Study of the Multilevel Fuzzy Comprehensive Evaluation of Rock Burst Risk

Abstract

Rock burst is a multifaceted phenomenon that involves various intricate factors. A precise evaluation of its risk encounters numerous challenges. To address this issue, the present paper proposed a multilevel fuzzy comprehensive evaluation model based on the Analytic Hierarchy Process–Fuzzy Comprehensive Evaluation (AHP-FCE) method. Three primary influencing factors and twelve secondary influencing factors that impact the rock burst risk were identified. The mechanisms by which each influencing factor affects the rock burst were analyzed and the membership degree for each factor was calculated accordingly. The weight of each influencing factor was determined through the AHP. To obtain a quantitative evaluation result, the evaluation model was calculated using the second-order fuzzy mathematics calculation method. The application of the model was demonstrated on the 310 working face of the Tingnan Coal Mine, and the evaluation results were consistent with those achieved through the use of the comprehensive index method and the probability index method. All of the results exhibited consistent alignment with the actual circumstances. The verification process confirmed the scientific, effective, and practical nature of the model.

1. Introduction

Rock burst is a significant safety hazard in coal mining operations, and its frequency and intensity of occurrence escalate in correlation with the progressive depth and intensity of the coal mining activities [1]. The rock burst risk encompasses both the probability of occurrence and the degree of damage caused by rock bursts during coal mining, which are influenced by the geological, mining technique, and safety management conditions [2]. The complexity of rock burst arises from the fact that it is a dynamic and unpredictable process that can be influenced by multiple factors and interactions. For example, changes in the stress conditions, rock properties, and mining operations can trigger or exacerbate rock burst. Also, rock burst can occur in different locations and scales, making it difficult to predict and control. Conducting risk evaluations for rock burst is crucial for ensuring the safety and productivity of mining operations. By conducting risk evaluations, mining operators can identify the areas and conditions that are most susceptible to rock burst and implement appropriate measures, such as reinforcement, support, and monitoring systems, to reduce the risks.

Currently, the evaluation of rock burst risk typically employs empirical analogy methods, represented by the comprehensive index method [3,4], as well as field measurement methods such as drilling cutting [5,6], stress monitoring [7,8], micro-seismic monitoring [9,10], electromagnetic radiation monitoring [9,11], and ground noise monitoring [9,12]. However, the empirical analogy methods have limitations on their ability to account for the interrelationships among the influencing factors, which can lead to a certain degree of bias in the analysis results. Additionally, the subjective judgments of the evaluators can also impact the accuracy of the method. While field measurement methods can produce more objective results, they are greatly affected by the complexity of geological conditions and mining technique conditions. Additionally, these methods provide an “in-process” evaluation, which can lead to time lags and hinder the effective guidance for mitigating rock burst occurrences during the early stages of working face mining or roadway excavation.

Rock burst is a multifaceted phenomenon that is influenced by the coupling of multiple factors. An accurate evaluation of the rock burst risk requires a comprehensive consideration of these multiple factors. Due to the uncertainty in the division of indicator values and the selection of influencing factors, it is difficult to use precise mathematical models and analytical methods to perform quantitative evaluations of rock burst risk. To address this issue, some scholars have achieved certain results by using fuzzy theory [13,14,15,16,17,18], Fisher discriminant analysis [19,20], and the BP neural network methods [21,22]. (“BP” refers to the “backpropagation” algorithm in neural networks, which is a commonly used optimization algorithm for training neural networks. It is employed to adjust the values of connection weights in the network, enabling the network to better adapt to the given training data.) However, due to the differences in the considered influencing factors, the methods for determining the weights of factors, and the division of evaluation levels, there are certain differences in the evaluation results. In summary, the evaluation of rock burst risk commonly relies on empirical analogy methods and field measurement techniques. However, empirical analogy methods have limitations in capturing the interrelationships between influencing factors, leading to potential biases. Field measurement methods offer more objective results but are influenced by complex geological conditions and mining techniques, leading to time lags in evaluating and mitigating rock burst occurrences. At the same time, the precise quantitative evaluation of rock burst risk is challenging due to uncertainties in the indicator values, factor selection, and the lack of standardized mathematical models.

After conducting a comprehensive analysis of the influencing factors, the present paper adopted the Analytic Hierarchy Process–Fuzzy Comprehensive Evaluation (AHP-FCE) method to establish a multilevel fuzzy comprehensive evaluation model. According to the principles of fuzzy transformation and maximum membership degree [23], the evaluation model is calculated to achieve a quantitative and effective evaluation of rock burst risk. Compared to the cases presented in previous studies [13,14,15,16,17,18], the developed model offers a heightened level of comprehensive factor consideration. By implementing a multilevel calculation approach, it effectively reduces the influence of subjective factors. Moreover, the model places significant emphasis on capturing the interrelationships among different influencing factors, leading to evaluation results that are more objective and precise. The accuracy of the model was validated through the evaluation of the 310 working face at Tingnan Coal Mine, Xianyang City, Shanxi Province, China.

Sustainability is a comprehensive concept aimed at balancing the needs and interests of society, economy, and the environment. Rock burst can lead to severe consequences, such as the destruction of coal mine tunnels and human casualties. Therefore, within the framework of sustainability, the primary task is to ensure the safety of personnel in mining engineering design and construction processes while considering the economic feasibility and benefits. By conducting a comprehensive evaluation of rock burst events, a deeper understanding of the influence of different factors on the risk level can be gained, enabling the implementation of suitable preventive and management strategies to minimize the potential harm to individuals and mitigate property damage. Furthermore, sustainable development requires the protection and maintain of the health and sustainability of the environment. By evaluating the hazards of rock burst events, the potential environmental impact of such events can be assessed, allowing for the implementation of measures during the design and execution of mining projects. These measures aim to minimize the negative consequences, foster sustainable land utilization, and safeguard ecological integrity. In summary, sustainable development is closely related to the multilevel fuzzy comprehensive evaluation of rock burst risk. By applying the principles of sustainability to the evaluation of rock burst risk, we can better ensure social safety, mitigate economic risks, protect the health of the environment, and ensure the sustainability of mining engineering.

2. Influencing Factors

The rock burst risk is influenced by a variety of factors. In order to evaluate the rock burst risk effectively, by consulting with relevant technical experts and reviewing the literature, three primary influencing factors of the evaluation model were selected [24,25,26]. These factors include geological conditions, mining technique, and safety management, all of which are crucial in determining the occurrence of rock burst events. The present study addresses the recognized main influencing factors in the field and incorporates insights from a diverse group of experts. As a result, the model can be applied to evaluate the rock burst risk in various mines, assuming the absence of exceptional and rare influencing factors. It primarily focuses on considering the widely acknowledged main influencing factors while acknowledging its limitations in not accounting for extremely rare and exceptional circumstances.

2.1. Geological Conditions

The identification of rock burst risk heavily relies on the geological conditions. In this research, five secondary factors were identified, including the roof strata, mining depth, geological structure, rock stress, and burst propensity. By incorporating these factors into the evaluation model, a better evaluation of the geological conditions that contribute to the occurrence of rock burst events can be obtained.

2.1.1. Roof Strata

The rock burst risk is strongly influenced by the properties and conditions of the roof strata. During the mining process, the probability of collapse is lower for roof strata that possess a high hardness and thickness, resulting the accumulation of high elastic deformation energy. The sudden release of the accumulated elastic deformation energy resulting from the fracture or slip of the roof strata can trigger rock burst events. Corresponding to the initial and periodic collapses of the roof strata, the formulas for calculating the elastic energy are as follows:

$$U_{W_i}=\frac{q^2{L}^5}{576EJ}$$

(1)

$$U_{W_c}=\frac{q^2{L}^5}{8EJ}$$

(2)

where q is the uniform load on the upper part of the roof; L is the overhanging length; E is the elastic modulus of the rock mass; J is the moment of inertia of the roof section.

According to Equations (1) and (2), it is known that the accumulated elastic deformation energy is proportional to the fifth power of the overhanging length L [18]. Consequently, it follows that the greater the overhanging length, the higher the accumulated elastic energy of the roof strata. Generally, a higher hardness of the roof strata is associated with a lower likelihood of collapse, which leads to an increase in the overhanging length, while a greater overhanging length increases the rock burst risk.

2.1.2. Mining Depth

Typically, rock burst events are triggered when the stress exerted on the roadway or coal seam surpasses a critical threshold. In the triaxial stress state without mining influence, the stress borne by a coal seam at a specific mining depth H can be mathematically described as follows:

$${\sigma }_1=\gamma H$$

(3)

$${\sigma }_2={\sigma }_3=\frac{\nu }{1-\nu }\gamma H$$

(4)

where ${\sigma }_1$ is the vertical stress; ${\sigma }_2$ and ${\sigma }_3$ are the two horizontal stresses; $\gamma$ is the average bulk density of the overlying strata; $\nu$ is Poisson’s ratio.

The elastic energy accumulated by coal due to volume deformation and shape deformation can be mathematically expressed as follows:

$$U_V=\frac{(1-2\nu ){(1+\nu )}^2}{6E{(1-\nu )}^2}{\gamma }^2{H}^2$$

(5)

$$U_S=\frac{(1+\nu ){(1-2\nu )}^2}{3E{(1+\nu )}^2}{\gamma }^2{H}^2$$

(6)

where E is the elastic modulus of the coal.

According to Equations (3)–(6), it is apparent that with an increase in the mining depth, the stress and the accumulated elastic energy also increase, resulting in a higher rock burst risk. Based on the relevant statistical data [27], there is a nonlinear relationship between the incidence of rock burst events per million tons of coal $W_t$ and the mining depth H, as shown in Figure 1. After the mining depth reaches 500 m, the incidence of events increases sharply with the increase in the mining depth.

2.1.3. Geological Structure

The presence of irregular geological structures can exert a substantial impact on the deformation characteristics and stress distribution of the coal. The occurrence of rock burst is mainly influenced by faults, folds, and variations in the coal seam thickness.

(1)

Faults

Faults are an important geological structure that significantly impact the occurrence of rock burst. As depicted in Figure 2, when mining activities are conducted in close proximity to a fault, the influence range of the advancing front abutment pressure expands continuously with the progression of the working face. When the working face approaches the region influenced by a fault, the fault tectonic stress will superimpose with the front abutment pressure, resulting in a new stress peak region between the fault and the working face, where a large amount of energy will accumulate. When the accumulated energy reaches a certain threshold, it may trigger the occurrence of rock burst.

Various studies [28,29] have demonstrated that fault throw is very important in determining the magnitude of structural stress in faults.

Table 1. The relationship between fault throw, unilateral influence range, and stress concentration factor.

Fault Throw/m	Unilateral Influence Range/m	Stress Concentration Factor K
0∼5	40	1.1
5∼10	50	1.2
10∼30	60	1.3
30∼50	70	1.4
50∼100	90	1.5
>100	100	1.7

presents the relationship between fault throw, the unilateral influence range, and the corresponding stress concentration factor.

(2)

Folds

Folding is a geological phenomenon resulting from the gradual deformation of rock strata caused by horizontal compression. When a roadway approaches an area of folding, there is a notable escalation in the frequency and intensity of rock burst events. Figure 3 illustrates the distribution pattern of rock burst risk in folding areas. Zone I refers to the area around the axial plane of the folded anticline, where the horizontal stress is compressive and the vertical stress is tensile, resulting in the maximum mining pressure. Zone II covers the area around the limbs of the fold, where both the vertical and horizontal stresses are compressive, making it the zone of maximum rock burst risk. Zone III is the area around the axial plane of the folded syncline, where the horizontal stress is tensile and the vertical stress is compressive, making it the zone most prone to roof caving and rock burst.

(3)

Variations in coal seam thickness

The variation in coal seam thickness is also a significant factor that can affect the occurrence of rock burst. Figure 4 illustrates that the concentration of stress increases significantly at the location of coal seam thickness variation due to the effect of tectonic stresses, which enhances the rock burst risk. The degree of stress concentration is proportional to the extent of the variation in coal seam thickness, with greater variation resulting in higher stress concentrations.

2.1.4. Rock Stress

The rock stress is composed of both the original rock stress and mining-induced stress. The original rock stress is the inherent stress in the crust that has not been disturbed by engineering activities, including gravity stress and tectonic stress. Gravity stress is the stress generated in a rock mass as a result of its own weight. Tectonic stress is the stress generated in the crust under various tectonic movements. Mining-induced stress is the stress formed by mining activities. Rock burst is the result of stress exceeding the critical strength of the coal and rock mass.

2.1.5. Burst Propensity

The burst propensity, an intrinsic characteristic of coal and rock mass itself, is a key internal factor that contributes to rock burst events. There are four commonly used burst propensity indexes in China, including the uniaxial compressive strength, elastic strain energy index, bursting energy index, and duration of dynamic fracture [30,31,32,33,34,35,36,37].

The uniaxial compressive strength ($R_C$) is defined as the ratio of the maximum load borne by a standard coal sample to its compressed surface area during the uniaxial compression progress [30,31], as shown in Figure 5a. The calculation method is as follows:

$$\frac{}{}{R}_C=\frac{P}{A}$$

(7)

where P is the maximum load, N; A is the compressed surface area, m${}^2$.

The elastic strain energy index ($W_{ET}$) is defined as the ratio of the accumulated elastic strain energy to the consumed plastic strain energy when the sample is unloaded after being loaded to 75∼85% of its uniaxial compressive strength [32,33,34], as shown in Figure 5b. The calculation method is as follows:

$$W_{ET}=\frac{{\varphi }_{SE}}{{\varphi }_{SP}}$$

(8)

where ${\varphi }_{SE}$ is the accumulated elastic strain energy, kJ; ${\varphi }_{SP}$ is the consumed plastic strain energy, kJ.

The bursting energy index ($K_E$) is the ratio of the accumulated strain energy before the peak to the consumed strain energy after the peak in the stress–strain curve of a coal sample during uniaxial compression [35,36], as shown in Figure 5c. The calculation method is as follows:

$$K_E=\frac{A_S}{A_X}$$

(9)

where $A_S$ is the accumulated strain energy, kJ; $A_X$ is the consumed strain energy, kJ.

The duration of dynamic fracture ($DT$) is defined as the time elapsed from the ultimate strength to complete failure of a coal sample under uniaxial compression and reflects the burst propensity from a temporal perspective [37], as shown in Figure 5d.

2.2. Mining Technique

Among mining techniques, the coal pillars, mining layout, goaf, and mining disturbance were selected as the secondary influencing factors [24,25,26]. Understanding the impact of these factors on rock burst events is crucial for developing effective prevention and control measures in mining.

2.2.1. Coal Pillars

The support provided by coal pillars is critical for maintaining the stability of underground mining operations. However, coal pillars are also the locations where stress concentration occurs, which can increase the rock burst risk. Statistics have indicated that approximately 60% of rock burst events were caused by an inadequate design of coal pillars [18]. The proper determination of coal pillar widths is of the utmost importance in mitigating the occurrence and controlling the severity of rock bursts. Figure 6 presents the vertical stress distribution of various coal pillar widths [18]. The maximum vertical stress exhibits a non-linear trend with the width. Initially, the stress increases with increasing width, eventually reaching a peak in the range of 12∼20 m. Beyond this range, the stress decreases again. These findings illustrate the importance of appropriate coal pillar design to ensure optimal stress distribution and reduce the risk.

2.2.2. Mining Layout

The mining layout directly affects the stress distribution, and a reasonable layout can effectively minimize the rock burst risk. Especially for the coal seams with a high rock burst potential, the mining layout should be optimized to prevent the formation of stress concentration zones near roadways and working faces.

2.2.3. Goaf

Following the coal seam mining, the overlying roof strata will experience fractures and movement. The three-dimensional spatial structure resulting from the fracturing of the overlying rock in the goaf exhibits variations across different goaf areas. Figure 7 illustrates the impact of the number of mined-out faces on the spatial structures. Specifically, a single mined-out face creates an unsupported “O”-shaped structure at the center (Figure 7a), while two mined-out faces result in an “S”-shaped structure (Figure 7b). Three mined-out faces lead to the formation of a “C”-shaped structure (Figure 7c), and four mined-out faces result in a supported “$\theta$”-shaped structure at the center (Figure 7d). The “O”-shaped structure has less influence on the support system and the stability of the surrounding rock. Because there is only one face of mining, the neighboring surrounding rock can provide support and constraints. “S”-shaped structures have a certain impact on the support system and the stability of the surrounding rock. Due to the two-face mining, the surrounding rock is subjected to a larger deformation and stress concentration, which increases the risk of rock burst. In order to maintain the stability of the surrounding rock, it is necessary to use stronger support measures. The “C”-shaped structure has a greater impact on the support system and the stability of the surrounding rock. The three-face mining leads to a serious deformation and stress concentration in the surrounding rock, and the risk of rock explosion increases significantly. “$\theta$”-shaped structures have the greatest influence on the support system and the stability of the surrounding rock. The four-face mining causes the surrounding rock to be in a high-stress state, and the risk of rock burst is very high. Inadequate and delayed support of the goaf can have severe consequences for the stability of the support system and surrounding rock. This can result in rock mass instability within the goaf, damage the support system, and ultimately induce rock burst.

2.2.4. Mining Disturbance

Mining disturbance can change the natural equilibrium of stress. The stress field of the rock will experience dynamic adjustment during the mining process, resulting in the redistribution of stress and elastic energy. These changes ultimately increase the rock burst risk. At the same time, the plastic damage induced by mining disturbances is a key factor contributing to the increased risk of rock burst. In the context of mining operations, various factors such as blasting, excavation, and stress redistribution can lead to significant disturbances in the surrounding rock mass. These disturbances often result in localized plastic deformation and damage accumulation. The accumulation of plastic damage weakens the structural integrity of the rock mass and increases its susceptibility to rock burst. To analyze and quantify the impact of plastic damage on the risk of rock burst, advanced modeling techniques are utilized. Coupled thermal–elastic–plastic damage models provide a comprehensive framework for capturing the complex behavior of the rock mass subjected to mining-induced disturbances [38,39,40]. These models incorporate the effects of plastic deformation, damage evolution, and energy dissipation mechanisms, offering a more accurate representation of the rock mass response under dynamic loading conditions. In conclusion, the plastic damage induced by mining disturbances plays a crucial role in increasing the risk of rock burst. Understanding and quantifying the extent of plastic damage in the rock mass is essential for effective risk assessment and mitigation strategies. By incorporating advanced modeling techniques, researchers can better evaluate the potential for rock burst events and develop proactive measures to ensure the safety and stability of mining operations.

2.3. Safety Management

For safety management, monitoring and warning for rock bursts, personnel training, and emergency plans for dealing with rock bursts were selected as the secondary influencing factors.

2.3.1. Monitoring and Warning

Monitoring and warning for rock bursts is an important part of safety management. Establishing an effective monitoring and warning system can promptly detect abnormal situations such as stress changes and take effective measures to prevent or mitigate the harm caused by rock bursts. Additionally, the monitoring and warning system can facilitate the collection of pertinent data and information, which can be analyzed to identify stress distribution characteristics. Based on such analyses, the mining plan can be optimized to diminish the rock burst risk.

2.3.2. Personnel Training

Providing training to the individuals responsible for preventing rock burst is an effective approach to enhance their awareness and knowledge of prevention measures. This training can include instruction on warning methods, prevention and control techniques, and other relevant knowledge to better equip them to carry out the work. Another purpose of training is to improve workers’ emergency response capabilities, enabling them to take prompt and effective measures in the event of rock burst accidents to prevent the situation from escalating and minimize losses.

2.3.3. Emergency Plan

A reasonable and effective emergency plan for rock burst accidents can guide coal mining enterprises to respond quickly and effectively in emergency situations. The emergency plan should explicitly outline the response process, organization structure, and division of responsibilities, ensuring that coal mining enterprises can promptly and effectively take emergency measures when accidents occur, improve their ability to deal with accidents, and minimize losses.

3. Establishment of the Multilevel Fuzzy Comprehensive Evaluation Model

According to the principles of fuzzy transformation and maximum membership degree [23], a hierarchical structure and judgment matrix of the AHP-FCE model is constructed, taking into consideration various factors affecting the evaluated object. The weights of each influencing factor are then calculated, and a comprehensive evaluation is achieved through quantitative calculation. The establishment of the model mainly includes the following key steps.

3.1. Establishment of the Evaluation Set

The rock burst risk consists of four distinct levels. Establishing the evaluation set for the rock burst risk is $\mathbf{V}=\{v_1,v_2,v_3,v_4\}$, where $v_1$ represents no risk of rock burst, $v_2$ represents weak risk, $v_3$ represents moderate risk, and $v_4$ represents strong risk.

3.2. Establishment of the Influencing Factors Set

According to the analysis of influencing factors in Section 2, a primary set of factors $\mathbf{S}=\{S_1,S_2,S_3\}$, secondary sets ${\mathbf{S}}_{\mathbf{1}}=\{S_{11},S_{12},S_{13},S_{14},S_{15}\}$, ${\mathbf{S}}_{\mathbf{2}}=\{S_{21},S_{22},S_{23},S_{24}\}$, and ${\mathbf{S}}_{\mathbf{3}}=\{S_{31},S_{32},S_{33}\}$, and a hierarchical structure of influencing factors (as shown in Figure 8) were established. The structure includes three primary evaluation indicators and twelve secondary evaluation indicators. Each element in the set of influencing factors corresponds to a specific factor within the hierarchical structure.

3.3. Establishment of the Fuzzy Relation Matrix

To effectively quantify and compare the various influencing factors in the evaluation process, it is essential to establish a fuzzy relation matrix. By performing weighted operations on the fuzzy relation matrix, the final evaluation result can be obtained. First, it is necessary to establish the level division standards for each influencing factor. By referring to the relevant literature [16,41,42,43,44], the level division standards for each influencing factor were determined, as presented in

Table 2. The standards for level division of influencing factors.

Influencing Factors	I	II	III	IV
Roof strata	$L_{st}\le 50$	$50<L_{st}\le 70$	$70<L_{st}\le 90$	$L_{st}>90$
Mining depth	$h\le 400$	$400<h\le 600$	$600<h\le 800$	$h>800$
Geological structure	No geological structures	Simple geological structure, slight influence	Relatively complex geological structure, moderate influence	Complex geological structure, significant influence
Rock stress	No influence	Slight influence	Moderate influence	Significant influence
Burst propensity	No burst propensity	Weak burst propensity	Moderate burst propensity	Strong burst propensity
Coal pillars	No influence	Slight influence	Moderate influence	Significant influence
Mining layout	Reasonable	Relatively reasonable	Partially reasonable	Unreasonable
Goaf	Solid coal face	One-side goaf	Two-side goaf	Three-side or more goaf
Mining disturbance	Weak	Moderate	Relatively strong	Strong
Monitoring and warning	Good effect	Relatively good effect	Moderate effect	No monitoring and warning
Personnel training	Good effect	Relatively good effect	Moderate effect	No training
Emergency plan	Good practicality	Relatively good practicality	Moderate practicality	No emergency plan

Note: $L_{st}$—The thickness characteristic parameter of the roof strata within 100 m above the coal seam.

. Corresponding to no risk, weak risk, moderate risk, and strong risk, the risk levels are expressed as I, II, III, and IV, respectively.

To evaluate the rock burst risk, it is imperative to determine the membership degree for each influencing factor at distinct levels, based on the established level division standards. The membership degree refers to the degree that each influencing factor belongs to a particular evaluation level and plays a critical role in the fuzzy comprehensive evaluation of rock burst risk. The roof strata and mining depth are two continuous quantitative influencing factors. To ascertain the membership degree of these factors, the present paper adopts the triangular membership distribution function [45], as presented in Equations (10) and (11). The triangular membership function is a widely used fuzzy set membership function that is easy to implement. It requires only three parameters to define a triangular fuzzy set, making it computationally efficient and accessible for practical applications. At the same time, the triangular membership function provides a smooth and continuous transition between the fuzzy set’s membership values. This property ensures that small changes in the input values result in gradual adjustments of the membership degrees, improving the system’s robustness. The other influencing factors are discrete qualitative factors that cannot be quantified directly. These factors are evaluated through an expert grading method [16,41,42,43,44], as shown in

Table 3. The membership degree of discrete qualitative factors.

Influencing Factors	Levels	Membership Degree
		No Risk	Weak Risk	Moderate Risk	Strong Risk
Geological structure	I	0.9	0.1	0	0
	II	0	0.6	0.3	0.1
	III	0	0.1	0.5	0.4
	IV	0	0	0	1
Rock stress	I	0.9	0.1	0	0
	II	0	0.9	0.1	0
	III	0	0	0.8	0.2
	IV	0	0	0.1	0.9
Burst propensity	I	0.9	0.1	0	0
	II	0.1	0.8	0.1	0
	III	0	0.1	0.8	0.1
	IV	0	0	0.1	0.9
Coal pillars	I	0.8	0.1	0.1	0
	II	0.2	0.7	0.1	0
	III	0	0.2	0.7	0.1
	IV	0	0.1	0.1	0.8
Mining layout	I	0.8	0.2	0	0
	II	0.1	0.6	0.3	0
	III	0.1	0.2	0.6	0.1
	IV	0	0.1	0.2	0.7
Goaf	I	0.7	0.3	0	0
	II	0.1	0.5	0.3	0.1
	III	0	0	0.6	0.4
	IV	0	0	0.2	0.8
Mining disturbance	I	0.8	0.2	0	0
	II	0.4	0.5	0.1	0
	III	0	0.2	0.6	0.2
	IV	0	0.1	0.3	0.6
Monitoring and warning	I	0.8	0.2	0	0
	II	0.1	0.7	0.2	0
	III	0	0.1	0.6	0.3
	IV	0	0	0.2	0.8
Personnel training	I	0.7	0.2	0.1	0
	II	0.2	0.6	0.2	0
	III	0	0.1	0.5	0.4
	IV	0	0.1	0.2	0.7
Emergency plan	I	0.8	0.1	0.1	0
	II	0.2	0.5	0.3	0
	III	0	0.2	0.5	0.3
	IV	0	0.1	0.1	0.8

.

$$\begin{array}{c}{T}_1\left(x_i\right)=\left\{\begin{array}{cc}1& x_i\le b_{i1}\\ \left(b_{i2}-x_i\right)/\left(b_{i2}-b_{i1}\right)& b_{i1}<x_i\le b_{i2}\\ 0& x_i>b_{i2}\end{array}\right.\hfill \\ T_2\left(x_i\right)=\left\{\begin{array}{cc}0& x_i\le b_{i1}\mathrm{or}{x}_i>b_{i3}\\ -\left(b_{i1}-x_i\right)/\left(b_{i2}-b_{i1}\right)& b_{i1}<x_i\le b_{i2}\\ \left(b_{i3}-x_i\right)/\left(b_{i3}-b_{i2}\right)& b_{i2}<x_i\le b_{i3}\end{array}\right.\hfill \\ T_3\left(x_i\right)=\left\{\begin{array}{cc}0& x_i\le b_{i2}\mathrm{or}{x}_i>b_{i4}\\ -\left(b_{i2}-x_i\right)/\left(b_{i3}-b_{i2}\right)& b_{i2}<x_i\le b_{i3}\\ \left(b_{i4}-x_i\right)/\left(b_{i4}-b_{i3}\right)& b_{i3}<x_i\le b_{i4}\end{array}\right.\hfill \\ T_4\left(x_i\right)=\left\{\begin{array}{cc}0& x_i<b_{i3}\\ -\left(b_{i3}-x_i\right)/\left(b_{i4}-b_{i3}\right)& b_{i3}\le x_i<b_{i4}\\ 1& x_i\ge b_{i4}\end{array}\right.\hfill \end{array}$$

(10)

$$\begin{array}{c}{b}_{i1}=\left(0+a_{i1}\right)/2\hfill \\ b_{ij}=\left(a_{ij-1}+a_{ij}\right)/2(j=2,3)\hfill \\ b_{i4}=\left(a_{i3}+a_{i3}\right)/2\hfill \end{array}$$

(11)

where $T_1\left(x_i\right)$, $T_2\left(x_i\right)$, $T_3\left(x_i\right)$, and $T_4\left(x_i\right)$ represent the membership degrees; $x_i$ is the index value of the factor; $a_{i1}$, $a_{ij}$ ($j=2,3$) are the standard values for the division of risk levels.

Based on the membership distribution described above, a fuzzy mapping from the influencing factor set S to the evaluation set V is established as follows: f:$S\to V$. Selecting the influencing factor $s\in S$, the fuzzy relation matrix R can be acquired through mapping as follows:

$$R=\left[\begin{array}{cccc}{r}_{11}& r_{12}& r_{13}& r_{14}\\ r_{21}& r_{22}& r_{23}& r_{24}\\ ⋮& ⋮& ⋮& ⋮\\ r_{m1}& r_{m2}& r_{m3}& r_{m4}\end{array}\right]$$

(12)

3.4. Establishment of the Set of Weights for Influencing Factors

The degree of impact from each influencing factor on the risk of rock burst varies, and it is crucial to accurately quantify their respective importance. The AHP is a systematic method that can be employed to calculate the weight of each factor precisely. The key steps of the AHP are as follows:

3.4.1. Establishment of the Judgment Matrix

We consulted experts with rich experience and used the 1∼9 scale method (

Table 4. The 1∼9 scale method.

Scale Value u	Meaning
1	$x_i$ and $x_j$ have equal influence
3	$x_i$ is slightly more important than $x_j$
5	$x_i$ is significantly more important than $x_j$
7	$x_i$ is strongly more important than $x_j$
9	$x_i$ is extremely more important than $x_j$
2, 4, 6, 8	Intermediate values can be used if a compromise between factors $x_i$ and $x_j$ is required

) to quantify the importance of the influencing factors. By comparing each factor pairwise, the judgment matrix M can be obtained as follows:

$$M=\left[\begin{array}{cccc}{m}_{11}& m_{12}& \cdots & m_{1j}\\ m_{21}& m_{22}& \cdots & m_{2j}\\ ⋮& ⋮& & ⋮\\ m_{i1}& m_{i2}& \cdots & m_{ij}\end{array}\right]$$

(13)

where $m_{ij}$ = 1/$m_{ji}$, $m_{ij}$ = 1 ($i=j$). Element $m_{ij}$ represents the relative importance of factor $x_i$ to factor $x_j$, that is:

$$m_{ij}=u_i/u_j$$

(14)

where $u_i$ and $u_j$ represent the importance scale values of factors $x_i$ and $x_j$, respectively.

3.4.2. Calculation of Weight and Solution of Maximum Eigenvalue

The judgment matrix M is normalized by column, and the normalized elements $\overline{m_{ij}}$ can be described as follows:

$$\begin{array}{c}\overline{m_{ij}}=m_{ij}/{\displaystyle \sum _{i=1}^n}{m}_{ij}(i,j=1,2,\cdots ,n)\hfill \end{array}$$

(15)

Using the normalized elements $\overline{m_{ij}}$ to calculate the $\overline{w_i}$:

$$\begin{array}{c}\overline{w_i}=\frac{1}{n}{\displaystyle \sum _{j=1}^n}\overline{m_{ij}}(i=1,2,\cdots ,n)\hfill \end{array}$$

(16)

According to Equation (14), the vector $\overline{W}=\left[\overline{w_1},\overline{w_2},\cdots ,\overline{w_n}\right]$ can be normalized to obtain the influence factor weight vector $W=\left[w_1,w_2,\cdots ,w_n\right]$.

$$\begin{array}{c}{w}_i=\overline{w_i}/{\displaystyle \sum _{i=1}^n}\overline{w_i}(i=1,2,\cdots ,n)\hfill \end{array}$$

(17)

The maximum eigenvalue ${\lambda }_{max}$ of the judgment matrix can be calculated as follows:

$$\begin{array}{c}{\lambda }_{max}=\frac{1}{n}{\displaystyle \sum _{i=1}^n}\left[\left({\displaystyle \sum _{j=1}^n}{m}_{ij}{w}_j\right)/w_i\right](i,j=1,2,\cdots ,n)\hfill \end{array}$$

(18)

3.4.3. Consistency Check of the Judgment Matrix

To ensure that the judgment matrix has reasonable consistency, AHP introduces the consistency index ($CI$) and the average random consistency index ($CR$). The smaller the value of $CI$, the higher the consistency level of the judgment matrix, indicating that the importance comparisons obtained from experts or decision makers are relatively reasonable. However, using $CI$ alone may not directly determine the consistency of the judgment matrix. Therefore, it is necessary to calculate the $CR$ to further evaluate the consistency. The $CR$ ranges from 0 to 1, and the closer it is to 0, the higher the consistency of the judgment matrix. When the $CR\le 0.10$, it indicates that the judgment matrix exhibits favorable consistency and the selection of weight is reasonable. Conversely, the judgment matrix needs to be reconstructed.

First, calculate the consistency index $CI$ as:

$$\begin{array}{c}CI=\frac{{\lambda }_{max}-n}{n-1}\hfill \end{array}$$

(19)

where n is the order of the judgment matrix.

Then, calculate the average random consistency index $CR$ as:

$$\begin{array}{c}CR=\frac{CI}{RI}\hfill \end{array}$$

(20)

where $RI$ is the average random consistency index of the same order. The specific values can be found in

Table 5. The random index ($RI$).

The Order of the Matrix n	1	2	3	4	5	6	7	8	9
$RI$	0	0	0.58	0.90	1.12	1.24	1.32	1.41	1.45

.

By following the above steps and using expert ratings and relevant literature sources [13,14,15,16,17,18], the judgment matrix can be established. The weight vector was calculated using Equations (12)–(14). The maximum eigenvalue ${\lambda }_{max}$ was calculated using Equation (15). The average random consistency index $CR$ was calculated using Equations (16) and (17). The results are presented in

Table 6. Primary influencing factor.

Judgment Matrix M	${\mathit{S}}_1$	${\mathit{S}}_2$	${\mathit{S}}_3$	Weight Vector W	${\mathit{\lambda }}_{\mathit{max}}$	$\mathbf{CR}$
$S_1$	1	2	5	0.58	3.01	0.009 < 0.1
$S_2$	1/2	1	3	0.31
$S_3$	1/5	1/3	1	0.11

,

Table 7. Secondary geological factor.

Judgment Matrix ${\mathit{M}}_1$	${\mathit{S}}_{11}$	${\mathit{S}}_{12}$	${\mathit{S}}_{13}$	${\mathit{S}}_{14}$	${\mathit{S}}_{15}$	Weight Vector ${\mathit{W}}_1$	${\mathit{\lambda }}_{\mathit{max}}$	$\mathbf{CR}$
$S_{11}$	1	1/3	1/4	2	3	0.12	5.24	0.054 < 0.1
$S_{12}$	3	1	1/2	4	5	0.28
$S_{13}$	4	2	1	6	7	0.45
$S_{14}$	1/2	1/4	1/6	1	4	0.10
$S_{15}$	1/3	1/5	1/7	1/4	1	0.05

,

Table 8. Secondary mining technique factor.

Judgment Matrix ${\mathit{M}}_2$	${\mathit{S}}_{21}$	${\mathit{S}}_{22}$	${\mathit{S}}_{23}$	${\mathit{S}}_{24}$	Weight Vector ${\mathit{W}}_2$	${\mathit{\lambda }}_{\mathit{max}}$	$\mathbf{CR}$
$S_{21}$	1	1/7	1/2	2	0.10	4.09	0.033 < 0.1
$S_{22}$	7	1	5	8	0.65
$S_{23}$	2	1/5	1	4	0.19
$S_{24}$	1/2	1/8	1/4	1	0.06

and

Table 9. Secondary safety management factor.

Judgment Matrix ${\mathit{M}}_3$	${\mathit{S}}_{31}$	${\mathit{S}}_{32}$	${\mathit{S}}_{33}$	Weight Vector ${\mathit{W}}_3$	${\mathit{\lambda }}_{\mathit{max}}$	$\mathbf{CR}$
$S_{31}$	1	3	1/2	0.32	3.02	0.017 < 0.1
$S_{32}$	1/3	1	1/4	0.12
$S_{33}$	2	4	1	0.56

.

3.5. Fuzzy Comprehensive Evaluation

By synthesizing the fuzzy relation matrix R and the weight vector W through the weighted average algorithm, the fuzzy subset B of the evaluation set V was established:

$$B=W·R=\left[\begin{array}{cccc}{b}_1& b_2& b_3& b_4\end{array}\right]$$

(21)

We define $b_k=max\left(b_1,b_2,\dots ,b_n\right)$ as the maximum membership degree, and the position of $b_k$ in the evaluation set V corresponds to the final evaluation level.

4. Application Example

The multilevel fuzzy comprehensive evaluation model was validated in the 310 working face of Tingnan Coal Mine, Xianyang City, Shanxi Province, China. The 310 working face is situated in the eastern wing of the third panel, which is located in the northern wing of the Lujia–Xiaolingtai anticline. The 308 goaf is positioned 6 m east of the return airway, while the 307 goaf is situated on the west side, with solid coal areas located to the south and east. The relative location of these areas is illustrated in Figure 9. The fully mechanized caving method with top coal caving is employed in the mining operations. The coal seam thickness is 4.3∼10.9 m, and the dip angle is 0∼$7^{\circ }$. The evaluation indicators are presented in

Table 10. The evaluation indicators of 310 working face.

Influencing Factors	Evaluation Indicators	Grading Situation
Roof strata	$L_{st}$ = 80.55	III
Mining depth	The maximum mining depth is 645 m	III
Geological structure	The working face is situated in the northern wing of the Lujia–Xiaolingtai anticline, without faults, and the geological structure is simple	II
Rock stress	Under the action of maximum horizontal principal stress, the working face exhibits floor heave and roof subsidence, which has a moderate influence	III
Burst propensity	Strong burst propensity	IV
Coal pillars	During the mining process, irregular coal pillars will be formed where the working face intersects with the roadway, leading to stress concentration, and the coal pillars have a significant influence	III
Mining layout	The overlying strata of the 310 working face may form an “O”-shaped or “S”-shaped structure, which has a moderate influence	III
Goaf	One-side goaf	II
Mining disturbance	Slight influence	II
Monitoring and warning	The micro-seismic and stress online monitoring and warning system have been effectively used in the mining operation	II
Personnel training	Regular training has been organized	II
Emergency plan	A special emergency plan for rock burst prevention has been developed, and it has proven to be practical	II

.

Based on the actual situation of each influencing factor, and utilizing Equations (10) and (11) in conjunction with Table 3, the secondary fuzzy relation matrices were obtained as follows:

$$R_1=\left[\begin{array}{cccc}0& 0& 0.95& 0.05\\ 0& 0.27& 0.73& 0\\ 0& 0.6& 0.3& 0.1\\ 0& 0& 0.8& 0.2\\ 0& 0& 0.1& 0.9\end{array}\right]$$

$$R_2=\left[\begin{array}{cccc}0& 0.2& 0.7& 0.1\\ 0.1& 0.2& 0.6& 0.1\\ 0.1& 0.5& 0.3& 0.1\\ 0.4& 0.5& 0.1& 0\end{array}\right]$$

$$R_3=\left[\begin{array}{cccc}0.1& 0.7& 0.2& 0\\ 0.2& 0.6& 0.2& 0\\ 0.2& 0.5& 0.3& 0\end{array}\right]$$

By calculating the secondary fuzzy relation matrices with corresponding weight vectors and normalizing them, the secondary fuzzy comprehensive evaluation subsets were obtained as follows:

$$B_1=W_1·R_1=\left[\begin{array}{cccc}0& 0.34& 0.54& 0.12\end{array}\right]$$

$$B_2=W_2·R_2=\left[\begin{array}{cccc}0.11& 0.27& 0.53& 0.09\end{array}\right]$$

$$B_3=W_3·R_3=\left[\begin{array}{cccc}0.17& 0.57& 0.26& 0\end{array}\right]$$

By constructing the secondary fuzzy comprehensive evaluation subsets, the primary fuzzy relation matrix R was obtained as follows:

$$R=\left[\begin{array}{c}{B}_1\\ B_2\\ B_3\end{array}\right]=\left[\begin{array}{cccc}0& 0.34& 0.54& 0.12\\ 0.11& 0.27& 0.53& 0.09\\ 0.17& 0.57& 0.26& 0\end{array}\right]$$

By performing the same calculation process, the primary fuzzy comprehensive evaluation set B was obtained as follows:

$$B=W·R=\left[\begin{array}{cccc}0.05& 0.34& 0.51& 0.10\end{array}\right]$$

The evaluation results are shown in Figure 10. ($v_1$, $v_2$, $v_3$, and $v_4$ correspond to no, weak, moderate, and strong risk, respectively). From Figure 10, it can be observed that the maximum membership index is $b_k$ = 0.51.

To ensure the reliability and accuracy of the evaluation results, the validity of the maximum membership degree of the results is verified according to Equation (22).

$$a=\frac{n\beta -1}{2\gamma (n-1)}$$

(22)

where a is the validity; n is the number of elements in the evaluation set; $\beta$ is the maximum membership degree in the target evaluation vector; $\gamma$ is the second maximum membership degree in the target evaluation vector.

The calculated value of $a=0.51$, which suggests that the evaluation results are stable and relatively effective. Based on the comprehensive evaluation, it has been determined that the rock burst risk level of the 310 working face is moderate. The results obtained using the proposed method are consistent with those obtained using the comprehensive index and the probability index method (Figure 11 and Figure 12) and are in agreement with the actual circumstances. It is proved that the multilevel fuzzy comprehensive evaluation model for rock burst risk is accurate and reliable and has significant practical value.

5. Conclusions

The evaluation of rock burst risk is a complex task that is impacted by various factors and characterized by high levels of uncertainty. Traditional evaluation methods often face difficulties in quantifying these uncertainties. To address this issue, the present paper developed a multilevel fuzzy comprehensive evaluation model for rock burst risk that integrates the AHP with the theory of fuzzy comprehensive evaluation. The model combines qualitative and quantitative methods, effectively integrating multiple influencing factors while reducing the subjective impact on the evaluation results. It has a wide range of applicability and can provide more accurate and reliable results for rock burst risk evaluation.

(1)

Identification of primary and secondary influencing factors: Based on previous research experience and expert opinions, three primary influencing factors and twelve secondary influencing factors that affect rock burst risk have been determined. A hierarchical structure is constructed based on the relationships between the primary influencing factors and secondary influencing factors to clearly present the hierarchical relationships among the factors.

(2)

Importance assessment of influencing factors: The AHP is employed to assess and compare the importance of each influencing factor. By performing a series of judgment matrix calculations and eigenvector computations, the weights for each influencing factor are determined. These weights reflect the relative importance of each factor in relation to the rock burst risk.

(3)

Integration of fuzzy comprehensive evaluation: The evaluation model is calculated using second-level fuzzy mathematical calculations to minimize the subjective influence of human factors in the evaluation process, making the evaluation results more objective and accurate.

(4)

Validation and applicability: The evaluation model has been validated in the 310 working face of the Tingnan Coal Mine. The obtained evaluation results are consistent with those obtained using the comprehensive index and probability index method and exhibit consistent alignment with the actual circumstances. The proposed approach is straightforward and can be widely promoted, with highly accurate and reliable results. This evaluation method provides a new and effective approach for assessing rock burst risk, with significant practical value.

(5)

Relation to sustainability: By evaluating the risk of rock burst, mining companies can take measures to reduce the likelihood of such events occurring, promoting a more sustainable approach to mining. Therefore, the study of the multilevel fuzzy comprehensive evaluation of rock burst risk is relevant to sustainability in the mining industry.