# A Status Evaluation of Rock Instability in Metal Mines Based on the SPA-IAHP-PCN Model
## 2 Methodology
### Conformation of PCN  
Utilizing the PCN, the status development trend of evaluation system can be forecasted. Moreover, the development trend of characteristics, namely identity degree, discrepancy degree and contradistinction degree, can be accurately reflected by the partial connection number of total connection number (TCN) [36]. For five-element connection number, \(\mu=a+b_{1}i_{1}+b_{2}i_{2}+b_{3}i_{3}+c_{j}\), \"a\" represents that the status of rock instability in the metal mine is level I (very good), while \"\(b_{1}\)\" represents that the status of rock instability in the metal mine is level II (good). Since the status level of rock instability in the metal mine changes over time, it can be considered that level I (very good) develops from level II (good). Therefore, the proportion of metal mine status developing from level II (good) to level I (very good) can be characterized by \(a/(a+b_{1})\) and \(\partial a=a/(a+b_{1})\). Similarly, the \(\partial b_{1}\), \(\partial b_{2}\) and \(\partial b_{3}\) are obtained. The \(\partial\mu\), the first partial connection number, is defined as Equation (17). At last, the second partial connection number, the third partial connection number and the fourth partial connection number can be calculated in a similar way (Equations (18)-(20)). The detailed calculation process of the PCN is below.  
The first-order PCN can be obtained on the basis of Equation (12), and that is expressed by Equation (14).  
\[\partial\mu=\partial a+\partial b_{1}i_{1}+\partial b_{2}i_{2}+\partial b_{3}i _{3} \tag{14}\]  
where \(\partial a=a/(a+b_{1})\), \(\partial b_{1}=b_{1}/(b_{1}+b_{2})\), \(\partial b_{2}=b_{2}/(b_{2}+b_{3})\), \(\partial b_{3}=b_{3}/(b_{3}+c)\), \(i_{1}\in[0,\,1]\), \(i_{2}\in[-1,\,0]\), \(i_{3}=-1\)  
The second partial connection number can be obtained on the basis of Equation (14), and that is expressed by Equation (The third partial connection number can be obtained on the basis of Equation (15), and that is expressed by Equation (16).  
\[\partial^{3}\mu=\partial^{2}(\partial\mu)=\partial^{3}a+\partial^{3}b_{1}i_{1} \tag{16}\]  
where \(\partial^{3}a=\partial^{2}a/(\partial^{2}a+\partial^{2}b_{1})\), \(\partial^{3}b_{1}=\partial^{2}b_{1}/(\partial^{2}b_{1}+\partial^{2}b_{2})\), \(i_{1}=-1\).  
The fourth partial connection number can be obtained on the basis of Equation (16), and that is expressed by Equation (17).  
\[\partial^{4}\mu=\partial^{3}(\partial\mu)=\partial^{4}a \tag{17}\]  
where \(\partial^{4}a=\partial^{3}a/(\partial^{3}a+\partial^{3}b_{1})\).  
It is particularly important to note that the PCN is a tool used to describe uncertainty, with its function focusing on the values of the connection coefficients. For example, in the ternary connection number \(a+b_{i}+c_{j}\), the value of \(i\) ranges between \([-1,\,1]\) and is uncertain. However, when the connection number expands to quaternary, quinary, \(\ldots\), n-ary connection numbers, the value range of each connection coefficient narrows as the number of connection components increases, thereby weakening the impact of uncertainty. In this paper, to ensure the reliability of the calculation results, we adopt the \"worst-case\" approach for the values of the connection coefficients. Therefore, calculating the first partial connection number, \(i_{1}=0\), \(i_{2}=0\), \(i_{3}=-1\) are defined; calculating the second partial connection number \(i_{1}=-1\), \(i_{2}=-1\), are defined; calculating the third partial connection number \(i_{1}=-1\), is defined. When \(\partial^{i}\mu>0\), the development trend of evaluation index is in ascending trend; and \(\partial^{i}\mu<0\), is in descending trend; then \(\partial^{i}\mu=0\), in a transitional state between ascending trend and descending trend. The process of the proposed method is illustrated in Figure 2 [43].  
Figure 2: Flowchart of proposed method.