Rockburst Hazard Evaluation Using an Extended COPRAS Method with Interval-Valued Fuzzy Information

Abstract

Rockburst is a major disaster in deep mining engineering, which can cause serious injury, death and economic losses. This study aims to solve rockburst hazard evaluation problems and determine the hazard levels of mines under uncertain circumstances. To this end, a novel multicriteria decision-making (MCDM) method was proposed in an interval-valued fuzzy context. The main contributions are three-fold. First, considering the heterogeneity of rock masses and the complexity of mining engineering, interval-valued fuzzy numbers (IVFNs) were adopted to express initial indicator information. Second, accounting for the uncertainty of indicator weights, the decision-making trial and evaluation laboratory (DEMATEL) and entropy methods were extended with IVFNs to determine fuzzy indicator weights comprehensively. Third, the complex proportional assessment (COPRAS) approach was extended with IVFNs to determine the rockburst hazard level. Finally, the proposed method was applied to evaluate rockburst hazards in the Jiaojia gold mine, and the ranking results were consistent with field status. Meanwhile, sensitivity and comparison analyses were performed to reveal the stability and effectiveness of the proposed method. The results indicated that the extended COPRAS method was reliable for rockburst hazard evaluation in deep mining engineering.

1. Introduction

Rockburst is a sudden and violent rock failure with a rapid release of energy, and is characterized by the ejection and spalling of rock fragments [1]. It is common in deep hard rock mines, and has become a major threat to mining engineering [2]. A large number of mines have suffered from rockbursts, including the Western Deep Levels gold mine [3] and Kopanang gold–uranium mine [4] in South Africa, the Macassa gold mine and Brunswick lead–zinc mine in Canada [5], the Sunshine and Galena silver–lead–zinc mines in the United States [6], the Junction gold mine and Black Swan nickel mine in Australia [6] and the Zhazixi antimony mine [7] and Hongtoushan copper mine [8] in China. Rockburst is not only widely distributed, but also poses a serious threat to people, property and engineering. For example, a violent rockburst occurred in the Linglong gold mine, causing severe damage to the transportation track and electric wires [9]. An intense rockburst of magnitude 4.1 occurred in the Lucky Friday mine, causing serious damage to bolts and chain link meshes [10]. A series of severe rockbursts of magnitude 4.0 or greater occurred in the Lake-shore gold mine, causing 21 fatalities and complete blockage of levels [5]. Therefore, evaluating rockburst hazards is of great significance for the safety of deep mining engineering.

Numerous methods have been proposed for rockburst evaluation, which can be summarized as three categories: empirical criteria, numerical simulation methods and mathematical algorithms. The empirical criteria were developed based on the understanding of rockburst mechanisms and field experience. They can be classified into four types: energy-based, brittleness-based, strength-based and stiffness-based criteria. The typical criteria include the peak elastic strain energy index [11], coefficient of deformation for brittleness [12], the geological strength index [13], the roof–pillar stiffness ratio [14] and so on [15,16]. Because of their simplicity and efficiency, the empirical criteria are widely used in practical engineering. However, only using a single criterion is insufficient to tackle the complexity of rockbursts.

Due to the progress in rock mechanics and computational science, numerical simulation methods have become effective methods for rockburst hazard evaluation. Many prominent types of simulation software have been applied for rockburst hazard analysis, such as fast Lagrangian analysis of continua [17], realistic failure process analysis [18], particle flow code [19] and so on [20,21]. Many indicators have been proposed for the simulation of rockbursts, including the absolute local energy release rate [22], energy dissipation rate [23], burst potential index [24] and so on [25,26]. The numerical simulation method can consider the effects of in situ stress, rock parameters and excavation activities simultaneously. However, it is difficult to precisely determine the input and constitutive relationship of the model.

Recently, various mathematical algorithms have been developed to evaluate rockburst hazards, and these algorithms can be classified into three types. The first one is the machine learning algorithm. Many machine learning algorithms have been applied to analyze rockburst hazards, such as decision trees [27], ensemble learning [28] and artificial neural networks [29]. Machine learning algorithms can build a complex and nonlinear relationship between indicators and hazard levels based on existing rockburst cases. However, the accuracy of this kind of methodology can only be ensured by using large amounts of reliable data. The second one is the uncertainty method. Typical approaches include extensible comprehensive evaluation [30], unascertained measure theory [31], the fuzzy comprehensive assessment method [32] and so on [33,34]. These methods can characterize the uncertainties of rockburst hazards by using membership functions. However, the reliable membership functions are difficult to determine. The third one is the multicriteria decision-making method (MCDM), which not only can simultaneously consider multiple indicators, but can also deal with uncertainties by coupling with fuzzy theory [35,36]. For example, Yin et al. [37] extended the preference ranking organization method for enrichment evaluation (PROMETHEE) with trapezoid fuzzy numbers to assess rockburst hazards. Xue et al. [38] combined the technique for order preference by similarity to an ideal solution (TOPSIS) and rough set theory to evaluate rockburst hazards. Liang et al. [39] extended the multiattributive border approximation area comparison (MABAC) method with triangular fuzzy numbers to assess rockburst hazards. Moreover, a novel MCDM method, the complex proportional assessment (COPRAS) approach, was proposed by Zavadskas et al. [40,41]. This is an effective method that considers both the positive-ideal and negative-ideal solutions [42]. The benefit- and cost-type indicators are first considered separately, and then combined to determine the significance performance of each sample. Due to the simplicity and flexibility of the COPRAS approach, it has been applied in many fields. For example, Abdel-Basset et al. [43] applied the COPRAS method to sustainability assessment of the optimal locations of electric vehicle charge stations. Mao et al. [44] used the COPRAS method to select sustainable hydrogen fuel cell suppliers of new energy vehicles.

Notably, the indicator values are an important basis for rockburst hazard evaluation. Considering the heterogeneity and anisotropy of rock masses, the indicator values have some uncertainties. For example, one could suppose that the values of unconfined compressive strength are determined as 97 MPa, 105 MPa and 110 MPa, and the average value is 104 MPa. However, if the values are determined as 95 MPa, 103 MPa and 114 MPa, the average value is still 104 MPa. Obviously, the obtained crisp values smooth out the difference. Therefore, crisp numbers are insufficient to fully characterize the indicator information in rockburst evaluation. Fuzzy numbers are capable of considering the uncertainties adequately, and have been successfully applied in different engineering fields [45,46]. As typical fuzzy numbers, interval-valued fuzzy numbers (IVFNs) are suggested to fully describe the uncertain information. The tangential stress index can be taken as an example. Supposing there are determined values of 0.3, 0.4 and 0.5 for sample one, and 0.35, 0.4 and 0.45 for sample two, they can be represented by two IVFNs, [0.3, 0.5] and [0.35, 0.45]. It is clear that the uncertainties are better considered than the average value of 0.4. Therefore, IVFNs may be a more informative and convenient way to describe indicator values in rockburst evaluation. To the best of our knowledge, IVFNs have not been combined with the COPRAS method for rockburst hazard evaluation.

The main purpose of this study is to develop a novel methodology integrating IVFNs with the COPRAS, DEMATEL and entropy methods. The key contributions are summarized as follows: First, IVFNs are adopted to better capture the uncertainties of indicator values. Second, evaluating rockburst hazards with fuzzy indicator weights has rarely been considered in previous studies. The DEMATEL and entropy methods are extended with IVFNs to determine the fuzzy indicator weights. Third, the COPRAS method is extended with IVFNs to evaluate the rockburst hazard.

In summary, the rest of this paper is organized as follows. The preliminaries of IVFNs are indicated in the section “Preliminaries”. The details of the proposed methodology are introduced in the section “Methodology”. The proposed methodology is adopted to evaluate the rockburst hazard in Jiaojia gold mine in the section “Case study”. The section “Discussion” further demonstrates the stability and effectiveness of the proposed methodology. Finally, the “Conclusions” are made.

2. Preliminaries

In this section, the preliminaries of IVFNs are first introduced as follows.

(1) Definition of IVFNs

Let $\tilde{\gamma }=[{\gamma }^L,{\gamma }^U]$, which is called an interval-valued fuzzy number (IVFN). ${\gamma }^L$ and ${\gamma }^U$ are the lower and upper limits of $\tilde{\gamma }$, respectively. If $0<{\gamma }^L<{\gamma }^U$, then $\tilde{\gamma }$ is called a positive IVFN. The middle point and half-width of $\tilde{\gamma }$ are defined as $\pi (\tilde{\gamma })=\frac{{\gamma }^L+{\gamma }^U}{2}$ and $\omega (\tilde{\gamma })=\frac{{\gamma }^U-{\gamma }^L}{2}$, respectively [47].

(2) Basic operations of IVFNs

Supposing $\tilde{a}=[a^L,a^U]$ and $\tilde{b}=[b^L,b^U]$ are two positive IVFNs, and $\lambda$ is a positive real number, their basic operations can be performed as follows [44,48]:

$$\tilde{a}+\tilde{b}=[a^L+b^L,a^U+b^U],$$

(1)

$$\lambda \tilde{a}=[\lambda a^L,\lambda a^U],$$

(2)

$${\tilde{a}}^{\lambda }=[{(a^L)}^{\lambda },{(a^U)}^{\lambda }],$$

(3)

$$\tilde{a}\times \tilde{b}=[\min (a^L{b}^L,a^L{b}^U,a^U{b}^L,a^U{b}^U),\max (a^L{b}^L,a^L{b}^U,a^U{b}^L,a^U{b}^U)],$$

(4)

$$\frac{\tilde{a}}{\tilde{b}}=[\min (\frac{a^L}{b^L},\frac{a^L}{b^U},\frac{a^U}{b^L},\frac{a^U}{b^U}),\max (\frac{a^L}{b^L},\frac{a^L}{b^U},\frac{a^U}{b^L},\frac{a^U}{b^U})].$$

(5)

(3) Comparison method between two IVFNs

Let $len(\tilde{a})=a^U-a^L$ and $len(\tilde{b})=b^U-b^L$; the possibility degree of $\tilde{a}\ge \tilde{b}$ is defined as follows [45,49]:

$$p(\tilde{a}\ge \tilde{b})=\max \left\{1-\max \left\{\frac{b^U-a^L}{len(\tilde{a})+len(\tilde{b})},0\right\},0\right\}.$$

(6)

Similarly, the possibility degree of $\tilde{b}\ge \tilde{a}$ is defined as

$$p(\tilde{b}\ge \tilde{a})=\max \left\{1-\max \left\{\frac{a^U-b^L}{len(\tilde{a})+len(\tilde{b})},0\right\},0\right\}.$$

(7)

On this basis, there are the following properties:

$$0<p(\tilde{a}\ge \tilde{b})<1, 0<p(\tilde{b}\ge \tilde{a})<1,$$

(8)

$$p(\tilde{a}\ge \tilde{b})+p(\tilde{b}\ge \tilde{a})=1,$$

(9)

$$p(\tilde{a}\ge \tilde{a})=p(\tilde{b}\ge \tilde{b})=0.5.$$

(10)

(4) Distances between IVFNs

The Euclidean distance ($d_E$) and Hamming distance ($d_H$) between two IVFNs can be calculated as follows:

$$d_E(\tilde{a},\tilde{b})=\sqrt{{(a^L-b^L)}^2+{(a^U-b^U)}^2},$$

(11)

$$d_H(\tilde{a},\tilde{b})=\left|a^L-b^L\right|+\left|a^U-b^U\right|.$$

(12)

3. Methodology

In this section, a novel methodology integrating IVFNs with COPRAS, DEMATEL and entropy methods is developed, including three phases: expressing the indicator values with IVFNs, determining the comprehensive indicator weights and evaluating the hazard level with extended COPRAS method. The structure of these phases is demonstrated in Figure 1.

3.1. Phase 1: Express the Indicator Values with IVFNs

Step 1: Obtain the initial indicator values

The initial indicator values are obtained through laboratory tests and on-site investigation, and are expressed as

$$x_{ij}=\left\{x_{ij}^k|k=1,2,\dots ,q\right\},$$

(13)

where $x_{ij}^k$ is the k-th crisp value of indicator $B_j(j=1,2,\dots ,n)$ for sample $C_i(i=1,2,\dots ,m)$.

Step 2: Express the indicator values using IVFNs

To better characterize the uncertainty of indicator values, the crisp values are converted to IVFNs. The conversion equation is indicated as

$${\tilde{x}}_{ij}=[x_{ij}^L,x_{ij}^U]=[\min (x_{ij}),\max (x_{ij})],$$

(14)

where ${\tilde{x}}_{ij}$ is the IVFN corresponding to $x_{ij}$.

Then, the fuzzy decision-making matrix can be obtained as

$$\tilde{X}=\left[\begin{array}{cccc}{\tilde{x}}_{11}& {\tilde{x}}_{12}& \dots & {\tilde{x}}_{1n}\\ {\tilde{x}}_{21}& {\tilde{x}}_{22}& \dots & {\tilde{x}}_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ {\tilde{x}}_{m1}& {\tilde{x}}_{m2}& \dots & {\tilde{x}}_{mn}\end{array}\right].$$

(15)

Step 3: Normalize the indicator values

Considering that the dimension of indicators can bias the calculation, the indicator values should be normalized. To standardize the indicator values into $[0,1]$, the min–max normalization technique is employed as follows:

$${\tilde{a}}_{ij}=[\frac{x_{ij}^L-x_j^{-}}{x_j^{+}-x_j^{-}},\frac{x_{ij}^U-x_j^{-}}{x_j^{+}-x_j^{-}}],$$

(16)

where $x_j^{+}$ and $x_j^{-}$ are the maximum and minimum values of each indicator, respectively.

Then, the normalized decision-making matrix can be obtained as

$$\tilde{A}=\left[\begin{array}{cccc}{\tilde{a}}_{11}& {\tilde{a}}_{12}& \dots & {\tilde{a}}_{1n}\\ {\tilde{a}}_{21}& {\tilde{a}}_{22}& \dots & {\tilde{a}}_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ {\tilde{a}}_{m1}& {\tilde{a}}_{2n}& \dots & {\tilde{a}}_{mn}\end{array}\right].$$

(17)

3.2. Phase 2: Determine the Indicator Weights with Comprehensive Weighting Method

Step 1: Calculate the subjective weights using extended DEMATEL method

The DEMATEL method is effective for identifying the cause–effect chain components of a complex system. It evaluates the interdependent relationships among indicators and finds the critical one through a visual structural model [50,51]. Therefore, it is used to determine the subjective weights.

First, the relationship between indicators is quantified using pairwise analysis. The relationship matrix can be expressed as

$$\tilde{G}=\left[\begin{array}{cccc}{\tilde{g}}_{11}& {\tilde{g}}_{12}& \dots & {\tilde{g}}_{1n}\\ {\tilde{g}}_{21}& {\tilde{g}}_{22}& \dots & {\tilde{g}}_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ {\tilde{g}}_{n1}& {\tilde{g}}_{n2}& \dots & {\tilde{g}}_{nn}\end{array}\right],$$

(18)

where ${\tilde{g}}_{ij}=[g_{ij}^L,g_{ij}^U],i,j=1,2,\dots ,n$, and its upper and lower bounds can be determined with the analysis rules shown in

Table 1. Rules for quantifying relationship between indicators.

Influence of i-th to j-th Indicator	None	Low	Medium	High	Extremely High
$g_{ij}^L$ < $g_{ij}^U$	0	1	2	3	4

.

Then, the direct influence matrix is calculated as

$${(\tilde{P})}_{n\times n}=\frac{\tilde{G}}{\max ({\displaystyle \sum _{j=1}^n{\tilde{g}}_{ij}})}.$$

(19)

Afterwards, the combined influence matrix is determined as

$${(\tilde{Y})}_{n\times n}=\underset{k\to \infty }{\lim }(\tilde{P}+{\tilde{P}}^2+\dots +{\tilde{P}}^k)=\tilde{P}{(\tilde{I}+\tilde{P})}^{-1},$$

(20)

where $\tilde{I}$ is the identity matrix composed of [0, 0] and [1, 1].

Further, the attributes of indicators are quantified as the degree of influence ($\tilde{M}$) and degree of being influenced ($\tilde{N}$), which can be calculated as

$${\tilde{M}}_i={\left[{\displaystyle \sum _{i=1}^n{\tilde{Y}}_{ij}}\right]}_{n\times 1},$$

(21)

$${\tilde{N}}_j={\left[{\displaystyle \sum _{j=1}^n{\tilde{Y}}_{ij}}\right]}_{1\times n}^T.$$

(22)

Finally, the degree of centrality is determined as the sum of $\tilde{M}$ and $\tilde{N}$, and the subjective weights can be obtained as

$${\tilde{w}}_j^s=\frac{{\tilde{M}}_i+{\tilde{N}}_j}{{\displaystyle \sum _{i=j=1}^n({\tilde{M}}_i+{\tilde{N}}_j)}}.$$

(23)

Step 2: Calculate the objective weights using extended entropy method

The entropy method is capable of quantifying the information amounts of indicators using entropy values. It determines the importance of indicators by comparing the entropy values. A smaller entropy value indicates more information and larger influence on the results [52]. Therefore, it is used to determine the objective weights.

First, the information entropies of each indicator are calculated by

$${\tilde{E}}_j=-\frac{1}{\ln n}{\displaystyle \sum _{j=1}^n{\tilde{h}}_{ij}}\ln ({\tilde{h}}_{ij}),$$

(24)

$${\tilde{h}}_{ij}=\frac{{\tilde{a}}_{ij}}{{\displaystyle \sum _{i=1}^m{\tilde{a}}_{ij}}}.$$

(25)

Then, the objective indicator weights are obtained as

$${\tilde{w}}_j^o=\frac{1-{\tilde{E}}_j}{{\displaystyle \sum _{j=1}^n(1-{\tilde{E}}_j)}}.$$

(26)

Step 3: Obtain the comprehensive indicator weights

By combining the subjective and objective weights, the comprehensive indicator weights can be determined as

$${\tilde{w}}_j=\theta {\tilde{w}}_j^s+(1-\theta ){\tilde{w}}_j^o,$$

(27)

where $0\le \theta \le 1$ is a preference factor; $\theta =0.5$ is selected in this paper, which indicates that the importance of subjective and objective weights is equal.

3.3. Phase 3: Obtain the Ranking Results with Extended COPRAS Method

Step 1: Determine the weighted decision-making matrix

According to Equation (2), the weighted decision-making matrix is obtained as follows [48]:

$${\tilde{d}}_{ij}={\tilde{w}}_j{\tilde{a}}_{ij}.$$

(28)

Step 2: Calculate the benefit and cost scores

The benefit and cost scores of each sample can be calculated as follows [48]:

$${\tilde{S}}_i^{+}={\displaystyle \sum _{j\in k^{+}}{\tilde{d}}_{ij}},$$

(29)

$${\tilde{S}}_i^{-}={\displaystyle \sum _{j\in k^{-}}{\tilde{d}}_{ij}},$$

(30)

where $k^{+}$ and $k^{-}$ are serial number sets of benefit- and cost-type indicators, respectively.

Step 3: Determine the minimum value of cost scores

$${\tilde{S}}_{\min }^{-}=\min ({\tilde{S}}_i^{-}),i=1,2,\dots ,m.$$

(31)

Step 4: Calculate the significance scores

The significance scores of each sample can be calculated as follows [53]:

$${\tilde{S}}_i={\tilde{S}}_i^{+}+\frac{\exp ({\tilde{S}}_{\min }^{-}){\displaystyle \sum _{i=1}^m\exp ({\tilde{S}}_i^{-})}}{\exp ({\tilde{S}}_i^{-}){\displaystyle \sum _{i=1}^m\frac{\exp ({\tilde{S}}_{\min }^{-})}{\exp ({\tilde{S}}_i^{-})}}}={\tilde{S}}_i^{+}+\frac{{\displaystyle \sum _{i=1}^m\exp ({\tilde{S}}_i^{-})}}{\exp ({\tilde{S}}_i^{-}){\displaystyle \sum _{i=1}^m\frac{1}{\exp ({\tilde{S}}_i^{-})}}}.$$

(32)

Step 5: Rank the samples

Based on Equations (6) and (7), the complementary matrix for pairwise comparison of significance scores can be constructed as [46]

$$P=\left[\begin{array}{cccc}{p}_{11}& p_{12}& \dots & p_{1m}\\ p_{21}& p_{22}& \dots & p_{2m}\\ ⋮& ⋮& \ddots & ⋮\\ p_{m1}& p_{m2}& \dots & p_{mm}\end{array}\right],$$

(33)

where $p_{ij}=p({\tilde{S}}_i\ge {\tilde{S}}_j),i,j=1,2,\dots ,m$ is the possibility degree of ${\tilde{S}}_i\ge {\tilde{S}}_j$.

Therefore, the ranks of samples can be determined using significance scores. The larger ${\tilde{S}}_i$ is, the better the sample.

In addition, to facilitate understanding of the proposed methodology, the symbols used have been compiled as

Table A1. List of symbols.

$\tilde{\gamma }$	Interval-valued fuzzy number (IVFN)	$\tilde{P}$	Direct influence matrix
${\gamma }^L$	Lower limit of $\tilde{\gamma }$	$\tilde{Y}$	Combined influence matrix
${\gamma }^U$	Upper limit of $\tilde{\gamma }$	$\tilde{I}$	Identity matrix composed of [0, 0] and [1, 1]
$\pi (\tilde{\gamma })$	Middle point of $\tilde{\gamma }$	$\tilde{M}$	Degree of influence
$\omega (\tilde{\gamma })$	Half-width of $\tilde{\gamma }$	$\tilde{N}$	Degree of being influenced
$\tilde{a}$, $\tilde{b}$	Example IVFNs	${\tilde{w}}_j^s$	Subjective indicator weight
$\lambda$	Real number	${\tilde{E}}_j$	Information entropy
$len(\tilde{a})$	Length of $\tilde{a}$	${\tilde{w}}_j^o$	Objective indicator weight
$p(\tilde{a}\ge \tilde{b})$	$\mathrm{Possibility}\mathrm{degree}\mathrm{of}\tilde{b}\ge \tilde{a}$	$\theta$	Preference factor of subjective and objective weights
$d_E(\tilde{a},\tilde{b})$	Euclidean distance between $\tilde{a}$ and $\tilde{b}$	${\tilde{w}}_j$	Comprehensive indicator weights
$d_H(\tilde{a},\tilde{b})$	Hamming distance between $\tilde{a}$ and $\tilde{b}$	${\tilde{d}}_{ij}$	Element of weighted decision-making matrix
$B_j$	The j-th indicator	${\tilde{S}}_i^{+}$	The benefit score
$C_i$	The i-th sample (alternative)	${\tilde{S}}_i^{-}$	The cost score
$x_{ij}$	$\mathrm{Data}\mathrm{collection}\mathrm{of}{B}_j$$\mathrm{for}{C}_i$	${\tilde{S}}_i$	The significance score
$x_{ij}^k$	$\mathrm{The}k-\mathrm{th}\mathrm{value}\mathrm{of}{x}_{ij}$	$P$	Complementary matrix
${\tilde{x}}_{ij}$	$\mathrm{Converted}\mathrm{IVFN}\mathrm{corresponding}\mathrm{to}{x}_{ij}$	${\sigma }_c$	Uniaxial compressive strength
$\tilde{X}$	Fuzzy decision-making matrix	${\sigma }_t$	Tensile strength
$x_j^{+}$	Maximum value of j-th indicator	${\sigma }_{\theta }$	The tangential stress
$x_j^{-}$	Minimum value of j-th indicator	${\sigma }_1$	The maximum principal stress
${\tilde{a}}_{ij}$	$\mathrm{Normalized}\mathrm{IVFN}\mathrm{of}{\tilde{x}}_{ij}$	$E$	Elastic modulus
$\tilde{A}$	Normalized decision-making matrix	${\phi }_1$	Stored elastic energy
$\tilde{G}$	Relationship matrix	${\phi }_2$	Dissipated elastic energy
$g_{ij}^L$	Lower influence of i-th to j-th indicator	$\epsilon$	Total strain
$g_{ij}^U$	Upper influence of i-th to j-th indicator	${\epsilon }_1$	Irreversible strain
${\tilde{g}}_{ij}$	Influence of i-th to j-th indicator	$H$	Depth

in Appendix A.

4. Case Study

In this section, the proposed methodology is applied to evaluate the rockburst hazard in the Jiaojia gold mine.

4.1. Project Profile

The Jiaojia gold mine is located in Laizhou City, Shandong Province, China, with the geographic location of 120°07′–120°10′ E, 37°23′–37°26′ N. It consists of three mining areas: Sizhuang, Jiaojia and Wangershan. As a member of the extra-large gold-rich area in the Laizhou–Zhaoyuan area, Jiaojia gold mine has been in production for over 40 years. The lithologies in the mining area are mainly granites, which have high rockburst proneness. Moreover, the Jiaojia fault belt traverses and affects most of the mining area, with a length of about 27 kilometers and a width of 70–250 meters. With the increase in mining depth, the stability of the rock mass becomes worse, and the rockburst hazard rises accordingly. The phenomena of rock mass fracture in the Jiaojia and Sizhuang districts are shown in Figure 2 and Figure 3, respectively. It can be seen that the spalling of the rock masses is evident, and the fractures are clear and serious. Therefore, evaluating the rockburst hazard is critical for the development of the Jiaojia gold mine.

4.2. Determination of the Evaluation Indicators

Determining appropriate indicators is necessary for the hazard evaluation of rockbursts. However, due to the diversity of indicators and the complexity of engineering cases, there is no standard indicator system yet. In this study, six indicators were selected based on the aspects of energy, strength, brittleness and in situ stress, including the linear elastic energy index ($B_1$), elastic strain energy index ($B_2$), coefficient of deformation for brittleness ($B_3$), coefficient of strength for brittleness ($B_4$), tangential stress index ($B_5$) and principal stress index ($B_6$). The detailed descriptions of these indicators are listed in

Table 2. Indicators for rockburst hazard evaluation.

Indicators	Types	Description	Perspective
$B_1$ [54,55]	Benefit	$\mathrm{Indicates}\mathrm{the}\mathrm{elastic}\mathrm{energy}\mathrm{stored}\mathrm{in}\mathrm{rock}\mathrm{specimen}\mathrm{before}\mathrm{reaching}\mathrm{peak}\mathrm{strength}\mathrm{in}\mathrm{the}\mathrm{uniaxial}\mathrm{compression}\mathrm{test},\mathrm{which}\mathrm{can}\mathrm{be}\mathrm{calculated}\mathrm{using}{B}_1=\frac{{\sigma }_c^2}{2E}$.	Energy
$B_2$ [56]	Benefit	$\mathrm{Reflects}\mathrm{the}\mathrm{characteristic}\mathrm{of}\mathrm{energy}\mathrm{accumulation}\mathrm{and}\mathrm{dissipation},\mathrm{which}\mathrm{can}\mathrm{be}\mathrm{calculated}\mathrm{using}{B}_2=\frac{{\phi }_1}{{\phi }_2}$.	Energy
$B_3$ [57]	Benefit	$\mathrm{Describes}\mathrm{the}\mathrm{characteristic}\mathrm{of}\mathrm{rock}\mathrm{brittleness}\mathrm{from}\mathrm{the}\mathrm{perspective}\mathrm{of}\mathrm{deformation},\mathrm{which}\mathrm{can}\mathrm{be}\mathrm{calculated}\mathrm{using}{B}_3=\frac{\epsilon }{{\epsilon }_1}$.	Brittleness
$B_4$ [58,59]	Benefit	$\mathrm{Describes}\mathrm{the}\mathrm{characteristic}\mathrm{of}\mathrm{rock}\mathrm{brittleness}\mathrm{from}\mathrm{the}\mathrm{perspective}\mathrm{of}\mathrm{strength},\mathrm{which}\mathrm{can}\mathrm{be}\mathrm{calculated}\mathrm{using}{B}_4=\frac{{\sigma }_c}{{\sigma }_t}$.	Brittleness
$B_5$ [60]	Benefit	$\mathrm{Represents}\mathrm{the}\mathrm{relationship}\mathrm{of}\tan \mathrm{gential}\mathrm{stress}\mathrm{and}\mathrm{strength}\mathrm{of}\mathrm{surrounding}\mathrm{rock},\mathrm{which}\mathrm{can}\mathrm{be}\mathrm{calculated}\mathrm{using}{B}_5=\frac{{\sigma }_{\theta }}{{\sigma }_c}$.	Stress and strength
$B_6$ [61,62]	Cost	$\mathrm{Represents}\mathrm{the}\mathrm{relationship}\mathrm{of}\mathrm{in}\mathrm{situ}\mathrm{stress}\mathrm{and}\mathrm{strength}\mathrm{of}\mathrm{surrounding}\mathrm{rock},\mathrm{which}\mathrm{can}\mathrm{be}\mathrm{calculated}\mathrm{using}{B}_6=\frac{{\sigma }_c}{{\sigma }_1}$.	Stress and strength

Note: ${\sigma }_c$ is the uniaxial compressive strength; $E$ is the elastic modulus; ${\phi }_1$ is the stored elastic energy and ${\phi }_2$ is the dissipated elastic energy; $\epsilon$ and ${\epsilon }_1$ are the total strain and irreversible strain of the rock before peak strength, respectively; ${\sigma }_t$ is the tensile strength; ${\sigma }_{\theta }$ is the tangential stress; ${\sigma }_1$ is the maximum principal stress.

.

According to the severity of consequences, the rockburst hazards were classified into four levels: none (I), weak (II), moderate (III) and strong (IV) [63,64]. Correspondingly, the indicators were divided into four levels, which are shown in

Table 3. Grading standards of each indicator.

Indicators	Levels
	I	II	III	IV
$B_1$	<40	40–100	100–200	>200
$B_2$	<2	2–3.5	3.5–5	>5
$B_3$	<2	2–6	6–9	>9
$B_4$	<10	10–14	14–18	>18
$B_5$	<0.2	0.2–0.3	0.3–0.55	>0.55
$B_6$	>14.5	5.5–14.5	2.5–5.5	<2.5

.

4.3. Data Gathering

To obtain the indicator values, field investigations were conducted in three mining areas. Then, the rock specimens were collected from −630 m, −670 m and −710 m levels in the Sizhuang district, −630 m and −710 m levels in the Jiaojia district and −630 m and −790 m levels in the Wangershan district. A series of laboratory tests were conducted to obtain the key parameters, including unconfined compressive strength tests, Brazilian indirect tension tests, unconfined compressive loading–unloading tests and in situ stress tests. By fitting the in situ stress test results of eight sites at four levels, the in situ stress field of the Jiaojia gold mine was obtained as

$$\begin{array}{l}{\sigma }_1=0.0252H+13.484\\ {\sigma }_2=0.0148H+3.5883\\ {\sigma }_3=0.0292H\end{array}$$

(34)

where ${\sigma }_1$ and ${\sigma }_2$ are the maximum and minimum horizontal principal stress, respectively; ${\sigma }_3$ is the vertical principal stress.

Therefore, the indicator values for rockburst hazard evaluation were obtained. Based on Equation (14), the fuzzy indicator values expressed by IVFNs were determined, as shown in

Table 4. Fuzzy indicator values.

Mining Area	Depth (m)	Lithology	${\mathit{B}}_{\mathbf{1}}$	${\mathit{B}}_{\mathbf{2}}$	${\mathit{B}}_{\mathbf{3}}$	${\mathit{B}}_{\mathbf{4}}$	${\mathit{B}}_{\mathbf{5}}$	${\mathit{B}}_{\mathbf{6}}$
Sizhuang	−660	Sericite	[119.74, 143.88]	[3.01, 9.67]	[5.42, 13.87]	[7.91, 11.23]	[0.52, 0.55]	[4.26, 4.50]
	−700	Sericitized granite	[124.30, 239.42]	[8.61, 18.97]	[8.22, 21.19]	[10.15, 19.96]	[0.42, 0.54]	[4.29, 5.63]
	−740	Sericitized granite	[127.03, 188.42]	[5.93, 28.72]	[8.15, 37.47]	[7.84, 9.15]	[0.52, 0.55]	[4.24, 4.47]
Jiaojia	−660	Weakly potassic granite	[94.46, 100.81]	[2.75, 5.61]	[9.86, 16.86]	[12.44, 15.17]	[0.68, 0.69]	[3.39, 3.46]
	−740	Weakly potassic granite	[56.45, 255.54]	[4.52, 23.29]	[8.49, 32.92]	[4.94, 23.61]	[0.45, 0.93]	[2.50, 5.20]
Wangershan	−660	Pyritized granite	[52.03, 228.88]	[2.32, 2.60]	[4.98, 5.98]	[8.68, 20.92]	[0.48, 0.92]	[2.56, 4.94]
	−820	Granite	[110.49, 487.07]	[7.06, 9.96]	[9.52, 23.18]	[15.78, 33.90]	[0.33, 0.67]	[3.43, 7.00]

. For example, the coefficients of strength for brittleness at the depth of 670 m in the Sizhuang district were calculated as {9.60, 11.23, 8.35, 9.09, 10.63, 7.91}, and then they were transformed into IVFNs [7.91, 11.23].

4.4. Evaluation of Rockburst Hazards

4.4.1. Express the Indicator Values with IVFNs

In order to distinguish the hazard levels of seven sites, the samples with known hazard levels were determined according to the grading standards in Table 3. The standard samples with levels I, II, III and IV were denoted as $C_{\mathrm{I}}^{*}$, $C_{\mathrm{II}}^{*}$, $C_{\mathrm{III}}^{*}$ and $C_{\mathrm{IV}}^{*}$, respectively, as shown in

Table 5. Fuzzy decision-making matrix.

	$B_1$	$B_2$	$B_3$	$B_4$	$B_5$	$B_6$
$C_{\mathrm{I}}^{*}$	[0, 0]	[0, 0]	[0, 0]	[0, 0]	[0, 0]	[14.5, 14.5]
$C_{\mathrm{II}}^{*}$	[40, 40]	[2, 2]	[2, 2]	[10, 10]	[0.2, 0.2]	[5.5, 5.5]
$C_{\mathrm{III}}^{*}$	[100, 100]	[3.5, 3.5]	[6, 6]	[14, 14]	[0.3, 0.3]	[2.5, 2.5]
$C_{\mathrm{IV}}^{*}$	[200, 200]	[5, 5]	[9, 9]	[18, 18]	[0.55, 0.55]	[0, 0]
$C_1$	[119.74, 143.88]	[3.01, 9.67]	[5.42, 13.87]	[7.91, 11.23]	[0.52, 0.55]	[4.26, 4.50]
$C_2$	[124.30, 239.42]	[8.61, 18.97]	[8.22, 21.19]	[10.15, 19.96]	[0.42, 0.54]	[4.29, 5.63]
$C_3$	[127.03, 188.42]	[5.93, 28.72]	[8.15, 37.47]	[7.84, 9.15]	[0.52, 0.55]	[4.24, 4.47]
$C_4$	[94.46, 100.81]	[2.75, 5.61]	[9.86, 16.86]	[12.44, 15.17]	[0.68, 0.69]	[3.39, 3.46]
$C_5$	[56.45, 255.54]	[4.52, 23.29]	[8.49, 32.92]	[4.94, 23.61]	[0.45, 0.93]	[2.50, 5.20]
$C_6$	[52.03, 228.88]	[2.32, 2.60]	[4.98, 5.98]	[8.68, 20.92]	[0.48, 0.92]	[2.56, 4.94]
$C_7$	[110.49, 487.07]	[7.06, 9.96]	[9.52, 23.18]	[15.78, 33.90]	[0.33, 0.67]	[3.43, 7.00]

.

Then, according to Equations (16) and (17), the normalized decision-making matrix was obtained, as shown in

Table 6. Normalized decision-making matrix.

	$B_1$	$B_2$	$B_3$	$B_4$	$B_5$	$B_6$
$C_{\mathrm{I}}^{*}$	[0, 0]	[0, 0]	[0, 0]	[0, 0]	[0, 0]	[1, 1]
$C_{\mathrm{II}}^{*}$	[0.0821, 0.0821]	[0.0696, 0.0696]	[0.0534, 0.0534]	[0.2950, 0.2950]	[0.2149, 0.2149]	[0.3793, 0.3793]
$C_{\mathrm{III}}^{*}$	[0.2053, 0.2053]	[0.1219, 0.1219]	[0.1601, 0.1601]	[0.4129, 0.4129]	[0.3223, 0.3223]	[0.1724, 0.1724]
$C_{\mathrm{IV}}^{*}$	[0.4106, 0.4106]	[0.1741, 0.1741]	[0.2402, 0.2402]	[0.5309, 0.5309]	[0.5908, 0.5908]	[0, 0]
$C_1$	[0.2458, 0.2954]	[0.1048, 0.3367]	[0.1446, 0.3702]	[0.2332, 0.3313]	[0.5622, 0.5938]	[0.2939, 0.3104]
$C_2$	[0.2552, 0.4915]	[0.2998, 0.6605]	[0.2194, 0.5655]	[0.2993, 0.5886]	[0.4464, 0.5852]	[0.2962, 0.3882]
$C_3$	[0.2608, 0.3869]	[0.2065, 1]	[0.2175, 1]	[0.2314, 0.2699]	[0.5585, 0.5889]	[0.2924, 0.3083]
$C_4$	[0.1939, 0.2070]	[0.0958, 0.1953]	[0.2631, 0.4500]	[0.3671, 0.4473]	[0.7323, 0.7458]	[0.2340, 0.2383]
$C_5$	[0.1159, 0.5247]	[0.1574, 0.8109]	[0.2266, 0.8786]	[0.1457, 0.6963]	[0.4796, 1]	[0.1722, 0.3589]
$C_6$	[0.1068, 0.4699]	[0.0808, 0.0905]	[0.1329, 0.1596]	[0.2560, 0.6171]	[0.5123, 0.9871]	[0.1768, 0.3407]
$C_7$	[0.2269, 1]	[0.2458, 0.3468]	[0.2541, 0.6186]	[0.4654, 1]	[0.3524, 0.7197]	[0.2363, 0.4826]

.

4.4.2. Determine the Comprehensive Indicator Weights

The relationship matrix of the indicators was determined via pairwise analysis, as shown in

Table 7. Relationship matrix of indicators.

	$B_1$	$B_2$	$B_3$	$B_4$	$B_5$	$B_6$
B1	[0.0, 0.0]	[3.1, 3.7]	[2.6, 3.2]	[2.7, 3.3]	[2.5, 2.8]	[2.4, 2.9]
$B_2$	[3.0, 3.5]	[0.0, 0.0]	[2.8, 3.3]	[2.3, 2.6]	[1.2, 1.5]	[1.1, 1.5]
$B_3$	[1.8, 2.1]	[2.0, 2.4]	[0.0, 0.0]	[3.1, 3.8]	[0.9, 1.3]	[0.9, 1.4]
$B_4$	[1.8, 2.1]	[1.5, 1.9]	[2.9, 3.5]	[0.0, 0.0]	[1.3, 1.8]	[1.5, 2.0]
$B_5$	[2.2, 2.5]	[2.0, 2.5]	[1.6, 1.8]	[1.8, 2.3]	[0.0, 0.0]	[3.2, 3.9]
$B_6$	[2.0, 2.8]	[1.8, 2.4]	[0.8, 1.5]	[1.6, 2.5]	[3.0, 3.8]	[0.0, 0.0]

. Then, the direct influence matrix was determined based on Equation (19), as shown in

Table 8. Direct influence matrix.

	$B_1$	$B_2$	$B_3$	$B_4$	$B_5$	$B_6$
$B_1$	[0.0000, 0.0000]	[0.2327, 0.2331]	[0.1955, 0.2013]	[0.2030, 0.2075]	[0.1761, 0.1880]	[0.1805, 0.1824]
$B_2$	[0.2201, 0.2256]	[0.0000, 0.0000]	[0.2075, 0.2105]	[0.1635, 0.1729]	[0.0902, 0.0943]	[0.0827, 0.0943]
$B_3$	[0.1321, 0.1353]	[0.1504, 0.1509]	[0.0000, 0.0000]	[0.2331, 0.2390]	[0.0677, 0.0818]	[0.0677, 0.0881]
$B_4$	[0.1321, 0.1353]	[0.1128, 0.1195]	[0.2180, 0.2201]	[0.0000, 0.0000]	[0.0977, 0.1132]	[0.1128, 0.1258]
$B_5$	[0.1572, 0.1654]	[0.1504, 0.1572]	[0.1132, 0.1203]	[0.1353, 0.1447]	[0.0000, 0.0000]	[0.2406, 0.2453]
$B_6$	[0.1504, 0.1761]	[0.1353, 0.1509]	[0.0602, 0.0943]	[0.1203, 0.1572]	[0.2256, 0.2390]	[0.0000, 0.0000]

. Afterwards, the combined influence matrix was determined based on Equation (20), as shown in

Table 9. Combined influence matrix.

	$B_1$	$B_2$	$B_3$	$B_4$	$B_5$	$B_6$
$B_1$	[0.5819, 0.6773]	[0.7547, 0.8633]	[0.7562, 0.8752]	[0.7916, 0.9244]	[0.6388, 0.7349]	[0.6420, 0.7609]
$B_2$	[0.6601, 0.7306]	[0.4645, 0.5496]	[0.6676, 0.7533]	[0.6640, 0.7571]	[0.4732, 0.5604]	[0.4742, 0.5784]
$B_3$	[0.5168, 0.6011]	[0.5162, 0.6125]	[0.4167, 0.5155]	[0.6248, 0.7396]	[0.3900, 0.4944]	[0.3961, 0.5152]
$B_4$	[0.5264, 0.6138]	[0.4998, 0.6035]	[0.6008, 0.7049]	[0.4441, 0.5592]	[0.4256, 0.5325]	[0.4423, 0.5577]
$B_5$	[0.6300, 0.7144]	[0.6045, 0.7097]	[0.5966, 0.6985]	[0.6371, 0.7636]	[0.4146, 0.5096]	[0.6163, 0.7251]
$B_6$	[0.5637, 0.7296]	[0.5392, 0.7082]	[0.4961, 0.6886]	[0.5639, 0.7745]	[0.5539, 0.7051]	[0.3768, 0.5309]

.

According to Equation (21), the degrees of influence were calculated as ${\tilde{M}}_i$ = {[4.1651, 4.8359], [3.4035, 3.9294], [2.8608, 3.4784], [2.9389, 3.5715], [3.4990, 4.1209], [3.0936, 4.1369]}. Based on Equation (22), the degrees of being influenced were calculated as ${\tilde{N}}_j$ = {[3.4790, 4.0668], [3.3789, 4.0468], [3.5340, 4.2360], [3.7255, 4.5185], [2.8961, 3.5369], [2.9476, 3.6682]}. Then, the subjective weights were determined using Equation (23), as listed in

Table 10. Indicator weights.

	$B_1$	$B_2$	$B_3$	$B_4$	$B_5$	$B_6$
${\tilde{w}}_j^s$	[0.1849, 0.1915]	[0.1657, 0.1699]	[0.1602, 0.1602]	[0.1669, 0.1680]	[0.1591, 0.1602]	[0.1513, 0.1621]
${\tilde{w}}_j^o$	[0.1660, 0.1670]	[0.1764, 0.2660]	[0.1412, 0.2106]	[0.1107, 0.1383]	[0.1091, 0.1260]	[0.1376, 0.2511]
${\tilde{w}}_j$	[0.1755, 0.1792]	[0.1710, 0.2179]	[0.1507, 0.1854]	[0.1388, 0.1532]	[0.1341, 0.1431]	[0.1445, 0.2066]

.

Based on Equations (24) and (25), the entropy values of each indicator were calculated as ${\tilde{E}}_j$ = {[0.8920, 0.9214], [0.8270, 0.9169], [0.8630, 0.9335], [0.9280, 0.9349], [0.9290, 0.9406], [0.8817, 0.9105]}. Then, the objective weights were determined using Equation (26), as listed in Table 10.

Finally, the comprehensive indicator weights were calculated using Equation (27), as shown in Table 10.

4.4.3. Evaluate the Hazard Level with Extended COPRAS Method

According to Equation (27), the weighted decision-making matrix was determined, as shown in

Table 11. Weighted decision-making matrix.

	$B_1$	$B_2$	$B_3$	$B_4$	$B_5$	$B_6$
$C_{\mathrm{I}}^{*}$	[0, 0]	[0, 0]	[0, 0]	[0, 0]	[0, 0]	[0.1445, 0.2066]
$C_{\mathrm{II}}^{*}$	[0.0144, 0.0147]	[0.0119, 0.0152]	[0.0080, 0.0099]	[0.0410, 0.0452]	[0.0288, 0.0308]	[0.0548, 0.0784]
$C_{\mathrm{III}}^{*}$	[0.0360, 0.0368]	[0.0208, 0.0266]	[0.0241, 0.0297]	[0.0573, 0.0632]	[0.0432, 0.0461]	[0.0249, 0.0356]
$C_{\mathrm{IV}}^{*}$	[0.0720, 0.0736]	[0.0298, 0.0379]	[0.0362, 0.0445]	[0.0737, 0.0813]	[0.0792, 0.0845]	[0, 0]
$C_1$	[0.0431, 0.0529]	[0.0179, 0.0734]	[0.0218, 0.0686]	[0.0324, 0.0507]	[0.0754, 0.0850]	[0.0425, 0.0641]
$C_2$	[0.0448, 0.0881]	[0.0513, 0.1439]	[0.0331, 0.1049]	[0.0415, 0.0902]	[0.0598, 0.0837]	[0.0428, 0.0802]
$C_3$	[0.0458, 0.0693]	[0.0353, 0.2179]	[0.0328, 0.1854]	[0.0321, 0.0413]	[0.0749, 0.0843]	[0.0422, 0.0637]
$C_4$	[0.0340, 0.0371]	[0.0164, 0.0426]	[0.0396, 0.0834]	[0.0510, 0.0685]	[0.0982, 0.1067]	[0.0338, 0.0492]
$C_5$	[0.0203, 0.0940]	[0.0269, 0.1767]	[0.0341, 0.1629]	[0.0202, 0.1066]	[0.0643, 0.1431]	[0.0249, 0.0742]
$C_6$	[0.0187, 0.0842]	[0.0138, 0.0197]	[0.0200, 0.0296]	[0.0355, 0.0945]	[0.0687, 0.1413]	[0.0255, 0.0704]
$C_7$	[0.0398, 0.1792]	[0.0420, 0.0756]	[0.0383, 0.1147]	[0.0646, 0.1532]	[0.0472, 0.1030]	[0.0341, 0.0997]

.

Based on Equations (29) and (30), the benefit and cost scores of each sample were obtained, as shown in

Table 12. Scores of samples.

	${\tilde{S}}_i^{+}$	${\tilde{S}}_i^{-}$	${\tilde{S}}_i$
$C_{\mathrm{I}}^{*}$	[0, 0]	[0.1445, 0.2066]	[0.9427, 0.9445]
$C_{\mathrm{II}}^{*}$	[0.1041, 0.1157]	[0.0548, 0.0784]	[1.1353, 1.1895]
$C_{\mathrm{III}}^{*}$	[0.1815, 0.2024]	[0.0249, 0.0356]	[1.2440, 1.3231]
$C_{\mathrm{IV}}^{*}$	[0.2909, 0.3219]	[0, 0]	[1.3802, 1.4832]
$C_1$	[0.1906, 0.3307]	[0.0425, 0.0641]	[1.2346, 1.4198]
$C_2$	[0.2305, 0.5108]	[0.0428, 0.0802]	[1.2741, 1.5826]
$C_3$	[0.2209, 0.5983]	[0.0422, 0.0637]	[1.2650, 1.6879]
$C_4$	[0.2392, 0.3383]	[0.0338, 0.0492]	[1.2922, 1.4438]
$C_5$	[0.1659, 0.6834]	[0.0249, 0.0742]	[1.2284, 1.7617]
$C_6$	[0.1568, 0.3693]	[0.0255, 0.0704]	[1.2186, 1.4517]
$C_7$	[0.2320, 0.6257]	[0.0341, 0.0997]	[1.2847, 1.6768]

. The minimum value of the cost score was determined as [0, 0]. Then, the significance scores of the samples were calculated using Equation (31), as listed in Table 12. Afterwards, the complementary matrix by pairwise comparison of significance scores was constructed as

Table 13. Complementary matrix.

	$C_{\mathrm{I}}^{*}$	$C_{\mathrm{II}}^{*}$	$C_{\mathrm{III}}^{*}$	$C_{\mathrm{IV}}^{*}$	$C_1$	$C_2$	$C_3$	C4	$C_5$	$C_6$	$C_7$
$C_{\mathrm{I}}^{*}$	0.5	0	0	0	0	0	0	0	0	0	0
$C_{\mathrm{II}}^{*}$	1.0	0.5	0	0	0	0	0	0	0	0	0
$C_{\mathrm{III}}^{*}$	1.0	1.0	0.5	0	0.3348	0.1263	0.1156	0.1337	0.1546	0.3347	0.0815
$C_{\mathrm{IV}}^{*}$	1.0	1.0	1.0	0.5	0.8624	0.5081	0.4148	0.7500	0.4004	0.7873	0.4010
$C_1$	1.0	1.0	0.6652	0.1376	0.5	0.2951	0.2545	0.3788	0.2664	0.4810	0.2341
$C_2$	1.0	1.0	0.8737	0.4919	0.7049	0.5	0.4342	0.6311	0.4207	0.6722	0.4252
$C_3$	1.0	1.0	0.8844	0.5852	0.7455	0.5658	0.5	0.6888	0.4806	0.7155	0.4948
$C_4$	1.0	1.0	0.8663	0.2500	0.6212	0.3689	0.3112	0.5	0.3145	0.5855	0.2927
$C_5$	1.0	1.0	0.8454	0.5996	0.7336	0.5793	0.5194	0.6855	0.5	0.7087	0.5155
$C_6$	1.0	1.0	0.6653	0.2127	0.5190	0.3278	0.2845	0.4145	0.2913	0.5	0.2671
$C_7$	1.0	1.0	0.9185	0.5990	0.7659	0.5748	0.5052	0.7073	0.4845	0.7329	0.5

.

Finally, the ranking results of the samples were determined as $C_5$ > $C_7$ > $C_3$ > $C_{\mathrm{IV}}^{*}$ > $C_2$ > $C_4$ > $C_6$ > $C_1$ > $C_{\mathrm{III}}^{*}$ > $C_{\mathrm{II}}^{*}$ > $C_{\mathrm{I}}^{*}$. Therefore, the rockburst hazards of $C_3$, $C_5$ and $C_7$ belonged to level IV, and those of $C_1$, $C_2$, $C_4$ and $C_6$ belonged to level III.

Actually, several rockburst phenomena have been reported from the −630 m level of Sizhuang district, the −630 m level of Jiaojia district and the −670 m, −710 m and −750 m levels of Wangershan district. The on-site investigation revealed that, with the increase in mining depth, rockburst phenomena such as rock ejection, spalling of rock masses and deformation of tunnels, were more and more severe. In addition, since the rockburst hazard was greater at higher depths under the same conditions, the ranking results of $C_7$ > $C_6$, $C_5$ > $C_4$ and $C_3$ > $C_2$ > $C_1$ were reliable. Therefore, it can be inferred that the hazard levels determined with the proposed method were consistent with the field status.

5. Discussion

5.1. Sensitivity Analysis

In this study, the indicator weights were determined by the combination of the extended DEMATEL and entropy methods with a preference factor of $\theta =0.5$. However, since the understanding of the indicators may differ, other preference factors may be selected. To explore the impact of the preference factor on the evaluation results, a sensitivity analysis was conducted.

First, the values of $\theta$ were assigned as 0 to 1, respectively. Then, the significance scores under different values of $\theta$ were obtained. To investigate the variation in significance scores, the middle points and half-widths of ${\tilde{S}}_i$ were calculated, as shown in Figure 4 and Figure 5, respectively. It can be seen that only minor changes occurred in the middle points when different values of $\theta$ were selected. The fluctuating trends of the curves were highly identical with each other. Although the half-widths varied when different values of $\theta$ were selected, the fluctuating trends of the curves were still sufficiently consistent.

Afterwards, the ranking results under different values of $\theta$ were determined, as listed in

Table 14. Scores of samples.

$\mathit{\theta }$	Ranking Results
0	$C_5$$>C_3$$>C_7$$>C_2$$>C_{\mathrm{IV}}^{*}$$>C_4$$>C_1$$>C_6$$>C_{\mathrm{III}}^{*}$$>C_{\mathrm{II}}^{*}$$>C_{\mathrm{I}}^{*}$
0.1	$C_5$$>C_3$$>C_7$$>C_2$$>C_{\mathrm{IV}}^{*}$$>C_4$$>C_1$$>C_6$$>C_{\mathrm{III}}^{*}$$>C_{\mathrm{II}}^{*}$$>C_{\mathrm{I}}^{*}$
0.2	$C_5$$>C_3$$>C_7$$>C_2$$>C_{\mathrm{IV}}^{*}$$>C_4$$>C_1$$>C_6$$>C_{\mathrm{III}}^{*}$$>C_{\mathrm{II}}^{*}$$>C_{\mathrm{I}}^{*}$
0.3	$C_5$$>C_3$$>C_7$$>C_2$$>C_{\mathrm{IV}}^{*}$$>C_4$$>C_6$$>C_1$$>C_{\mathrm{III}}^{*}$$>C_{\mathrm{II}}^{*}$$>C_{\mathrm{I}}^{*}$
0.4	$C_5$$>C_3$$>C_7$$>C_{\mathrm{IV}}^{*}$$>C_2$$>C_4$$>C_6$$>C_1$$>C_{\mathrm{III}}^{*}$$>C_{\mathrm{II}}^{*}$$>C_{\mathrm{I}}^{*}$
0.5	$C_5$$>C_7$$>C_3$$>C_{\mathrm{IV}}^{*}$$>C_2$$>C_4$$>C_6$$>C_1$$>C_{\mathrm{III}}^{*}$$>C_{\mathrm{II}}^{*}$$>C_{\mathrm{I}}^{*}$
0.6	$C_5$$>C_7$$>C_3$$>C_{\mathrm{IV}}^{*}$$>C_2$$>C_4$$>C_6$$>C_1$$>C_{\mathrm{III}}^{*}$$>C_{\mathrm{II}}^{*}$$>C_{\mathrm{I}}^{*}$
0.7	$C_5$$>C_7$$>C_3$$>C_{\mathrm{IV}}^{*}$$>C_2$$>C_4$$>C_6$$>C_1$$>C_{\mathrm{III}}^{*}$$>C_{\mathrm{II}}^{*}$$>C_{\mathrm{I}}^{*}$
0.8	$C_5$$>C_7$$>C_3$$>C_{\mathrm{IV}}^{*}$$>C_2$$>C_4$$>C_6$$>C_1$$>C_{\mathrm{III}}^{*}$$>C_{\mathrm{II}}^{*}$$>C_{\mathrm{I}}^{*}$
0.9	$C_7$$>C_5$$>C_3$$>C_{\mathrm{IV}}^{*}$$>C_2$$>C_4$$>C_6$$>C_1$$>C_{\mathrm{III}}^{*}$$>C_{\mathrm{II}}^{*}$$>C_{\mathrm{I}}^{*}$
1	$C_7$$>C_5$$>C_3$$>C_{\mathrm{IV}}^{*}$$>C_2$$>C_4$$>C_6$$>C_1$$>C_{\mathrm{III}}^{*}$$>C_{\mathrm{II}}^{*}$$>C_{\mathrm{I}}^{*}$

. It can be seen that the variances occurred when different values of preference factors were allocated. For example, when $\theta$ was specified as 0, 0.1 and 0.2, the ranking results were $C_5$ > $C_3$ > $C_7$ > $C_2$ > $C_{\mathrm{IV}}^{*}$ > $C_4$ > $C_1$ > $C_6$ > $C_{\mathrm{III}}^{*}$ > $C_{\mathrm{II}}^{*}$ > $C_{\mathrm{I}}^{*}$. When $\theta$ was specified as 0.9 and 1, the ranking results were $C_7$ > $C_5$ > $C_3$ > $C_{\mathrm{IV}}^{*}$ > $C_2$ > $C_4$ > $C_6$ > $C_1$ > $C_{\mathrm{III}}^{*}$ > $C_{\mathrm{II}}^{*}$ > $C_{\mathrm{I}}^{*}$. When $\theta$ was specified as 0.5, 0.6, 0.7 and 0.8, the ranking results were the same as $C_5$ > $C_7$ > $C_3$ > $C_{\mathrm{IV}}^{*}$ > $C_2$ > $C_4$ > $C_6$ > $C_1$ > $C_{\mathrm{III}}^{*}$ > $C_{\mathrm{II}}^{*}$ > $C_{\mathrm{I}}^{*}$. Compared with $\theta =0$ and $\theta =1$, the comprehensive weights had better stability. Although there are some differences in the evaluation results under different preference factors, the ranking results of $C_7$ > $C_6$, $C_5$ > $C_4$ and $C_3$ > $C_2$ > $C_1$ were stable. Therefore, the reliability of the evaluation results with $\theta =0.5$ was demonstrated.

5.2. Comparison Analysis

In this study, the COPRAS method was extended with IVFNs for rockburst hazard evaluation. To further verify the effectiveness of the proposed method, different methods were used for comparison.

The extended TOPSIS method with IVFNs and extended VIKOR method with IVFNs were used to compare with the proposed method. The evaluation results are shown in

Table 15. Evaluation results with different methods.

Methods	Ranking Results	Hazard Levels
		${\mathit{C}}_{\mathbf{1}}$	${\mathit{C}}_{\mathbf{2}}$	${\mathit{C}}_{\mathbf{3}}$	${\mathit{C}}_{\mathbf{4}}$	${\mathit{C}}_{\mathbf{5}}$	${\mathit{C}}_{\mathbf{6}}$	${\mathit{C}}_{\mathbf{7}}$
Proposed method	$C_5$$>C_7$$>C_3$$>C_{\mathrm{IV}}^{*}$$>C_2$$>C_4$$>C_6$$>C_1$$>C_{\mathrm{III}}^{*}$$>C_{\mathrm{II}}^{*}$$>C_{\mathrm{I}}^{*}$	III	III	IV	III	IV	III	IV
Extended TOPSIS	$C_5$$>C_7$$>C_3$$>C_2$$>C_{\mathrm{IV}}^{*}$$>C_4$$>C_6$$>C_1$$>C_{\mathrm{III}}^{*}$$>C_{\mathrm{II}}^{*}$$>C_{\mathrm{I}}^{*}$	III	IV	IV	III	IV	III	IV
Extended VIKOR	$C_5$$>C_2$$>C_3$$>C_7$$>C_1$$>C_{\mathrm{IV}}^{*}$$>C_4$$>C_6$$>C_{\mathrm{III}}^{*}$$>C_{\mathrm{II}}^{*}$$>C_{\mathrm{I}}^{*}$	IV	IV	IV	III	IV	III	IV

, and the detailed calculation processes are indicated as follows.

When the extended TOPSIS method with IVFNs was applied, the fuzzy decision-making matrix was first obtained. Then, the ideal solution was determined as ${\tilde{a}}_j^{+}$ = {[0.4106, 1], [0.2998, 1], [0.2631, 1], [0.5309, 1], [0.7323, 1], [0, 0]}, and the anti-ideal solution was determined as ${\tilde{a}}_j^{-}$ = {[0, 0], [0, 0], [0, 0], [0, 0], [1, 1]}. Afterwards, the weighted distances of each sample from the ideal solution were calculated as ${\tilde{d}}_i^{+}$= {[1.0517, 1.2560], [0.7878, 0.9226], [0.6493, 0.7547], [0.4840, 0.5625], [0.5642, 0.6626], [0.4208, 0.4956], [0.3643, 0.4144], [0.5315, 0.6239], [0.3172, 0.3689], [0.5458, 0.6541], [0.3518, 0.4351]}, and the distances from the anti-ideal solution were calculated as ${\tilde{d}}_i^{-}$ = {[0, 0], [0.2741, 0.3450], [0.4258, 0.5281], [0.6157, 0.7475], [0.4961, 0.6024], [0.6377, 0.7689], [0.7118, 0.8710], [0.5420, 0.6563], [0.7710, 0.9286], [0.5308, 0.6307], [0.7472, 0.8746]}. Finally, the relative closeness of the samples was determined as ${\tilde{R}}_i$ = {[0, 0], [0.2581, 0.2722], [0.3961, 0.4117], [0.5599, 0.5706], [0.4679, 0.4762], [0.6025, 0.6081], [0.6614, 0.6776], [0.5049, 0.5126], [0.7085, 0.7157], [0.4909, 0.4930], [0.6678, 0.6799]}. Therefore, the corresponding ranking result was $C_5$ > $C_7$ > $C_3$ > $C_2$ > $C_{\mathrm{IV}}^{*}$ > $C_4$ > $C_6$ > $C_1$ > $C_{\mathrm{III}}^{*}$ > $C_{\mathrm{II}}^{*}$ > $C_{\mathrm{I}}^{*}$.

When the extended VIKOR method with IVFNs was applied, the fuzzy decision-making matrix was first obtained. The ideal and anti-ideal solutions were the same as those determined with the extended TOPSIS method. Then, the global benefit values of the samples were calculated as ${\tilde{G}}_i$ = {[0.9145, 1.0855], [0.7042, 0.8231], [0.5873, 0.6829], [0.4477, 0.5219], [0.5052, 0.5915], [0.3678, 0.4307], [0.3128, 0.3527], [0.4819, 0.5643], [0.2762, 0.3186], [0.4972, 0.5944], [0.3068, 0.3776]}, and the individual regret values of the samples were calculated as ${\tilde{H}}_i$ = {[0.1755, 0.2179], [0.1582, 0.2001], [0.1468, 0.1870], [0.1369, 0.1744], [0.1175, 0.1443], [0.0863, 0.0882], [0.1024, 0.1068], [0.1360, 0.1733], [0.0908, 0.0927], [0.1533, 0.1953], [0.1074, 0.1368]}. Afterwards, by using the weight for the strategy of maximum group utility $\beta =0.5$, the compromise sorting scores were calculated as ${\tilde{Z}}_i$ = {[1.0000, 1.0000], [0.7387, 0.7601], [0.5829, 0.6185], [0.4178, 0.4648], [0.3540, 0.3944], [0.0718, 0.0731], [0.0939, 0.1192], [0.4398, 0.4883], [0.0176, 0.0251], [0.5486, 0.5926], [0.1422, 0.2260]}. Finally, the corresponding ranking result was $C_5$ > $C_2$ > $C_3$ > $C_7$ > $C_1$ > $C_{\mathrm{IV}}^{*}$ > $C_4$ > $C_6$ > $C_{\mathrm{III}}^{*}$ > $C_{\mathrm{II}}^{*}$ > $C_{\mathrm{I}}^{*}$.

It can be seen that the ranking results and corresponding hazard levels under the extended COPRAS and TOPSIS methods were similar. The ranking results of $C_5$ > $C_7$ > $C_3$ > $C_2$ > $C_4$ > $C_6$ > $C_1$ were identical, and difference only occurred in $C_2$. As for the extended VIKOR method, deviations arose from $C_1$, $C_2$ and $C_7$. However, the hazard levels of $C_3$, $C_4$, $C_5$, $C_6$ and $C_7$ were still the same as those under the extended COPRAS and TOPSIS methods. Therefore, the effectiveness of the extended COPRAS method was demonstrated.

In summary, the highlights of the proposed method are concluded to be as follows:

(1) The indicator values are expressed by IVFNs instead of crisp numbers, and thus can better capture the intrinsic uncertainty of indicator values.

(2) The extended DEMATEL and entropy methods are combined to obtain comprehensive weights, which can guarantee the stability of the evaluation results.

(3) The COPRAS method is extended with IVFNs, which can cope well with the evaluation problems of rockburst hazards under uncertainty.

(4) The extended COPRAS method was applied in the Jiaojia gold mine, and credible ranking results and hazard levels were obtained, which demonstrates the effectiveness and reliability of the proposed methodology.

6. Conclusions

Evaluating rockburst hazards is of great significance for the development of deep mining engineering. In this study, an integrated methodology was developed for rockburst hazard evaluation. First, considering that evaluation indicators are subject to some uncertainties, the IVFNs were used to express indicator values, so that the inherent uncertainties of indicator values were better captured. Second, the uncertainties of indicator weights were also taken into account. By extending the DEMATEL and entropy methods with IVFNs, the comprehensive fuzzy indicator weights were determined. Third, the COPRAS method was extended with IVFNs to evaluate the hazard levels of rockbursts under uncertain environments. Subsequently, an example of evaluating rockburst hazards in the Jiaojia gold mine was illustrated. The results showed that rockburst hazards in three critical areas belonged to the strong level (IV), and those of the other four areas belonged to the moderate level (III). The determined levels were consistent with the field status, and showed a tendency to increase with depth in the same district. Furthermore, a sensitivity analysis was conducted by assigning different values to preference factors, and the fluctuating trends of middle-point curves and half-width curves demonstrated the stability of the proposed methodology. Compared with the evaluation results obtained using the extended TOPSIS and VIKOR methods, the effectiveness of the proposed approach was demonstrated. The evaluation results can provide an effective reference for rockburst prevention. Specifically, the prevention measures should be specially designed according to different hazard levels.

It should be noted that the proposed methodology still has some limitations. Potential outliers in data collection are capable of distorting IVFNs. Whenever possible, it is suggested to collect high-quality data or to conduct data cleaning with poor-quality data. Moreover, due to the complex geological environments of deep mining engineering, the mechanism and influencing factors of rockbursts are complicated. It is necessary to develop some appropriate principles for indicator selection in the future.