the operating time)

By comparing the energy consumption and operating time before and after optimization (shown in Figure 12), it can be found that the total energy consumption of AGVs is significantly reduced after the AGV scheduling scheme is optimized using VNS, with the decrease stabilized at about 26%, and the decrease does not change much under the condition of different numbers of AGVs. It is calculated that approximately 7 kg of CO₂ emissions can be reduced by optimization of VNS. Additionally, the operating time of AGVs is also remarkably reduced, and the decrease is more than 10%, which plays a significant role in improving the operation efficiency of AGVs.

6.2.2. Comparison of Algorithms in Small-Scale

Most of the AGV scheduling problems related to energy consumption have been solved by genetic algorithms (GAs) in the existing literature. In order to compare the optimization effect and efficiency of GA and VNS used in this paper, experiments were designed for comparative analysis.

The experimental setup is the same as that of the experiment in Section 6.2.1. The GA from reference [15] and the VNS proposed in this study were utilized to optimize the AGV scheduling scheme. A total of 10 experimental trials were conducted. The average total energy consumption and CPU time for each number of AGVs is detailed in

Table 3. Results on comparison of algorithms in small-scale.

Number of AGVs	Initial EC (kWh)	GA [15]			VNS			Gap
		EC (kWh)	OR	CPU Time (s)	EC (kWh)	OR	CPU Time (s)	EC	CPU Time
2	27.96	23.71	15.20%	18.03	20.54	26.54%	12.91	13.37%	28.40%
3	28.07	23.79	15.25%	17.76	20.69	26.29%	12.34	13.03%	30.52%
4	28.21	23.98	14.99%	18.23	20.86	26.05%	12.68	13.01%	30.44%
5	28.29	24.14	14.67%	18.35	20.97	25.87%	12.66	13.13%	31.01%

.

In the small-scale container transportation experiment, by comparing the total energy consumption of different numbers of AGVs optimized by GA and VNS (shown in Figure 13), it can be seen that the total energy consumption of AGVs optimized by VNS is significantly lower than that of GA, with an average increase of about 13%, and the optimized stability of VNS is better, with less fluctuation in a single experiment (shown in Figure 14). Under the conditions of small-scale experiments, GA and VNS reduce CO₂ emissions by about 4.2 kg and 7.4 kg, respectively. At the same time, it can be concluded through the comparison of algorithm execution times that VNS has less computing time than GA, reducing the time by about 30% on average. It proves that the VNS designed in this paper has a better optimization effect, stability, and computational efficiency compared with GA.

Figure 15 illustrates the percentages of energy consumption across three states. The energy consumption during the loading state of AGVs constitutes the majority of the total energy consumption. As a matter of fact, when the number of containers is certain, the energy consumption of AGVs in the loading state remains constant and does not vary with the optimization of AGV scheduling. From Figure 15, it can be observed that the energy reduction achieved through algorithm optimization is primarily attributed to the energy consumption of AGVs in the unloading state. Additionally, there is a slight decrease in the energy consumption of AGVs in the waiting state, but the reduction is not obvious because this part of the energy consumption is small. This indicates that VNS significantly reduces the unnecessary unloading movements of AGVs and the waiting times for AGVs under quay and yard cranes.

6.2.3. Convergence Analysis in Small-Scale

The convergence process of the two algorithms is analyzed using an example from the small-scale experiment with two AGVs. Both algorithms start iterating from the same initial solution, with an initial energy consumption of 27.96 kWh. In the first iteration, both algorithms achieve a reduction in energy consumption, but their optimization effects differ markedly: GA reduces the total energy consumption to 26.10 kWh, while VNS achieves a more substantial reduction, bringing it down to 23.31 kWh. Subsequently, the total energy consumption of both algorithms decreases gradually as the number of iterations increases. However, their decreasing trends and convergence speeds differ significantly. VNS optimizes more quickly, with total energy consumption rapidly decreasing to approximately 21 kWh and converging after about the 10th iteration. In contrast, GA optimizes more slowly, with total energy consumption decreasing gradually and stabilizing after approximately the 15th iteration.

Overall, the optimization effect of VNS is better than GA in this problem, with a faster convergence speed and lower total energy consumption in the end. The convergence process is shown in Figure 16.

6.3. Large-Scale Example

6.3.1. Comparison of the Number of AGVs in Large-Scale

Each group of experiments randomly generates 500 container pick-up and drop points. Under this condition, all container tasks are operated by two, three, four, and five AGVs, respectively. The total energy consumption is calculated for each number of AGVs. The parameters of AGV speed and energy consumption in three states are taken according to Table 1. A total of 10 experimental groups are conducted. The average total energy consumption for each number of AGVs is shown in

Table 4. Results on comparison of the number of AGVs in large-scale.

Number of AGVs	Pre-Optimization		Post-Optimization		Average OR
	EC (kWh)	OT (s)	EC (kWh)	OT (s)	EC	OT
2	139.01	32,013.01	101.66	27,330.87	26.87%	14.63%
3	139.18	21,603.92	101.75	18,374.90	26.89%	14.95%
4	139.20	16,201.16	101.86	13,726.88	26.82%	15.27%
5	139.36	13,224.88	101.97	11,539.90	26.83%	12.74%

.

By comparing the energy consumption and operating time before and after optimization (shown in Figure 17), it can be found that after using VNS to optimize the AGV scheduling scheme, the total energy consumption of the AGVs decreases notably, with a decrease of about 26.8%, and the decrease remains stable with the change in the number of AGVs. By calculation, VNS can reduce approximately 37 kg of carbon dioxide emissions. Meanwhile, the operation time of AGVs is also greatly reduced, with a decrease rate between 12% and 16%, which plays an important role in improving the operation efficiency of AGVs.

By comparing the decrease in energy consumption in the small- and large-scale experiments, it is evident that, with the same number of AGVs, the decrease in total energy consumption is significant when the number of containers is larger. However, the gap between the different scale experiments is only about 1%, which proves that VNS shows outstanding stability in different scales. Meanwhile, the reduction of operating time increases with the increase in the containers’ scale, which further enhances the operational efficiency of AGVs. Overall, regardless of the scale of containers, the variable neighborhood search algorithm consistently and effectively optimizes both the total energy consumption and operating time of AGVs.

6.3.2. Comparison of Algorithms in Large-Scale

The experimental setup is the same as that of the experiment in Section 6.3.1. The GA from reference [15] and the VNS proposed in this study were utilized to optimize the AGV scheduling scheme. A total of 10 experimental trials were conducted. The average total energy consumption and CPU time for each number of AGVs is detailed in

Table 5. Results on comparison of algorithms in large-scale.

Number of AGVs	Initial EC (kWh)	GA [15]			VNS			Gap
		EC (kWh)	OR	CPU Time (s)	EC (kWh)	OR	CPU Time (s)	EC	CPU Time
2	139.03	122.67	11.77%	58.55	101.41	27.06%	41.04	17.33%	29.91%
3	139.23	123.00	11.66%	62.12	101.51	27.09%	42.87	17.45%	30.83%
4	139.32	123.09	11.65%	59.34	101.54	27.12%	42.35	17.51%	28.63%
5	139.39	123.13	11.67%	61.24	101.73	27.02%	43.42	17.38%	29.10%

.

In the large-scale container transportation experiment, by comparing the total energy consumption of different numbers of AGVs optimized by GA and VNS (shown in Figure 18), the results show that the decrease in VNS is also considerably better than that in GA, with an average decrease of about 17.4%. Additionally, VNS demonstrates better stability in experiments of varying scales, and the fluctuation in individual experiments is smaller (shown in Figure 19). Under the conditions of large-scale experiments, GA and VNS reduce carbon dioxide emissions by around 16.2 kg and 37.5 kg, respectively. At the same time, comparison of the algorithm execution times reveals that VNS has a shorter computing time than GA, with an average reduction of about 30%.

According to Figure 20, the percentages of energy consumption in large-scale experiments is similar to that in small-scale experiments. However, in large-scale experiments, the initial energy consumption in the loading state is higher than that in small-scale experiments, and the energy consumption in the unloading and waiting states decreased compared to small-scale experiments. This may be due to the fact that with a larger number of containers, the selection range of containers for AGVs increases, thus reducing the idle running and waiting times of AGVs. In addition, after optimization by GA and VNS, there is a significant reduction in the percentage of energy consumption in the unloading and waiting states in the large-scale experiments. Notably, the VNS algorithm proposed in this paper enhances the percentage of energy consumption in the loading state by more than 90%, which significantly improves the utilization rate of AGVs and reduces their ineffective operations.

When combining the results of the small-scale and large-scale experiments, the reduction of GA decreases when handling large-scale container tasks compared to small-scale tasks. In contrast, VNS maintains stability and even shows a slight increase in reduction. This demonstrates that VNS has superior adaptability and stability in addressing this problem, as well as higher computational efficiency.

6.3.3. Convergence Analysis in Large-Scale

The convergence process of the two algorithms is analyzed using an example from the large-scale experiment with two AGVs. Both algorithms start iterating from the same initial solution, with an initial energy consumption of 139.03 kWh. In the first iteration, the optimization effects of the two algorithms already show significant differences: GA reduces the total energy consumption to 133.79 kWh, while VNS achieves a greater reduction by lowering the total energy consumption to 114.47 kWh, with a more pronounced gap. As the iterations proceed, the total energy consumption of both algorithms decreases gradually. However, their decreasing trends and convergence speeds differ markedly. VNS optimizes more quickly, converging after about the 40th iteration and ultimately reducing the total energy consumption to around 101 kWh. In contrast, GA approaches stability after about the 50th iteration, with a relatively smaller reduction in total energy consumption.

Overall, in the context of the large-scale experiment, the optimization capability of VNS is significantly superior to that of GA, as evidenced by its faster convergence speed and lower final total energy consumption. The convergence process is depicted in Figure 21.

6.4. Analysis of AGV Speed Change

AGV speed is a key factor influencing the total energy consumption of AGVs. Changes in speed lead to corresponding alterations in both energy consumption per unit time and operating time. These parameters significantly impact energy consumption and, in turn, may affect the optimization performance of the algorithm. Therefore, in addition to the original AGV parameters listed in Table 1, three additional sets of AGV parameters were selected for experimentation. These parameters are detailed in

Table 6. Different AGV speeds and corresponding energy consumption.

Loading Speed (m/s)	Unloading Speed (m/s)	Loading EC/(kWh/s)	Unloading EC/(kWh/s)
3.5	6	0.006	0.004
4.2	7.2	0.00864	0.00576
5.25	9	0.0135	0.009
7	12	0.024	0.016

.

The four sets of AGV parameters are applied in each group of experiments, with each group randomly generating 100 container pick-up and drop points. Under these conditions, all container tasks are operated using two AGVs for each set of parameters. The total energy consumption is calculated for each set of AGV parameters. A total of 10 experimental groups were conducted. The average total energy consumption for each set of AGV parameters is shown in

Table 7. Results on different AGV speeds.

Loading Speed (m/s)	Unloading Speed (m/s)	Pre-Optimization EC (kWh)	Post-Optimization EC (kWh)	OR
3.5	6	28.22	20.69	26.68%
4.2	7.2	33.81	24.79	26.66%
5.25	9	42.18	30.99	26.53%
7	12	56.12	41.26	26.48%

.

The experimental results indicate that as the AGV speed increases, the calculated total energy consumption rises sequentially. These findings are in line with the actual production situation in the port. Additionally, the total energy consumption of AGVs optimized using VNS is significantly reduced. The decrease for the four sets of speeds remains relatively stable at around 26.5%, although it decreases slightly with increasing speed. This demonstrates that the optimization effect of VNS is both substantial and stable when AGV speeds are adjusted. The results confirm that the model and VNS proposed in this study maintain strong optimization performance, adaptability, and stability under the normal range of AGV speed variation.

A comparison of the average energy consumption generated by AGVs operating at different speeds and optimized by VNS is shown in Figure 22 (x-axis is four sets of AGV speed, and y-axis is the average energy consumption):

In this study, the actual operation of container terminals is simulated through the above two different scales experiments. The model and VNS proposed in this paper are employed to solve these experiments. The optimization performance of VNS in practical applications is evaluated by comparing some key metrics, such as energy consumption before and after optimization. Additionally, considering the variability in AGV speeds during real operations, corresponding experiments were designed to analyze AGV operations under different speed conditions. The results show that the AGV scheduling scheme optimized by VNS significantly outperforms the initial scheduling scheme in terms of operational efficiency and total energy consumption. This fully demonstrates the effectiveness, adaptability, and superiority of the model and VNS proposed in this paper.

Energy consumption optimization is one of the core objectives in AGV intelligent scheduling in green automated container terminals. VNS optimizes the total energy consumption during AGV operation to a great extent through continuous dynamic evaluation and AGV task allocation adjustment during the search process. The algorithm not only takes into account the dynamic issues of the AGV waiting time and task allocation, but also can adapt to varying scales of container operation and AGV parameters, ensuring the robustness of energy consumption optimization. This provides an effective basis and reference for AGV energy consumption optimization in automated container terminals.

7. Conclusions

This paper focuses on the intelligent scheduling optimization problem of AGVs in automated container terminals, incorporating cutting-edge research findings in this domain. The optimization objective is to minimize the total energy consumption of AGV operations, thereby reducing terminal carbon emissions and enhancing operational efficiency. Initially, based on the operational workflow of AGVs in actual automated container terminals, the energy consumption of AGVs is segmented into three states: loading, unloading, and waiting. This analysis strategy ensures the comprehensiveness of the total energy consumption calculation while significantly simplifying the model’s complexity. Next, the import/export operation process of the container is described, clearly demonstrating the cooperative mechanisms and potential conflict points among terminal operation equipment. On this basis, a mathematical model aimed at minimizing AGV total energy consumption is constructed. The model takes into account the specific impacts of various AGV states and scheduling arrangements on energy consumption. The variable neighborhood search (VNS) algorithm is proposed to solve this model, leveraging its ability to switch between different neighborhood structures to escape local optima and improve the solution quality. Finally, the model and algorithm proposed in this study are analyzed and compared through specific numerical experiments. The results indicate that the VNS algorithm exhibits a satisfactory optimization effect and stability in addressing the AGV scheduling problem considering energy consumption in automated container terminals.

Currently, the mainstream approach to addressing the AGV scheduling problem in container terminals is genetic algorithms. However, this paper takes an alternative route by introducing the Variable Neighborhood Search (VNS) algorithm to tackle this issue. The VNS algorithm is not only capable of effectively handling complex constraints and objective functions but is also particularly well-suited for terminal models aimed at minimizing total energy consumption. This provides a new solution pathway for optimizing AGV scheduling in container terminals. The research results indicate that the Variable Neighborhood Search (VNS) algorithm demonstrates good adaptability and stability. However, when addressing large-scale problems, the solution efficiency of VNS is somewhat limited, often requiring the support of other algorithms to enhance its performance. Looking ahead, with further optimization, the VNS algorithm is expected to be effectively applied in the actual intelligent scheduling of automated container terminals, thereby facilitating efficient operation and management.