# An Integrated Decision Support System for Low-Disturbance Surface Mining
## 2 Methodology
### Integrated Methodological Approach
#### 2.2.2 Grey Clustering Method  
Grey clustering is a clustering method based on grey system theory. It works by using the grey correlation matrix or grey whitening weight function to classify the objects to be observed into defined classes and effectively rank them [38].  
(1) Determination of the grey level  
Grey clustering coefficients were calculated using the centroid triangle whitening weight function (CTWWF) [23]; the weight of a factor in different grey levels is evaluated comprehensively. The evaluation indexes are divided into five LD levels, and the centroids are identified \(\lambda_{k}\). The order of the centroids is shown in Figure 6.  
Figure 6: Centroid triangular whitening weight function.  
According to the evaluation criteria in this paper, the evaluation indicators are categorized into five LD levels, which are denoted as \(k=1\),..., s. These LDM levels correspond to \(x\in[0,1]\) \"very low level (\(k=1\))\"; \(x\in[1,2]\) \"low level (\(k=2\))\"; \(x\in[2,3]\). \"medium level (\(k=3\))\"; \(x\in[3,4]\) \"high level (\(k=4\))\"; and \(x\in[4,5]\) \"very high level (\(k=5\))\".  
(2) Define the grey whitening weight function  
The grey clustering coefficient of LD class \(k\) is calculated by CTWWF, and the grey whitening weight function is defined as shown in Equation (10):  
\[f_{j}^{k}(x)=\begin{cases}0&x\
otin[\lambda_{k-1},\lambda_{k+1}]\\ \left(x-\lambda_{k-1}\right)/\left(\lambda_{k}-\lambda_{k-1}\right)&x\in[ \lambda_{k-1},\lambda_{k}]\\ \left(\lambda_{k-1}-x\right)/\left(\lambda_{k+1}-\lambda_{k}\right)&x\in[ \lambda_{k},\lambda_{k+1}]\end{cases} \tag{10}\]  
The weighting functions [22] used describe the five grey classifications as follows:  
Function 1: Very low-rank level, when the grey number is \(\otimes 1\in[0,1,2]\) the function shown in Equation (11):  
\[f_{i}^{1}(x)=\begin{cases}0&x\
otin[0,2]\\ 1&x\in[0,1]\\ 2-xx\in[1,2]\end{cases} \tag{11}\]  
Function 2: Low-rank level, when the grey number is \(\otimes 1\in[1,2,3]\) the function shown in Equation (12):  
\[f_{i}^{2}(x)=\begin{cases}0&x\
otin[1,3]\\ x-1x\in[1,2]\\ 3-xx\in[2,3]\end{cases} \tag{12}\]  
Function 3: Medium rank level, when the grey number is \(\otimes 1\in[2,3,4]\) the function shown in Equation (13):  
\[f_{i}^{3}(x)=\begin{cases}0&x\
otin[2,4]\\ x-2x\in[2,3]\\ 4-xx\in[3,4]\end{cases} \tag{13}\]  
Function 4: High-rank level, when the grey number is \(\otimes 1\in[3,4,5]\) the function shown in Equation (14):  
\[f_{i}^{4}(x)=\begin{cases}0&x\
otin[3,5]\\ x-3x\in[3,4]\\ 5-xx\in[4,5]\end{cases} \tag{14}\]  
Function 5: Very high-rank level, when the grey number is \(\otimes 1\in[3,4,5]\) the function shown in Equation (15):  
\[f_{i}^{5}(x)=\begin{cases}0&x\
otin[4,6]\\ x-4x\in[4,5]\\ 6-xx\in[5,6]\end{cases} \tag{15}\]  
The combined clustering coefficient \(\sigma_{i}^{k}\) is calculated using the above function as shown in Equation (16):  
\[\sigma_{i}^{k}=\sum_{j=1}^{m}f_{j}^{k}(x_{ij})\cdot w \tag{16}\]  
If so \(\max_{1\leq k\leq s}\left\{\sigma_{i}^{k}\right\}=\sigma_{i}^{k^{\star}}\), then indicator \(i\) belongs to the \(k^{\star}\) tier level, and its priority is determined by the overall clustering coefficient. In short, the largest element of the grey clustering coefficient is selected as the clustering result.