# An Integrated Decision Support System for Low-Disturbance Surface Mining
## 2 Methodology
### Integrated Methodological Approach
#### 2.2.1 F-AHP
The representation of the comparison matrix \(\tilde{A}\) is shown in Equation (2):  
\[\widetilde{A}=(\widetilde{a}_{ij})_{\text{m}\times\text{n}}= \begin{bmatrix} \widetilde{a}_{11}&\widetilde{a}_{12}&\cdots&\widetilde{a}_{1n}\\ \widetilde{a}_{21}&\widetilde{a}_{22}&\cdots&\widetilde{a}_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ \widetilde{a}_{\text{m}1}&\widetilde{a}_{\text{m}2}&\cdots&\widetilde{a}_{ \text{mn}}\end{bmatrix}  \tag{2}\]  
where: if \(i=j\), then \(\widetilde{a}_{ij}=1\); If \(i\) is more important than \(j\), then \(\widetilde{a}_{ij}=\widetilde{1},\widetilde{2}\cdots,\widetilde{9}\); If \(i\) is not as important as \(j\), then \(\widetilde{a}_{ij}=\widetilde{1}^{-1},\widetilde{2}^{-1}\cdots,\widetilde{9}^ {-1}\). In the comparison matrix, the clear values given to the linguistic terms are converted into triangular fuzzy numbers. It is performed by entering these triads of triangular fuzzy numbers, which are used as membership functions. The geometric mean method was used to calculate the fuzzy comparison matrix. \[ \begin{cases}\widetilde{r}_{i}=(\widetilde{a}_{ij}^{1}\otimes\widetilde{a}_{ ij}^{2}\cdots\widetilde{a}_{ij}^{n})^{1/n}\\ \widetilde{w}_{i}=\widetilde{r}_{i}\otimes(\widetilde{r}_{1}\oplus\widetilde{ r}_{2}\cdots\widetilde{r}_{n})^{-1}\end{cases}  \tag{3}\]  
where: The \(\widetilde{a}_{ij}\) is the fuzzy comparison value of indicator \(i\) to indicator \(\widetilde{r}_{i}\) is the geometric mean of the fuzzy comparison values of indicator \(i\) for each indicator; the \(\widetilde{w}_{i}\) is the fuzzy weight of the ith indicator;  
The fuzzy weights \(\widetilde{w}_{i}\) are expressed as triangular fuzzy numbers:  
\[\widetilde{w}_{i}=\widetilde{r}_{i}\otimes(lw_{i}\oplus mw_{i}\oplus uw_{i}) \tag{4}\]  
where: \(uw_{i}\) are the lower limit value, the middle value, and the upper limit value of the fuzzy weights of the \(i\)th indicator, respectively. Since the output is in the form of fuzzy weights, defuzzification is performed by locating the best non-fuzzy performance value (_BNP_) to convert the fuzzy weights into clear numbers. Using Equation (5) to determine the _BNP_, the _BNP_ for each weight (\(l\), \(m\), \(u\)) is determined as:  
\[BNP_{w}=[(u_{w}-l_{w})+(m_{w}-l_{w})]/3+l_{w} \tag{5}\]  
The normalized _BNP_ values are considered relative weights (_N-BNP_), i.e., the _BNP_ values are divided by the sum of the _BNP_ values of all metrics to obtain a total _N-BNP_ weighting of 1. Figure 5: The membership functions for linguistic variables. To test the consistency of the results, it is crucial to calculate The Consistency Ratio (\(CR\)) based on the two sets of random indicators [36]. Take the middle values of the elements of matrix \(A\) to form matrix \(A_{m}\). Take the square root of the product of the lower and upper bounded values of the elements of matrix \(A\) to form matrix \(A_{g}\). That is:  
\[\left\{ \begin{aligned} A_{m}&=\left[a_{jm}\right]\\ A_{g}&=\left[\sqrt{a_{ij}a_{ijl}}\right]\end{aligned} \right. \tag{6}\]  
According to [PERSON]'s method, the maximum eigenvalues of matrix \(A_{m}\) and matrix \(A_{g}\) can be calculated by Equation (7) as follows:  
\[\left\{ \begin{aligned} \lambda_{\max}^{m}&=\frac{1}{n} \sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n}a_{ijm}(w_{j}^{m}/w_{i}^{m})\\ \lambda_{\max}^{g}&=\frac{1}{n}\sum\limits_{i=1}^{ n}\sum\limits_{j=1}^{n}\sqrt{a_{iju}a_{ijl}}(w_{j}^{g}/w_{i}^{g})\end{aligned} \right. \tag{7}\]  
The Consistency Index (\(CI\)) of matrix \(A_{m}\) and \(A_{g}\) is calculated as follows:  
\[\left\{ \begin{aligned} & CI_{m}=\frac{\lambda_{\max}^{m}-n}{n-1}\\ & CI_{g}=\frac{\lambda_{\max}^{m}-n}{n-1}\end{aligned} \right. \tag{8}\]  
Finally, the Consistency Ratio (\(CR\)) of each matrix is calculated according to Equations (8) and (9) with the following formula:  
\[\left\{ \begin{aligned} & CR_{m}=\frac{CI_{m}}{RI_{m}}\\ & CR_{g}=\frac{CI_{g}}{RI_{g}}\end{aligned} \right.