## 2 Newly Proposed Evaluation Method and Analysis Procedure
### Typical Procedure of Newly Proposed CRITIC-GRA Evaluation Model
#### 2.1.1 Weight Analysis of Safety and Economic Indicators via the CRITIC Method  
The CRITIC method [26] allows one to objectively analyze the weight values of each considered safety and economic indicator, which represents different importance to the dimension optimization of open stopes, and to statistically calculate the Variability Index and Conflict Index for each indicator. This avoids the subjectivity of artificial selection of weights based on experience, thus obtaining a more reasonable and impersonal assessment within the assigned ranges of each considered indicator among the preset schemes.  
Figure 2: Typical procedure of CRITIC-GRA evaluation model to select the optimal scheme.  
The above-mentioned value of Variability Index [27] for each indicator is determined by calculating the standard deviation of their preset ranges. The value of the Variability Index is positively correlated with the weight value assigned to each indicator. In addition, the value of the Conflict Index, which is represented by the correlation among all the evaluated indicators, is negatively correlated with the weight value allocated to each indicator.  
It is assumed that all preset schemes and considered indicators form an \(m\times n\) matrix as shown in Equation (7), where the number of all preset schemes to be evaluated is \(m\) and the number of all selected influential indicators is \(n\).  
\[X=\begin{pmatrix}x_{11}&\dots&x_{1n}\\ \vdots&\ddots&\vdots\\ x_{m1}&\dots&x_{mn}\end{pmatrix} \tag{7}\]  
where, \(x_{ij}\) represents the safety or economic value of the scheme \(i\) and the indicator \(j\) which affects the optimization of slope dimensions.  
Due to the noncomparability of all the considered influential indicators which have different units, it is necessary to normalize the values of these indicators for integrated comparisons. Equation (8) gives a benefit-oriented normalization method for assessing parameters positively related to the open stoping stability or mining costs. Conversely, Equation (9) represents a cost-oriented normalization method applicable to the considered indicators that are inversely linked to the open stoping stability or mining costs.  
\[X_{ij}^{\prime}=\frac{X_{j}-X_{\min}}{X_{\max}-X_{\min}} \tag{8}\]  
\[X_{ij}^{\prime\prime}=\frac{X_{\max}-X_{j}}{X_{\max}-X_{\min}} \tag{9}\]  
The Variability Index \(S_{j}\) is represented by the standard deviation of Equation (10), which reflects the fluctuation of differences in the range of values taken by each indicator. The standard deviation is positively correlated with the weight value of the indicator.  
\[S_{j}=\sqrt{\frac{\sum\limits_{i=1}^{m}\left(X_{ij}-\overline{X}_{j}\right)^{ 2}}{m-1}} \tag{10}\]  
\[\overline{X}_{j}=\frac{1}{m}{\sum\limits_{i=1}^{m}X_{ij}} \tag{11}\]  
The Conflict Index \(R_{j}\) between indicators is measured by the correlation coefficient shown in Equation (12), which indicates the degree of association between indicators. The correlation coefficient exhibits a negative correlation with the weight value assigned to the indicators.  
\[R_{j}=\sum\limits_{i=1}^{n}\left(1-r_{ij}\right) \tag{12}\]  
where, \(r_{ij}\) represents the correlation coefficient between evaluated indicators \(i\) and \(j\).  
After that, by calculating the weight value \(w_{j}\) for each indicator using Equation (13), the weight values matrix \(W\) shown in Equation (14) can be obtained, which applies to all possible influential indicators considered in all the preset schemes to obtain an optimal scheme with suitable stope dimensions for the following GRA analysis.  
\[w_{j}=\frac{S_{j}\times R_{j}}{\sum\limits_{j=1}^{n}\left(S_{j}\times R_{j} \right)} \tag{13}\]\[W=(w_{1}w_{2}\cdots w_{m}) \tag{14}\]