Scheduling of Container Transportation Vehicles in Surface Coal Mines Based on the GA–GWO Hybrid Algorithm

Abstract

The coal loading operation of the coal preparation plant of an open pit coal mine causes chaos in coal mine vehicle scheduling due to the unreasonable arrival times of outgoing and container transportation vehicles. To further reduce the length of time that vehicle transportation equipment waits for each other and to reduce the total cost of container transportation, the optimisation model of container transportation vehicle scheduling in an open pit coal mine is constructed to minimise the minimum sum of the shortest time of container reversal and the lowest cost of container transportation. To accurately measure the total cost of container backward transportation, waiting time and unit waiting time cost parameters are introduced, and the total cost of container transportation is measured using the transportation cost and the waiting time cost transformation method. An improved grey wolf algorithm is proposed to speed up the convergence of the algorithm and improve the quality of the solution. When employing the genetic algorithm (GA) and grey wolf optimisation algorithm (GWO) for optimising the scheduling of container transport vehicles in coal mines, it is noted that while the GA can achieve the global optimum, its convergence speed is relatively slow. Conversely, the GWO converges more quickly, but it tends to be trapped in local optima. To accelerate the convergence speed of the algorithm and improve the solution quality, a hybrid GA−GWO algorithm is proposed, which introduces three genetic operations of selection, crossover, and mutation of GA into the GWO algorithm to prevent the algorithm from falling into the local optimum due to the fall; at the same time, it introduces hunting and attacking operations into the elite retention strategy of GA, which improves the stability of the algorithm’s global convergence. Analysis indicates that, compared to SA, GWO, and GA, the hybrid algorithm enhances optimisation speed by 43.1%, 46.2%, and 43.7%, increases optimisation accuracy by 4.12%, 6.1%, and 3.2%, respectively, and reduces the total container reversal time by 35.46, 22, and 31 h. The total cost of container transportation is reduced by 2437 RMB, 3512 RMB, and 1334 RMB, respectively.

1. Introduction

Due to the large area and complex terrain of surface coal mines, different sections of coal mining points and coal preparation plants loading coal require a corresponding number of coal mining vehicles and vehicles, so the scheduling and management of these coal mine vehicles have become a severe problem. The existing transport system presents slow-running vehicles, an extended waiting time, chaos in management, and other shortcomings in a modern mine; the modern production needs of coordinated scheduling, safe and efficient transportation, and intelligent resource deployment characteristics have become a hot spot for many coal companies and many scholars to pay attention to and research [1,2]. Research statistics indicate that transportation costs of trucks in surface coal mines represent 45% to 60% of the total operating expenses. Consequently, effectively reducing these transportation costs and enhancing production efficiency are crucial for improving the overall productivity of mines. The study of transportation scheduling is a crucial method to enhance transportation efficiency, and optimising transportation scheduling can save 3−20% of the cost [3]. Given this, improving the efficiency of coal mine vehicle operation and effectively reducing the transportation cost of vehicles in the transportation process has become an essential issue for the surface coal mine’s imminent solution.

The vehicle scheduling problem involves managing a fleet of multiple vehicles, a distribution centre (or warehouse), and a set number of customers. It requires planning optimal driving routes while adhering to various constraints, such as goods quantity, time limitations, distance restrictions, and vehicle load limits. The objective is to achieve optimal outcomes such as minimising costs, reducing travel distances, decreasing delivery times, and using the fewest number of vehicles possible [4]. Surface coal mine truck scheduling is a complex system engineering problem; the truck scheduling model scheduling objectives of the unitary are an important reason affecting the accuracy of the model; the actual coal transportation is a multi-factor interaction of engineering; multiple optimisation scheduling objectives interact with each other to influence the constraints of the distribution of trucks in a motorised distribution [5] model. In 2017, Sun Xiaoyu et al. [6] examined the congestion levels of surface mining transport routes and their road capacities. This study aims to address the existing gaps in knowledge regarding road resistance, establish a uniform distribution model of the open pit mine traffic road network considering road access saturation, realise the re-optimisation of the scheduling scheme when the road is congested, and improve the production applicability of the scheduling model. Ali M A et al. [7] studied the scheduling model to minimise shovelling empty time, truck waiting time, and deviation from the production optimisation path. They proposed a real-time truck scheduling multi-objective transportation model that makes the optimal decision to handle the material through path production optimisation and real-time truck scheduling, which meets 85% of the production requirements of truck operation in the actual production state. In contrast, the number of mining trucks is reduced by 30% compared to the number required. Menfei et al. [8] established a corresponding mathematical model for the low-carbon transportation scheduling problem of open-pit mines to minimise the sum of carbon emission costs and transportation costs.

The coal mine transportation scheduling problem is primarily a multi-objective function optimisation issue focusing on maximising value. In recent years, inspired by natural organisms, many scholars have used intelligent swarm search algorithms to solve the coal mine truck transportation scheduling problem. Ying Sun [9] and Qing Huagu [10] designed improved ant colony algorithms to solve the underground vehicle path problem. Kai Su et al. [11] constructed a mathematical model of open pit mine transportation scheduling to minimise the transportation cost. They solved it using an adaptive fruit fly optimisation algorithm, effectively reducing transportation costs. Alvarenga G B et al. [12] investigated the time-window scheduling model for the vehicle path problem, and through practical genetic algorithms and set partitioning formulas, they proposed a distance travelled as the main objective of the VRPTW robust heuristic to shorten the minimum travel distance, which outperforms all previously known and published heuristics. Chang et al. [13] studied station siting and mixed fleet sizing problems based on a minimum cost flow model considering CO₂ emission constraints. Yue et al. [14] proposed an optimisation problem for multidimensional complex spaces based on the fireworks algorithm (FWA) and the grey wolf optimisation algorithm, which merges the superior developmental capabilities of the grey wolf algorithm with the robust exploratory prowess of FWA. Thirty tests were conducted on 16 benchmark functions using swarm intelligence algorithms such as the improved particle swarm algorithm, crow search algorithm, mothballing algorithm, and the proposed hybrid algorithm. The performance advantages of the hybrid algorithm are verified. Rongheng He et al. [15] studied the multi-story warehouse layout problem under uncertainty, where uncertain variables describe monthly demand and horizontal transportation distance. The relationship between the opportunity-constrained planning model and the opportunity maximisation planning model is discussed, which provides an effective method for finding the optimal solution for the opportunity maximisation planning model. The hybrid vehicle path optimisation model studied by Li, Y. et al. [16] incorporates a fleet allocation problem and sets up different cost calculation formulas for the differences between the two models. The influence of charging facilities on the planning results is also considered. The ant colony algorithm is improved and integrated with the idea of decentralised search, combined into a two-stage heuristic algorithm. The algorithm’s performance is demonstrated by comparing the proposed algorithm with the Cplex solver. The shortest path planning problem for mobile equipment, including auxiliary vehicles in underground coal mines, was investigated using differential evolutionary algorithms by Guoguo Han [17] and others. Cheng Ping et al. [18] constructed a multi-objective scheduling optimisation model. They introduced a multi-objective optimisation algorithm, NNIA, to solve the problem by comprehensively considering various constraints, such as the production capacity of the car shovels, the ore grade, and the vehicles’ loading capacity. Liu Changshi et al. [19] proposed a hybrid vehicle scheduling model that distinguishes between different vehicle types and considers factors such as traffic congestion, load limits, and driving distances. This model employs genetic and simulated annealing algorithms for solving the problem and experimentally compares the performance of these two algorithms. While these algorithms have partially resolved the transportation scheduling issues in surface coal mines, challenges remain, including limited global optimisation capabilities and the need for extensive parameter tuning.

Open pit mining card scheduling optimisation is mainly implemented using heuristic algorithms, genetic algorithms, annealing algorithms, grey wolf optimiser [20]. Genetic algorithms can effectively solve multi-objective optimization problems and have strong global optimization seeking ability [21]. The grey wolf optimisation (GWO) algorithm is characterised by robust convergence, minimal parameter requirements, straightforward programming implementation, and a strong local search capability [22]. GWO has excellent results in solving combinatorial optimisation problems such as vehicle scheduling, path planning, and job shop scheduling problems [23,24,25]. Still, GWO algorithms have the problem of quickly falling into the local optimum problem [26]. Tawhid et al. proposed a hybrid grey wolf optimisation genetic algorithm (HGWOGA), which balances the exploration and exploitation capabilities with the grey wolf optimisation algorithm, and then, after dividing the population, the crossover operator is used on the divided sub-populations to increase the diversity of the search; finally, the mutation operator of the genetic algorithm is applied to the whole population to avoid premature maturity [27]. Therefore, this paper takes the open pit coal mine container transport vehicle as the research object and establishes the open pit coal mine container transport vehicle scheduling optimisation model to minimise the sum of the shortest container transport time and the lowest container transport cost. The GWO algorithm is introduced in the late stage of GA to improve its slow convergence, and a hybrid GA–GWO algorithm is proposed and used to solve the scheduling optimisation model of surface coal mine container transport vehicles.

2. Problem Description and Modelling

2.1. Problem Description

Empty containers in the coal mine yard must be transported to the coal preparation plant to be loaded with coal, completing the day’s container transportation plan. The process known as ‘container reversal’ involves several steps: First, transport vehicles move the containers from the coal block truck to the loading station outside the coal preparation plant. At the loading station, containers are weighed empty, then filled with coal and levelled, weighed again when heavy, and finally returned to the yard via transport vehicles. This entire sequence of movements constitutes the container reversal process. The demand for containers is Q. The distance between different coal processing plants. The yard is D_i (i = 1, 2, …, N); the capacity efficiency of each coal preparation plant is C_i (i = 1, 2, …, N); the initial stockpiles are S_i (i = 1, 2, …, N); the yard owns M wagons to haul goods to the coal processing plant; and the maximum load of each wagon is Lj (j = 1, 2, …, M). The speed is V_j (j = 1, 2, …, M), and the unit transportation cost per truck is Pj (j = 1, 2, …, M). In container transportation, outbound K vehicles occasionally load coal at the coal processing plant. The target coal processing plant reached by the outbound vehicles is respectively recorded as G_k (k = 1, 2, …, K), the arrival time of each vehicle is denoted as T_k (k = 1, 2, …, K), the demand for each vehicle is denoted as q_k (k = 1, 2, …, K), it takes time to load coal at the coal preparation plant, and its loading speed is s. At the exact moment, a coal preparation plant can only provide a vehicle for coal loading operations. If more than one vehicle arrives at the coal preparation plant, there will be a queuing phenomenon when loading vehicles. Container vehicles in the coal preparation plant coal loading end after weighing and returning to the yard to unload the container; the vehicle can continue to pull the remaining containers to the coal preparation plant coal loading.

The requirement is to design a rational vehicle transportation scheme that minimises the total cost of completing the task and minimises the container transportation time while meeting the yard’s needs.

2.2. Basic Assumptions

The container transport vehicle scheduling optimisation model needs to comply with the following assumptions:

• All containers meet international standards, and all trucks and containers are free from fitness issues.
• A truck can transport a maximum of two containers at a time.
• The time taken by cranes to lift heavy or light boxes is negligible.
• There are no truck changes during a single container transportation process.
• The transportation route from the yard to the coal plant remains fixed and does not change midway.
• The weight of the loaded container must not exceed the vehicle’s maximum load capacity.
• The freight cost between nodes is stable, with no consideration for fluctuating transportation costs during off-peak seasons.
• During the operation period, equipment failures are not considered; thus, equipment is assumed to maintain regular operation throughout the production time.

2.3. Parameterization and Decision Variables

The parameters of the variables involved in the model are shown in

Table 1. Description of variables.

Notation	Instructions	Unit
Q	Total container requirements at the yard	t
N	Number of coal preparation plants	size
Di	Distance from the coal preparation plant to the freight yard	km
Ci	Capacity of coal preparation plant i	t/h
Si	Beginning stockpile at coal preparation plant i	t
M	Number of containerised vehicles in the yard	Number of vehicles
Lj	Maximum load of container transport vehicle j	t
Vj	Speed of container transport vehicles j	km/h
Pj	Unit transportation cost of container transport vehicles j	RMB/(t×km)
P2	Waiting cost per unit of time for container transport vehicles	RMB/h
K	Number of vehicles hauling coal for external transportation
Gk	Coal beneficiation plants visited by outbound vehicles k
Tk	Time of arrival of outbound vehicles k at the coal preparation plant	time slot
qk	Coal hauled by outbound vehicles k	t
s	Coal preparation plant loading speed	t/h
b	Coal unloading speed at the yard	t/h
$\mathrm{T}{1}_j^t$	Container transport vehicles j in t missions, time of departure from the yard	time slot
$\mathrm{T}{2}_j^t$	Container transport vehicles j at t mission, arrival time at coal processing plant	time slot
$\mathrm{T}{3}_j^t$	Container transport vehicles j at task t, time of start of coal loading	time slot
$\mathrm{T}{4}_j^t$	Container transport vehicles j at t missions, time of departure from coal plant	time slot
$\mathrm{T}{5}_j^t$	Container transport vehicles j in t missions, return time to yard	time slot
$\mathrm{T}{6}_j^t$	Container transport vehicles j in t missions, time to unload cargo	time slot
${\mathrm{waiting}\mathrm{T}}_j^t$	Container transport vehicles j waiting time at task t before coal loading begins	time slot
${\mathrm{working}\mathrm{T}}_j^t$	Length of time container transport vehicles j are out of and back to the yard in t missions	time slot
$total{Q}_i$	Total coal loading at coal preparation plant i	t
${\mathrm{X}}_j^t$	Decision variable, container transport vehicles j at t tasks to go to the target coal processing plant id
${\mathrm{A}}_j^t$	Decision variable, the amount of coal pulled by container transport vehicles j at t missions to the target coal preparation plant	t

.

2.4. Container Transport Vehicle Scheduling Optimisation Model Construction

The primary objective of scheduling container transport vehicles is to streamline the execution of the transportation plan. This involves ensuring seamless coordination between the container transport vehicles and outbound vehicles to minimise wait times for loading. By optimising these operations, the transport vehicles can operate more consistently, thus reducing the overall time required to complete the transportation plan. Additionally, it aims to complete the transportation tasks while minimising total costs and reducing the time taken for container transit. Therefore, the objective function is formulated to achieve the lowest possible total costs and the shortest container transportation time while ensuring the completion of the transportation tasks. Therefore, the objective function is established as follows.

$$\begin{array}{l}\min f1={\displaystyle \sum _{i=1}^N{\displaystyle \sum _{j=1}^M{\displaystyle \sum _{t=1}^T{\mathrm{X}}_{ij}^t\times A_{ij}^t\times D_{\mathrm{i}}\times P_j}}}+{\displaystyle \sum _{j=1}^M{\displaystyle \sum _{t=1}^T{\mathrm{waitingT}}_j^t\times P_2}}\\ \min f2={\displaystyle \sum _{j=1}^M{\displaystyle \sum _{t=1}^T{\mathrm{workingT}}_j^t}}\\ \min F=\alpha \times f1+\beta \times f2\end{array}$$

(1)

where the first item is the total cost of container transportation and is equal to the price of container transport vehicle transportation plus the cost of waiting time for container transport vehicle, and the second item is the container turnaround time. The third item is the total objective, requiring the lowest total cost and the shortest container turnaround time, and each of them is given a weight: α is the first and β is the second.

Constraints:

$${\mathrm{T}2}_j^t={\mathrm{T}1}_j^t+\frac{{\mathrm{D}}_j^t}{{\mathrm{V}}_j}$$

(2)

$${\mathrm{T}3}_j^t={\mathrm{T}2}_j^t+{\mathrm{waitingT}}_j^t$$

(3)

$${\mathrm{T}4}_j^t={\mathrm{T}3}_j^t+\frac{A_j^t}{s}$$

(4)

$${\mathrm{T}5}_j^t={\mathrm{T}4}_j^t+\frac{{\mathrm{D}}_j^t}{{\mathrm{V}}_j}$$

(5)

$${\mathrm{T}6}_j^t={\mathrm{T}5}_j^t+\frac{{\mathrm{D}}_j^t}{b}$$

(6)

$${\mathrm{T}1}_j^t=\{\begin{array}{cc}0& t=1\\ {\mathrm{T}6}_{j-1}^t& t>1\end{array}$$

(7)

$${\mathrm{waitingT}}_j^t={\mathrm{T}3}_j^t-{\mathrm{T}2}_j^t$$

(8)

$${\mathrm{workingT}}_j^t{=\mathrm{T}5}_j^t-{\mathrm{T}1}_j^t$$

(9)

$$A_j^t\le L_j$$

(10)

$${\mathrm{T}3}_j^t\notin [{\mathrm{T}3}_i^t,{\mathrm{T}4}_i^t]$$

(11)

$$S_i\le total{Q}_i$$

(12)

$$A_j^t=2W$$

(13)

(2) indicates that the time of arrival of the container transport vehicle at the coal preparation plant is equal to the time of departure from the yard plus the time of travel of the container transport. (3) indicates that when the coal preparation plant starts loading coal, it is equal to when the containerised vehicle arrives at the coal preparation plant plus the time when the containerised vehicle waits. (4) indicates that the time the container left the coal preparation plant is equal to the time the coal preparation plant started loading coal plus the time the coal preparation plant loaded coal. (5) indicates that the time for the container transport vehicle to return to the yard is equal to the time for the container transport vehicle to leave the coal preparation plant plus the time for the container transport vehicle to travel. (6) indicates that the unloading time of a containerised transport vehicle is equal to when the car returns to the yard plus the unloading time. (7) indicates that when the container transport vehicle leaves the yard, it is equal to when the cargo is unloaded. (8) indicates that Container Transportation Vehicle (CTV) Waiting Time is equal to CTV Starting Coal Loading Time minus CTV Arrival Time at the Coal Processing Plant. (9) indicates the container transportation time when the container transport vehicle finished unloading the cargo and when the container transport vehicle travelled to the coal processing plant. (10) indicates that the amount of coal loaded in the container is less than or equal to the maximum amount of coal loaded in the container. (11) indicates that the time when the containerised vehicle started loading coal is less than the time when the last vehicle was loaded (the time when the last container was loaded or the time when the outbound vehicle was loaded). (12) indicates that the initial stockpile of coal in the coal preparation plant is less than or equal to the amount of coal pulled by containers. (13) indicates that containerised vehicles can only be loaded with two containers simultaneously.

3. GA−GWO Hybrid Algorithm

3.1. GA Algorithm

GA is an RMB-inspired algorithm inspired by the laws of biological evolution in nature that simulates a computational model of the biological evolution process based on Darwin’s theory of biological evolution and Mendel’s laws of genetics. This algorithm was first proposed by Holland in 1992. It is a highly efficient, parallel, global search algorithm [28], which, after more than 50 years of development, has been widely applied to various optimisation problems, such as continuous optimisation, discrete optimisation, combinatorial optimisation, etc., and has also been used in many fields, such as image processing, automatic control, route planning, network optimisation, data mining, etc., and produced good results. Optimisation, discrete optimisation, combinatorial optimisation, etc. It has been used in various areas such as image processing, automation, route planning, network optimisation, data mining, etc., with good success. GA is classified in a larger field known as evolutionary computation, a computational model built on the assumption that natural operations of interactions between biological entities can be replicated as problem-solving strategies. [29]. Although genetic algorithms are also ineffective in avoiding falling into local optima, their crossover and mutation operations maintain population diversity and avoid premature convergence, thus making it easier for the algorithms to converge to the global optimum [30].

The general process of the genetic algorithm is as follows: problem definition, initialisation parameters, initialisation of population, selection operation, crossover operation, mutation operation, calculation of population fitness, elite strategy, and its algorithmic framework are shown in

Table 2. Genetic algorithm (GA) framework.

Step	Marginal Notes
model	Init Model Of Problem()
D	Decision variable dimensions
crossover rate
mutation rate
population	Initial Population (N, D)
Pop Fitness	Obtain Population Fitness (population, N)
Best Individual Set	Max gen × D-dimensional all-0 matrices
Best Fitness Set	1 × max Gen-dimensional all-0 vectors
for i	1: NexGen
Selection Operation (population, pop Fitness)
Crossover Operation (new Population, crossover Rate)
Mutation Operation (new Population, mutation Rate)
New Pop Fitness	Obtain Population Fitness (new population);
population, pop Fitness	Elite Selection Strategy
Best Individual Set, best Fitness Set	Updating the current optimal
return Population	Return to Final Evolutionary Population

.

3.2. GWO Algorithm

GWO is a recently developed meta-heuristic algorithm [31] inspired by the leadership and hunting behaviour of grey wolves in nature. The purpose of using GWO for the study is to explore its different search mechanisms based on grey wolf leadership behaviour. The grey wolf population is divided into four strata from top to bottom: α, β, δ, ω. Grey wolves in these four strata have different divisions of labour in hunting: α stratum is the grey wolf leadership, which is responsible for all kinds of decision-making matters of the pack; β stratum is the grey wolf management, which acts as the think-tank of the pack and is mainly responsible for assisting α wolves in making decisions and hunting matters; δ stratum is the grey wolf ordinary stratum, which is responsible for carrying out the hunting instructions issued by the α and β stratum; and ω is the grey wolf bottom stratum, which listens to the instructions given by α, β, δ stratum. δ layer grey wolves’ instructions and move towards the optimal direction for hunting. The flowchart of the GWO algorithm is shown in Figure 1. In GWO, the search direction is determined by the leading wolves (α, β, δ) during the whole iteration process, which helps to ensure fast convergence speed. GWO has a self-adaptive adjusting convergence factor and information feedback mechanism, which can realise the balance between local optimisation and global search and has good performance in terms of the accuracy of solving problems and convergence speed.

Steps for solving the GWO problem:

Step 1.

Initialise the grey wolf population: Establish the number and initial positions of grey wolves, randomly distributed across the search space.

Step 2.

Calculate the fitness value of the grey wolves: according to the optimisation objective of the problem, calculate the fitness value of each grey wolf, which measures the degree of superiority or inferiority of the individual in the current position.

Step 3.

Update alpha, beta, and delta grey wolves: Assign the grey wolf with the highest fitness as alpha, the second highest as beta, and the third highest as delta.

Step 4.

Group collaboration: According to the current positions of alpha, beta, and delta grey wolves, other grey wolves can approach alpha, beta, and delta by adjusting their positions.

Step 5.

Update grey wolf position: according to specific iteration rules and formulas, update the position of grey wolves to achieve better optimisation.

Step 6.

Adaptation evaluation: calculate the new position’s adaptation value and update each grey wolf’s adaptation.

Step 7.

Judge the termination condition: Judge whether the termination condition is satisfied, such as reaching the maximum number of iterations or finding the optimal solution that satisfies a specific condition.

Step 8.

Iterate steps 3–7: Repeat steps 3–7 until the termination condition is satisfied.

Step 9.

Output: Output the found optimal solution or near-optimal solution, and other statistical information can also be output.

Essential Elements of the Grey Wolf Optimisation Algorithm

(1)

Surrounding:

After determining the location of the prey, the wolf should first encircle the prey, during which the distance between the prey and the grey wolf can be expressed.

$$D=|C\cdot X_P(t)-X(t)|$$

(14)

$$X(t+1)=X_P\left(t\right)-A·D$$

(15)

where D is the distance between the grey wolf and the prey, t is the number of generations selected, X_p(t) is the position of the prey after it has been selected (i.e., the position of the optimal solution) in generation t, X(t) is the position of the grey wolf after it has been selected (i.e., the position of the potential solution) in generation t, and A and C are the coefficient factors, which are computed in the following formula:

$$A=2a\cdot r_1-a$$

(16)

$$C=2\cdot r_2$$

(17)

A decreases linearly from 2 to 0 as the number of selected generations increases, and r₁ and r₂ are random numbers between [0, 1].

(2)

Hunting:

Grey wolves recognise the location of their prey and surround them. When the grey wolf recognises the location of the prey, β-wolf and δ-wolf are α under the leadership of the wolf and guide the wolf to surround the prey. In the decision space of the optimisation problem, we do not know the optimal solution (location of the prey). Therefore, to model the hunting behaviour of grey wolves, we assume that α-wolf, β-wolf, and δ-wolf are better informed about the potential location of the prey. We have kept the three best solutions so far. We used the positions of these three solutions to determine the location of the prey while forcing other grey wolf individuals (including ω) to update their positions based on the position of the best grey wolf individual and gradually approach the prey.

After encircling the prey, β wolves and δ wolves chased the prey under the leadership of α wolves. During the hunting process, the positions of individual wolves change as the prey escapes, and then the prey is re-located based on the updated α, β, and δ positions. In this phase, the wolf positions are updated as follows:

$$\{\begin{array}{c}{D}_{\alpha }=|C_1{X}_{\alpha }(t)-X(t)|\\ D_{\beta }=|C_2{X}_{\beta }(t)-X(t)|\\ D_{\sigma }=|C_3{X}_{\sigma }(t)-X(t)|\end{array}$$

(18)

$$\{\begin{array}{c}{X}_1=X_{\alpha }(t)-A_1{D}_{\alpha }\\ X_2=X_{\rho }(t)-A_2{D}_{\rho }\\ X_3=X_{\delta }(t)-A_3{D}_{\delta }\end{array}$$

(19)

$$X_p(t+1)=\frac{(X_1+X_2+X_3)}{3}$$

(20)

where D_α, D_β, and D_δ denote the distance between α, β, and δ wolves and ω wolves (other individuals), respectively.

(3)

Attack

The optimal solution is obtained when the wolf pack attacks and captures the prey. This process is mainly realised by the decrease in the value of A in Equation (16). When the value of a decreases linearly from 2 to 0, the corresponding value of A also varies in the interval [−a, a]. In addition, when |A| < 1, i.e., the range of A values in [−1, 1], it indicates that the next position of the wolf pack will be closer to the position of the prey; when 1 < |A| < 2, the wolf pack will be dispersed in the direction of A, which is far away from the prey, resulting in the convergence of the GWO algorithm to the optimal solution, and thus entering into the process of A-local optimisation.

3.3. GA–GWO Hybrid Algorithm

The GWO algorithm offers robust search capabilities; however, its convergence properties mean that the GA algorithm can effectively address the issue of local optima through genetic variations and selection operations. Additionally, the competitive behaviour among GWO’s grey wolves facilitates a local search mechanism. Therefore, the two algorithms are mixed, through which the hybrid algorithm can better balance the global search and local search and take into account the solution needs of the global optimum and local optimum while optimising the hybrid algorithm:

• Piecewise chaotic mapping is used to initialise the total grey wolf population, and the expression of piecewise mapping is as follows:

$$x(t+1)=\{\begin{array}{c}\begin{array}{c}\begin{array}{c}\frac{x(t)}{p},0\le x(t)<p\\ \frac{x(t)-p}{0.5-p},p\le x(t)<0.5\end{array}\\ \frac{1-p-x(t)}{0.5-p},0.5\le x(t)<1-p\end{array}\\ \frac{1-x(t)}{p},1-p\le x(t)<1\end{array}$$

(21)

$P\in \left[0,1\right], p=0.4;x(1)=rand$.

2.

Convergence factor and adjustment strategy

$$a=\{\begin{array}{c}{a}_{final}+(a_{initial}-a_{final})\times {\cos }^{\Lambda }2\left[\frac{t-1}{2(T-1)}*\pi \right],t\le 0.5T\\ a_{intial}-\left[a_{final}+(a_{inttial}-a_{final})\times {\sin }^{\Lambda }2(\frac{t-1}{2(T-1)}*\pi )\right],0.5T\le t\le T\end{array}$$

(22)

$a_{final}=0;a_{initial}=2;$ t is the current iteration number, and T is the maximum iteration number.

3.

Position update formula

Proportional weights based on step size and Euclidean distance 128, 256, 512, and 512, respectively. Through four rounds of downsampling, the input river remote sensing image is condensed into a 32 pixels × 32 pixels × 512 size output image, effectively capturing the essential features of the river in the remote sensing image.

$$L_1=\frac{\left|X_1\right|}{\left|X_1\right|+\left|X_2\right|+\left|X_3\right|}$$

(23)

$$L_2=\frac{\left|X_2\right|}{\left|X_1\right|+\left|X_2\right|+\left|X_3\right|}$$

(24)

$$L_3=\frac{\left|X_3\right|}{\left|X_1\right|+\left|X_2\right|+\left|X_3\right|}$$

(25)

$$X(t+1)=\frac{L_1{X}_1+L_2{X}_2+L_3{X}_3}{3}{(\frac{t-1}{T-1})}^{{\lambda }_1}+{(X_1\frac{t}{T})}^{{\lambda }_2}$$

(26)

included among these is ${\lambda }_1 ,{\lambda }_2\in rand\left[0,1\right]$.

4.

GA parameter design based on hormone regulation

Typically, the population evolution process is divided into two phases: (1) the early phase, which features high crossover and low mutation probabilities, promoting rapid population convergence and preserving desirable traits; and (2) the late phase, characterised by low crossover and high mutation probabilities, which enhance search refinement and maintain population diversity. However, traditional GAs have fixed crossover and mutation probabilities, making parameter tuning challenging and impacting optimisation outcomes. Consequently, an improved adaptive genetic algorithm inspired by the regulatory mechanisms of endocrine hormones, which feature ascending and descending functions, is proposed.

Meanwhile, population diversity is also closely related to the selection operator, so it is necessary to set an adaptive selection threshold regulated by hormones.

The fundamental law of endocrine hormones was proposed by L.S. Farhy [32] in 2004, which reveals that hormones have monotonic and non-negative change characteristics. The ascending regularity function F_up(G) and the descending regularity function F_down(G) of hormone regulation follow Hill’s function law, which is expressed in the following equation:

$$F_{np}(G)=\frac{G^n}{D^n+G^n}$$

(27)

$$F_{down}(G)=\frac{D^n}{D^n+G^n}$$

(28)

G is the function independent variable; D is the threshold value; D > 0; n is the Hill coefficient; and n 1; n and D determine the slope of the upward descent curve.

If hormone x is regulated by hormone y, the relationship between the secretion rate S_x of hormone x and the concentration C_y of hormone y is as follows:

$$S_x=\alpha F(C_y)+S_{x0}$$

(29)

S_x0 is the basal secretion rate of hormone x, and α is a constant coefficient.

$$V_{up}=\alpha \frac{C_y^n}{D^n+C_y^n}+S_{x0}=S_{x0}(1+\frac{\alpha }{S_{x0}}·\frac{C_y^n}{D^n+C_y^n})$$

(30)

$$V_{down}=\alpha \frac{D^n}{D^n+C_y^n}+S_{x0}=S_{x0}(1-\frac{\alpha }{S_{x0}}\cdot \frac{C_y^n}{D^n+C_y^n})+\alpha$$

(31)

The population update in the grey wolf algorithm is that each approaches this optimal three wolves, which has the advantage of fast convergence. Still, it is easy to fall into the local optimum. The evolutionary strategy of the genetic algorithm can compensate for this point well. Therefore, the two algorithms are combined, and the population evolves in each generation with the updating strategy of the grey wolf algorithm and the updating strategy of the genetic algorithm, respectively. The flow of the hybrid GA–GWO algorithm (shown in Figure 2).

Steps of the GA–GWO hybrid algorithm to solve the problem:

• Import coal processing plant information, container information (container number, demand), and other fundamental data for loading data.
• Initialise the model: first of all, it is necessary to initialise the model of the problem, that is, to determine the objectives and constraints of the problem.
• Initialise aInitial and aFinal: aInitial and aFinal are used to control the range of parameter updates during the iteration process of the algorithm. aInitial and aFinal determine the minimum and maximum values for updating the grey wolf position.
• Initialise Hill coefficient n and threshold value D: Hill coefficient n is used to regulate the speed of grey wolf position update and control the degree of contraction of grey wolf search space, which is generally a positive integer. The threshold value D is used as a regulation parameter in repairing the population, and the threshold value is used to judge whether the grey wolf needs to carry out the positional repair.
• Initialise the grey wolf population with piecewise chaotic mapping: This method enhances algorithm randomness and search performance by generating the initial population.
• Repairing the population: for each grey wolf in the population, check whether its position is out of the range of the search space of the problem. If it is out of range, its position is repaired to a reasonable range. This ensures the legitimacy of the population location.
• Fitness evaluation pop Fitness: Calculate each grey wolf in the repaired population’s fitness value. The fitness function is based on the specific requirements of the problem and is used to evaluate the degree of merit of the grey wolf’s location. The higher the fitness value, the more excellent the location.
• Determine the termination conditions: During the iterative process, evaluate whether the conditions for termination, such as reaching the maximum number of iterations or achieving target accuracy, have been met. If so, select the optimal individual and conclude the algorithm.
• Update vUp and vDown in hormone regulation if termination conditions are not met: Adjust the grey wolf population’s search range using hormone-based regulators to enhance the search process.
• Update crossover probability Pc and mutation probability Pm: crossover probability and mutation probability are two critical parameters in genetic algorithms, and the search behaviour of genetic algorithms can be adjusted by updating them.
• Binary Tournament Selection: Through the binary tournament selection method, select two individuals in the population for comparison and select the individual with higher fitness as the parent to enter the crossover and mutation operations.
• Crossover and mutation operations to obtain a new population: Through crossover and mutation operations, operate on the selected individuals of the parent generation to generate a new population.
• Repair the population: for each grey wolf in the new population, position repair is performed to adjust the position beyond the search space range to a reasonable range.
• Adaptation evaluation new Pop Fitness: Adaptation evaluation is carried out for the repaired new population, and its adaptability value is calculated.
• Elite strategy to update the population: According to the fitness value, adopt an elite strategy to update the population and keep a part of the individuals with the highest fitness.
• Update convergence factor a: according to the convergence state of the algorithm, adjust the search speed and range of the algorithm by updating the convergence factor a.
• Obtaining the position of α, Β, and γ wolves: according to the current fitness value of the population, select the three individuals with the highest fitness as α, Β, and γ grey wolves from the population.
• Update the wolf population based on step length Euclidean distance proportional weight new Population: according to the step length Euclidean distance proportional weight formula, update the position of grey wolves in the latest population using the three grey wolves α, Β, and γ as references.
• Repair the population: repair the position of the updated new population and adjust the position beyond the range of search space to a reasonable range.
• Adaptation Evaluation New Pop Fitness: Evaluate the adaptability of the repaired new population and calculate its adaptability value.
• Elite strategy to update the population: According to the fitness value, adopt the elite strategy to update the population and keep a part of the individuals with the highest fitness.
• Iterate steps 7–20: Repeat steps 7–20 until the termination conditions are met.
• Output results: output the found optimal or near-optimal solution, which can also output other statistical information.

The pseudo-code of the GA–GWO hybrid algorithm is shown in

Table 3. GA–GWO hybrid algorithm pseudocode.

GA–GWO Hybrid Algorithm Pseudocode
	Input: N (population size), maxGen (maximum number of evolutionary generations)
	Output: Population (final evolved population)
1	model ← initModel Of Problem ();	Initialisation problems
2	aInitial, aFinal ← 2, 0;	Initialise aInitial, aFinal
3	n, D ← 1, 2;	Initialise Hill coefficient n, threshold D
4	Pc0, Pm0 ← 0.6, 0.02;	Initialise crossover, mutation probability
5	D ← Decision variable dimensions;	D is an integer
6	pop ← Initial Population By Piecewise (N, D);	Piecewise Chaos Mapping Initial Grey Wolf Population
7	pop ← repair Operation (pop, model);	population restoration
8	popFitness ← get Population Fitness (pop, model);	Calculating population fitness
9	Best IndividualSet ← maxGen × D-dimensional all-0 matrices;	The set of optimal individuals per generation
10	Best Fitness Set ← 1 × maxGen-dimensional all-0 vector;	Highest fitness set per generation
11	for t ← 1: maxGen
12	Vup, Vdown ← get Vup And Vdown (t, D, n, 0);	hormone renewal
13	Pc ← Pc0 × Vdown;	Hormone Renewal Cross Probability
14	Pm ← Pm0 × Vup;	Hormone Renewal Mutation Probability
15	newPop ← selection Operation (pop, popFitness);	Selection of operation
16	newPop ← crossover Operation (newPop, Pc);	crossover operation
17	newPop ← mutation Operation (newPop, Pm);	variant operation
18	newPop ← repair Operation (newPop, model);	population restoration
19	newPopFitness ← get Population Fitness (newPop);	Adaptation of new populations
20	pop, popFitness ← elite Strategy (pop, newPop, popFitness, newPopFitness);	Merger of child and parent populations, selection (elite strategy)
21	a ← updateA (aInitial, aFinal, t, maxGeneration);	Updating the convergence factor a
22	alphaWolf, betaWolf, deltaWolf ← getEliteWolf (pop, popFitness);	Obtain the first 3 elite wolves
23	newPop ← evolve Population GWO (alphaWolf, betaWolf, deltaWolf, pop, a, t, maxGen);	Updating wolf packs based on step size Euclidean distance proportional weights
24	newPop ← repair Operation (newPop, model);	population restoration
25	newPopFitness ← get Population Fitness (newPop);	Adaptation of new populations
26	pop, popFitness ← elite Strategy (pop, newPop, popFitness, newPopFitness);	Merger of child and parent populations, selection (elite strategy)
27	Best Individual Set, best Fitness Set;	Updating the current optimal
28	return Population;	Return to the Final Evolutionary Population

.

3.4. SA Algorithm

The simulated annealing algorithm is actually a greedy algorithm, which can easily fall into a local optimum solution while potentially failing to find the best solution globally. The simulated annealing algorithm can probabilistically jump out of the local optimal solution. The core idea is as follows: if moving in a certain direction, as long as the result is better than the current value, then continue to iterate forward, but if the result is worse than the current value, then there is a certain probability to accept this value, and then continue to iterate with this value, and then loop iteration, ultimately, may be able to find the global optimal solution.

As shown in Figure 3, the goal is to find the minimum value of the function in this interval. The initial value comes to point A, and after the above steps to go a period of time to the local optimal solution point B, at this time, and then go, the result will be worse than the current local optimal solution, but the simulated annealing algorithm will allow him to continue to the right with a certain probability, and ultimately there is a possibility that he will be able to find the global optimal solution point C. The simulated annealing algorithm generally has two layers of cycles: the external cycle on behalf of the temperature continues to fall, the internal cycle is at this temperature many times the perturbation to produce different states, the beginning of the temperature is relatively high, the probability of the perturbation will be relatively large, with the external cycle of the temperature getting lower and lower, and gradually stabilised in the region of the probability of the perturbation will be reduced, ultimately staying at a certain value. The simulated annealing algorithm’s internal cycle is to accept the new state according to the Metropolis criterion. According to the Metropolis criterion, the probability of the particle randomly perturbed at temperature T is exp (−∆E/KT), where E is the internal energy at temperature T, ∆E is the number of changes in the internal energy, and k is Boltzmann’s constant. The Metropolis criterion is the following equation:

$$\begin{array}{c}p=\{\begin{array}{ccc}1& ;& if E\left(X_{new}\right)=E\left(X_{old}\right)\\ e^{\frac{E\left(X_{new}\right)-E\left(X_{old}\right)}{T}}& ;& if E\left(X_{new}\right)>E\left(X_{old}\right)\end{array}\end{array}$$

(32)

Components of the simulated annealing algorithm

• Search space: Also known as the state space, it consists of all feasible solutions to the problem; each state in the space corresponds to a feasible solution.
• Energy function: In the search space, there must be a function as the optimisation objective; the minimum point of this function is the optimal solution to the required problem.
• State transfer rule: In the simulated annealing algorithm, the state transfer rule refers to the transfer probability, which is the probability of transitioning from one state to another, related to the temperature parameter.
• Cooling schedule: It is the table that manages the temperature drop and records the cooling process.

Specific implementation steps of the simulated annealing algorithm

Step 1.

Initialization. Randomly select an initial solution x₀, so that the current solution is x_old = x₀, the current iteration step is k = 0, the current annealing temperature is t_k = t_max, and calculate the objective function E(x_old);

Step 2.

Randomly select a neighbour from the neighbourhood of the current initial solution x_old as the new solution x_new; calculate the objective function E(x_new);

Step 3.

Calculate $△E_{old;new}=E\left(X_{new}\right)-E\left(X_{old}\right)$, if $△E_{old;new}=0$, then accept the new solution, $X_{new}=X_{old}$,$E\left(X_{new}\right)=E\left(X_{old}\right)$; otherwise, accept the new solution in the case of $e^{-\frac{△E_{old;new}}{t_k}}>rand\left(0;1\right)$ ($rand\left(0;1\right)$, which denotes a uniform random number between 0 and 1).

Step 4.

Judge whether the pre-set number of iterations of the algorithm is reached; if not, then jump to Step 2; if the pre-set number of iterations of the algorithm is reached, then continue to judge whether the set termination conditions are met; if the set convergence conditions are not met, then the annealing temperature is slowly lowered by the annealing speed function and the number of iterations is updated; and then return to Step 2; if the termination conditions are met, then the loop ends and returns to the optimal solution. If the termination conditions are met, the loop ends, and the optimal solution is returned.

The flow chart of the above-simulated annealing algorithm is shown in Figure 4.

4. Instance Analysis

To verify the effectiveness of the hybrid GA–GWO algorithm applied in a real scenario of scheduling container transportation vehicles in surface coal mines., the actual transportation scheduling data of container vehicles of a coal mine on a particular day are taken as an example, and the GA–GWO hybrid algorithm is applied to solve the optimisation model of coal mine container vehicle scheduling. The performance of the SA, GA, and GWO algorithms is compared and analysed using this hybrid approach.

The coal mine has one goods yard, four coal preparation plants, and coal preparation plant attributes including the distance between the goods yard and the coal preparation plant, capacity efficiency, initial inventory, coal loading speed, specific information as shown in

Table 4. Coal preparation plant information.

The Coal Preparation Plant Id	Distance (km)	Capacity Efficiency (t/h)	Initial Stockpile (t)	Coal Loading Rate (t/h)
No. 1 Coal Preparation Plant Information	60	1300	64	1830
No. 2 Coal Preparation Plant Information	62	1607	82	1090
No. 3 Coal Preparation Plant Information	56	1919	56	1330
No. 4 Coal Preparation Plant Information	92	1195	67	1740

, loading and unloading speed of 2000 t/h, and unit time waiting cost of 50 RMB/h. Containerised vehicles are 30, and the maximum coal loading capacity of the container is 34 t. The average speed of the cars travelling with heavy and empty loads is 22 and 34 km/h, respectively, and the average speed of the cars driving with heavy and empty loads is 22 and 34 km/h, respectively. The average speed of the vehicles travelling with heavy and empty loads is 22 and 34 km/h, respectively.

GA–GWO parameter settings: The search space dimension is 28, the initial population size is 20, the number of evolutionary iterations is 2000, the maximum number of elite iterations is 500, and the crossover probability and variance probability are 0.6 and 0.1, respectively. GA parameter settings refer to the literature [33], while GWO algorithm parameter settings refer to the literature [8]. To reduce the algorithm’s complexity in time, this paper directly takes the objective function as the fitness function. It uses Matlab 2022b to simulate and solve the problem of scheduling coal mine container vehicles, and the evolution curve is shown in Figure 5. Figure 5 illustrates that, compared to SA, GWO, and GA, the GA–GWO hybrid algorithm shows an improvement of 43.1%, 46.2%, and 43.7% in optimisation speed, respectively. The optimisation accuracy is enhanced by 4.12%, 6.1%, and 3.2%, respectively. Furthermore, the evolution curve of GA–GWO exhibits rapid changes prior to 500 iterations and stabilises thereafter. The evolutionary curve of GA–GWO changes rapidly before 500 iterations and stabilises after 500 iterations. This is because GA–GWO inherits the strong computational ability and global search ability of GA in the early stage, which makes the population converge to the global optimum quickly; in the middle and late stages, the addition of the GWO algorithm allows the population to concentrate around the most optimal population quickly, which reduces the probability of GA falling into the slow convergence in the middle and late stages and thus balances the global optimum with the local optimum.

The population evolution curve of the algorithm includes two indicators, the best objective value and the average objective value, and the population evolution curve is shown in Figure 6. In SA, the randomness of the initial value causes the algorithm to fall into a local optimal solution in the initial stage, which increases the optimisation difficulty and running time during the iteration process. In GA, the best and average objective values converge gradually with the number of iterations. However, as the genetic algorithm proceeds, due to the failure to utilise the feedback information from the network promptly, the search speed of the later algorithm slows down, and more training time is required to obtain a more accurate solution. In GWO, the grey wolves search and evolve with each other to find a better solution, and the whole population grows towards a better objective function. Because of the significant dependence on the initial solution, the quality of the initial solution will directly affect the algorithm’s convergence, and the final result will cause the algorithm to fall into the local optimal solution. The optimised GA–GWO hybrid algorithm integrates the strengths of both the genetic and grey wolf optimisation algorithms. It demonstrates potentially faster convergence and improved outcomes for both optimal and average objective values on the population evolution curve. By employing the genetic algorithm’s crossover and mutation operations, combined with the search strategy of the grey wolf optimisation algorithm, the algorithm enhances its global and local search capabilities during evolution. Consequently, this comprehensive approach allows the algorithm to converge more efficiently to the global optimum solution.

The waiting time of container vehicles serves as one of the objective functions, and the comparative waiting times under the three algorithms are displayed in Figure 7. GA selects ‘excellent’ individuals based on fitness function results and performs further operations on them. Through many iterations, SA calculates the objective function difference based on random selection within the domain of the current solution. The waiting time of container transportation vehicles is reduced to a certain extent by the number of iterations after the temperature is reduced to a certain threshold. GA will gradually optimise the vehicle scheduling; we can see that the waiting time of 30 vehicles in the last batch is getting less and less, and GWO searches and adjusts by simulating the collaborative and competitive behaviours of grey wolves and gradually converges to the optimal solution, which can effectively find the solution with less waiting time in the vehicle scheduling scheme. However, some vehicles still have a long waiting time. The GA–GWO hybrid algorithm considers both the global search and the optimisation of the process. The process finds both worldwide search and local search capabilities. Through the crossover and mutation operations of the genetic algorithm and the search strategy of the grey wolf optimisation algorithm, the solution space can be searched more finely, and the overall waiting time of the container transport vehicles can be found to be an optimised solution with less waiting time.

Container reversal time is an essential indicator of the model. The container reversal time of the three algorithms is shown in Figure 8. SA does not have a more reasonable scheduling optimisation for subsequent batches of vehicles after falling into a local optimal solution in the previous period. GA may be unable to optimise the subsequent batches’ transportation order, leading to an extended overall container reversal time. GWO may have a locally optimal solution problem in specific cases. In the following batches, there may be a possibility that some vehicles are not reasonably scheduled, which leads to a relatively extended container reversal time. The GA–GWO hybrid algorithm can better optimise the vehicle transportation order and make the overall container shortest time.

In summary, after analysis, the GA–GWO hybrid algorithm solved the container reversal-time problem better. In practical applications, the GA–GWO hybrid algorithm is chosen to optimise vehicle scheduling to minimise the short backward time and improve the efficiency of container transportation.

In the GA–GWO hybrid algorithm, 30 container vehicles are dispatched in a maximum batch of 7 times, and the results of vehicle dispatching are shown in

Table 5. Container transport vehicle scheduling results.

	Vehicle Lots	1	2	3	4	5	6	7
Vehicle ID
1		1	3	4	2	2	1	0
2		2	1	3	1	2	0	0
3		3	1	1	3	0	0	0
4		3	1	2	2	0	0	0
5		4	2	3	1	0	0	0
6		3	3	1	1	2	0	0
7		4	1	4	4	1	0	0
8		3	3	2	3	2	0	0
9		1	3	3	1	1	1	0
10		1	1	1	0	0	0	0
11		2	4	1	1	1	0	0
12		1	1	3	3	2	0	0
13		3	3	1	1	1	0	0
14		1	1	1	0	0	0	0
15		2	3	2	3	0	0	0
16		3	1	3	2	0	0	0
17		3	3	3	1	1	0	0
18		1	1	1	3	3	0	0
19		3	2	1	2	1	3	0
20		1	3	4	4	0	0	0
21		4	1	3	1	0	0	0
22		1	1	3	1	0	0	0
23		3	2	1	3	0	0	0
24		3	3	3	1	1	0	0
25		1	1	1	0	0	0	0
26		2	4	1	1	4	1	0
27		1	3	1	4	0	0	0
28		4	2	3	1	3	2	1
29		2	1	1	2	1	0	0
30		1	4	3	1	2	1	0

.

In the actual container vehicle scheduling, one shift requires 30 transport vehicles; combined with the specific parameters of the coal mine, the exact total cost of a change in container reversal operation is 9121.6 RMB, the reversal time is 72 h, and the container reversal completion time is 3.2 h, respectively, based on the SA, GA, GWO, and GA–GWO solutions for coal mine container transportation. The total cost of vehicle scheduling, container-to-short time, and container-to-short plan completion time are shown in

Table 6. Comparison of container transport vehicle scheduling results.

#	SA	GA	GWO	GA–GWO
total cost (RMB)	8794.428	8636.857	8948	8446.286
reversal time (h)	67.065	66.429	65.143	62
Finish time (h)	2.995	2.874	2.713	2.574

.

Table 5 shows that, compared with the actual transportation data, the total cost solved by the SA, GA, and GWO algorithms within the same shift only saves 327.171 RMB, 484.74 RMB, and 173.6 RMB, respectively. In contrast, the total cost solved by the hybrid GA–GWO algorithm saves 675.31 RMB, and the container’s backward time and planned completion time save ten h and 0.63 h.

5. Conclusions

• The container transportation cost and vehicle waiting time cost are incorporated into the container backwards short cost, forming an optimisation model aimed at minimising both container transportation time and backwards short cost.
• To address the slow convergence speed of the GA, a hybrid GA–GWO algorithm has been developed by integrating the GWO algorithm into the GA search process. This hybrid algorithm not only comprehensively searches the solution space, balancing global and local optimisation capabilities, but also significantly enhances convergence and stability.
• The GA–GWO hybrid algorithm has been applied to optimise the scheduling of container transport vehicles in coal mines. This application has improved the algorithm’s convergence accuracy and speed while also reducing total costs, container reversal time, and completion time of the container schedule, outperforming both the SA, GA, and GWO algorithms.

Although the hybrid GA–GWO algorithm provides new ideas for the solution of the inverted-short scheduling problem for coal mine containers, there is still room for improvement and depth in the scheduling strategy. Due to the carbon emissions from traditional fuel trucks in the container transport process, transportation costs are high, and with the release and implementation of the “double carbon” policy, as well as the initial application of pure electric trucks in the surface coal mine transport, low-carbon transportation will be the hot direction of the surface coal mine transport. Compared with traditional fuel trucks, pure electric trucks improve transportation efficiency, reduce vehicle queuing time, and greatly reduce carbon emissions and transportation costs. However, it is necessary to consider the choice of charging stations for pure electric trucks, battery charging time, and re-planning the vehicle transportation path.