## 3 Methodology
Thus, if \(n\) is the number of analysed criteria, then the AHP comparison procedure is as follows:  
* to construct a \(n\)\(\times\)\(n\) pairwise comparison matrix \(m\) for analysed criteria, where \(a_{ij}\) denotes entry in the \(i\) th row and the \(j\) th column of matrix \(m\),
* \(a_{ij}\) states the preference score of criterion \(i\) to criterion \(j\) using the nine-integer value scale suggested by [PERSON] [37], where:  
* the entries of preference score \(a_{ij}\) and \(a_{ji}\) must satisfy the following constraint of Equation (1): \[aij\times aji=1\] (1)
* to establish a normalized pairwise comparison matrix \(m\), the sum of each column must be equal to 1. This can be obtained using Equation (2) to calculate \(\overline{a}_{ij}\) for each entry of the matrix \(\overline{m}\)[38], \[\overline{\overline{a}_{ij}}=\frac{a_{ij}}{\sum_{d=1}^{n}a_{ij}},\] (2)  
Figure 3: Decision hierarchy for waste site selection (3 rd level repeats and is shown for one criterion only). * to obtain the relative weights, the average across rows is computed using the Equation (3); for each element, the relative weight is within the range of 0 to 1 and a higher weight shows a greater influence of a given element (criterion) [37], \[w_{i}=\frac{\sum_{l=1}^{n}\overline{a_{ij}}}{n},\] (3)  
A test of the degree of consistency of the derived weights is performed to check the consistency of the experts' judgements. It involves a calculation of the Consistency Ratio (CR), which indicates the probability that the matrix values have been randomly generated. According to [PERSON] [7], a matrix that has a consistency ratio greater than 0.10 should be re-evaluated. For controlling the consistency of the estimated weight values, the consistency ratio (CR) is calculated as follows [36]:  
* calculate the eigenvector and the maximum eigenvalue for matrix \(m\),
* next, calculate an approximation to the Consistency Index (\(CI\)) according to Equation (4): \[CI=\frac{\lambda_{max}-n}{n-1},\] (4) where
* \(\lambda_{max}\) is the maximum eigenvalue of the comparison matrix,
* \(n\) is the number of criteria. The Consistency Ratio is calculated from the Equation (5),  
\[CR=\frac{CI}{RI}, \tag{5}\]  
where,  
* \(RI\) is the random consistency index that varies according to the number of criteria in a comparison (\(n\)). As mentioned above for \(CR\leq 0.10\), the degree of consistency is considered satisfactory; otherwise, there are serious inconsistencies in the pairwise comparison and the AHP may not return meaningful results. The above procedure was used twice, for deriving weights of criteria (mining waste uses) and for deriving the local preferences (sites with deposited waste with respect for each criterion). Each of the authors and experts prepared matrix with pairwise comparison for criteria and six matrices for with pairwise comparison for alternatives with respect to each criterion. These matrices were used to establish the weighted scores of criteria and alternatives. Overall priorities of the alternatives were calculated as a weighted sum (\(S\)) of the products of criterion weight and local preference relative to this criterion, Equation (6):  
\[S=\sum_{i=1}^{n}Cw_{i}\times Aw_{i}, \tag{6}\]  
where,  
* \(Cw_{i}\) is the criterion weight,
* \(Aw_{i}\) is the preference of alternative relative to that criterion,
* \(n\) is the number of criteria.