# Rockburst Hazard Evaluation Using an Extended COPRAS Method with Interval-Valued Fuzzy Information
## 3 Methodology
### 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  
\[\widetilde{G}=\begin{bmatrix}\widetilde{\delta}_{11}&\widetilde{\delta}_{12}& \ldots&\widetilde{\delta}_{1n}\\ \widetilde{\delta}_{21}&\widetilde{\delta}_{22}&\ldots&\widetilde{\delta}_{2 n}\\ \vdots&\vdots&\ddots&\vdots\\ \widetilde{\delta}_{n1}&\widetilde{\delta}_{n2}&\ldots&\widetilde{\delta}_{ mn}\\ \end{bmatrix}, \tag{18}\]  
where \(\widetilde{\delta}_{ij}=[s_{ij}^{L},s_{ij}^{U}],i,j=1,2,\ldots,n\), and its upper and lower bounds can be determined with the analysis rules shown in Table 1.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Influence of \(i\)-th to \(j\)-th** & **None** & **Low** & **Medium** & **High** & **Extremely High** \\ \hline \(\widetilde{g}_{ij}^{L}<\widetilde{g}_{ij}^{U}\) & 0 & 1 & 2 & 3 & 4 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Rules for quantifying relationship between indicators.  
Then, the direct influence matrix is calculated as  
\[(\widetilde{P})_{n\times n}=\frac{\widetilde{G}}{\max(\sum\limits_{j=1}^{n} \widetilde{S}_{ij})}. \tag{19}\]  
Afterwards, the combined influence matrix is determined as  
\[(\widetilde{Y})_{n\times n}=\lim\limits_{k\rightarrow\infty}(\widetilde{P}+ \widetilde{P}^{2}+\ldots+\widetilde{P}^{k})=\widetilde{P}(\widetilde{I}+ \widetilde{P})^{-1}, \tag{20}\]  
where \(\widetilde{I}\) is the identity matrix composed of \([0,0]\) and \([1,1]\).  
Further, the attributes of indicators are quantified as the degree of influence (\(\widetilde{M}\)) and degree of being influenced (\(\widetilde{N}\)), which can be calculated as  
\[\widetilde{M}_{i}=\left[\sum\limits_{i=1}^{n}\widetilde{Y}_{ij}\right]_{n \times 1}, \tag{21}\]  
\[\widetilde{N}_{j}=\left[\sum\limits_{j=1}^{n}\widetilde{Y}_{ij}\right]_{1 \times n}^{\top}. \tag{22}\]  
Finally, the degree of centrality is determined as the sum of \(\widetilde{M}\) and \(\widetilde{N}\), and the subjective weights can be obtained as  
\[\widetilde{w}_{j}^{s}=\frac{\widetilde{M}_{i}+\widetilde{N}_{j}}{\sum\limits _{i=j=1}^{n}\left(\widetilde{M}_{i}+\widetilde{N}_{j}\right)}. \tag{23}\]  
**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  
\[\widetilde{E}_{j}=-\frac{1}{\ln n}\sum\limits_{j=1}^{n}\widetilde{h}_{ij}\ln( \widetilde{h}_{ij}), \tag{24}\]  
\[\widetilde{h}_{ij}=\frac{\widetilde{a}_{ij}}{\sum\limits_{i=1}^{m}\widetilde{ a}_{ij}}. \tag{25}\]  
Then, the objective indicator weights are obtained as  
\[\widetilde{w}_{j}^{o}=\frac{1-\widetilde{E}_{j}}{\sum\limits_{j=1}^{n}\left( 1-\widetilde{E}_{j}\right)}. \tag{26}\]  
**Step 3:** Obtain the comprehensive indicator weights  
By combining the subjective and objective weights, the comprehensive indicator weights can be determined as  
\[\widetilde{w}_{j}=\theta\widetilde{w}_{j}^{s}+(1-\theta)\widetilde{w}_{j}^{o}, \tag{27}\]where \(0\leq\theta\leq 1\) is a preference factor; \(\theta=0.5\) is selected in this paper, which indicates that the importance of subjective and objective weights is equal.