## 4 Methodology  
To solve the problem defined in this study, a new hybrid MCDM model was developed. It includes a combination of the BWM and ADAM method in a grey environment. The steps of the proposed model are described below and presented in Figure 1.  
Step 1: Define the problem structure--form the sets of alternatives and criteria for their evaluation. Step 2: Define the greyscale for the evaluations required by the grey BWM and grey ADAM method (Table 1). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Linguistic Term** & **Abbreviation** & **Grey Scale** \\ \hline \"None\" & “N” & [0, 2] \\ \"Very Low\" & “VL” & [1, 3] \\ \"Low\" & “L” & [2, 4] \\ \"Fairly Low\" & “FL” & [3, 5] \\ \"Moderate\" & “M” & [4, 6] \\ \"Fairly High\" & “FH\" & [5, 7] \\ \"High\" & “H\" & [6, 8] \\ \"Very High\" & “VH\" & [7, 9] \\ \"Extremely High\" & “EH\" & [8, 10] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Grey evaluation scale. Figure 1: Structure of the proposed MCDM model. Step 3: Obtain the criteria weights by using the grey BWM method [37]. Step 3.1: Decision-makers (DMs) \(d\) (\(d=1,\ldots,f\)), where \(f\) represents the DMs, select the best and the worst criteria (\(j_{B}\) and \(j_{W}\), \(j=1,\ldots,m\)), where \(m\) is the number of criteria. DMs evaluate other criteria in comparison with \(j_{B}\) and \(j_{W}\), by using the scale from Table 1, thus obtaining grey vectors: \"the best compared to others\"--\(\otimes A_{B}=(\otimes a_{B1},\otimes a_{B2},\ldots,\otimes a_{Bm})\), and \"others compared to the worst\"--\(\otimes A_{W}=(\otimes a_{1W},\otimes a_{2W},\ldots,\otimes a_{mW})\). Step 3.2: Obtain the optimal grey criteria weights, \(\otimes w_{d1},\otimes w_{d2},\ldots,\otimes w_{dm}\), \(\forall d=1,\ldots,f\), by solving the optimization problem:  
\[s.t. \begin{cases}\min\otimes\xi\\ \frac{\left|\frac{\otimes w_{B}}{\otimes w_{dj}}\right|}{\otimes w_{W}}\otimes a _{jW}\end{cases} \leq\otimes\xi\\ \frac{\left|\frac{\otimes w_{dj}}{\otimes w_{W}}\right|}{\sum_{j=1}^{m}W \left(\otimes w_{dj}\right)}=1\\ \frac{w_{dj}\leq\overline{w}_{dj}}{\underline{w}_{dj}}\\ \frac{w_{dj}\geq 0}{j=1,\ldots,m}\end{cases}, \tag{1}\]  
where \(\otimes\xi=\left[\xi,\xi\right]\), \(\otimes w_{B}=[\underline{w}_{B},\overline{w}_{B}]\) is the grey weight of the \"best\" criterion, \(\otimes w_{w}=[\underline{w}_{W},\overline{w}_{W}]\) is the grey weight of the \"worst\" element criterion, \(\otimes w_{dj}=\left[\underline{w}_{dj},\overline{w}_{dj}\right]\) is the grey weight of criterion \(j\), \(j=1,\ldots,m\), \(j\
eq j_{B},j_{W}\), \(\otimes a_{Bj}=\left[\underline{a}_{Bj},\overline{a}_{Bj}\right]\) is the grey preference of the \"best\" criterion over criterion \(j\), and \(\otimes a_{jW}=\left[\underline{a}_{jW},\overline{a}_{jW}\right]\) is the preference of criterion \(j\) over the \"worst\" criterion. \(W\big{(}\otimes w_{j}\big{)}\) is the white value of the grey number \(\otimes w_{j}\):  
\[W\Big{(}\otimes w_{dj}\Big{)}=\Big{(}\underline{w}_{dj}+\overline{w}_{dj} \Big{)}/2, \tag{2}\]  
Step 3.3: Check the comparison consistency by calculating the consistency ratio (\(CR\)) with  
\[CR=R(\otimes\xi)/CI, \tag{3}\]  
where \(R(\otimes\xi)\) is the white value of the grey number \(\otimes\xi\), obtained from (2), and \(CI\) is the consistency index:  
\[CI^{2}-(1+2\overline{a}_{BW})CI+\Big{(}\overline{a}_{BW}^{2}-\overline{a}_{BW }\Big{)}=0, \tag{4}\]  
where  
\[\otimes a_{BW}=\max_{j}\{\otimes a_{Bj},\otimes a_{jW}\}. \tag{5}\]  
The comparison is considered consistent if the \(CR\) value is close to 0. Step 3.4: Obtain the final criteria weights:  
\[\otimes w_{j}=\left[\underline{w}_{j},\overline{w}_{j}\right],\ \forall j=1,\ldots,m, \tag{6}\]  
\[\underline{w}_{j}=\left(\prod_{d=1}^{l}\underline{w}_{dj}\right)^{1/d}, \tag{7}\]  
\[\overline{w}_{j}=\left(\prod_{d=1}^{l}\overline{w}_{dj}\right)^{1/d}, \tag{8}\]  
Step 4: Rank the alternatives using the extension of the ADAM-MCDM method in the grey environment.