# A Status Evaluation of Rock Instability in Metal Mines Based on the SPA-IAHP-PCN Model
## 2 Methodology  
### Calculation of Index Weight Value  
Through a review of the relevant literature and consultation with engineers and professors specializing in the field of rock instability hazards in metal mines, an index evaluation system for metal mines is established based on the specific conditions of the assessed mines. The evaluation system is composed of the cumulative number of events, cumulative microseismic event energy, cumulative apparent stress, and cumulative logarithmic seismic moment. The evaluation index system for assessing the instability status of rock in metal mines, along with the definitions of its respective parameters, is shown in Figure 1.  
For establishment of index system, the index weight is calculated in this section. IAHP is employed to confirm the weight interval. Considering that the weight interval is not a specific weight value, SPA is introduced to convert the weight interval into a specific weight value. SPA is a connecting mathematical method proposed by Chinese scholar [PERSON] in 1989 [14]. SPA is superior to traditional analytical methods in the aspect of handling uncertain problems. Moreover, the uncertainty of two sets can be analyzed by the characteristics of set pair and construction of a connection degree to them [40]. The pair set \(H=(A,B)\) is made up of two sets A and B, and the total number of the characteristics in set H is N; then the amounts of identical characteristics is \(S\); in addition, the number of discrepant characteristics is \(F\); and the amount of contradictory characteristic is \(P\). Then,  
Figure 1: The status evaluation index system of rock instability of metal mine. the \(S\), \(F\), \(P\) and \(N\) satisfy \(N=S+F+P\). At last, the connection degree (CD) represented by \(\mu\) is calculated by Equation (1),  
\[\mu=\frac{S}{N}+\frac{F}{N}i+\frac{P}{N}j=a+bi+cj \tag{1}\]  
where \(\mu\) denotes connection degree (CD), \(a\) denotes identity degree, \(b\) denotes discrepancy degree, \(c\) denotes contradistinction degree, \(i\in[-1,\,1]\) denotes uncertainty coefficient of discrepancy degree, \(j=-1\) denotes the contradictory coefficient and the relation of a, b and c should satisfy \(a+b+c=1\)[41,42]. In the evaluation system, each index has a different level of impact on the system. Therefore, IAHP-SPA proposed in this paper is used to determine the weights of the index. The detailed calculation process of IAHP-SPA is presented as follows:  
Step 1: Establishment of the interval pairwise comparison matrix  
Based on the 1-9 scaling rules proposed by [PERSON] [43], the relative importance of each influencing index is evaluated by the famous experts, scholars and engineers invited in the field of rock instability of metal mines. The interval pairwise comparison matrix \(A\), therefore, is established as Equation (2). \[A= \begin{bmatrix}[1,1]&[a_{12}^{L},a_{12}^{U}]&\cdots&[a_{1n}^{L},a_{1n}^{U}] \\ [a_{21}^{L},a_{21}^{U}]&[1,1]&\cdots&[a_{2n}^{L},a_{2n}^{U}]\\ \vdots&\vdots&\ddots&\vdots\\ [a_{n1}^{L},a_{n1}^{U}]&[a_{n2}^{L},a_{n2}^{U}]&\cdots&[1,1]\end{bmatrix}  \tag{2}\]  
where \(a_{ij}^{L}\) and \(a_{ij}^{U}\) denote the relative importance of the \(i\)-th matrix over the \(j\)-th matrix and \(a_{ij}^{L}\) and \(a_{ij}^{U}\) are, respectively, the lower and upper boundary of interval number \([a_{ij}^{L},a_{ij}^{U}]\). Additionally, the \(A^{L}\) and \(A^{U}\) can be obtained on the basis of the interval number theory [38]. \[A^{L}= \begin{bmatrix}1&a_{12}^{L}&\cdots&a_{1n}^{L}\\ a_{21}^{L}&1&\cdots&a_{2n}^{L}\\ \vdots&\vdots&\ddots&\vdots\\ a_{n1}^{L}&a_{n2}^{L}&\cdots&1\end{bmatrix} ,\,A^{U}= \begin{bmatrix}1&a_{12}^{ U}&\cdots&a_{1n}^{U}\\ a_{21}^{U}&1&\cdots&a_{2n}^{U}\\ \vdots&\vdots&\ddots&\vdots\\ a_{n1}^{U}&a_{n2}^{U}&\cdots&1\end{bmatrix}  \tag{3}\]  
Step 2: Calculation of the maximum eigenvalues and maximum eigenvectors  
The maximum eigenvalue \(\gamma^{L}\) and \(\gamma^{U}\), corresponding to \(A^{L}\) and \(A^{U}\), can be calculated using Matlab software (R2023b), respectively.