## 3 Methodology
### Spherical Fuzzy DEMATEL-EDAS Method
\[\widetilde{\tau}_{j}=\sum_{l=1}^{I}\widetilde{t}_{jl} j=1\dots J \tag{30}\] \[\widetilde{c}_{l}=\sum_{j=1}^{I}\widetilde{t}_{jl} l=1\dots J \tag{31}\]  
_Step 8: Calculate the criteria' prominence, relation and weight_  
First, the SF row sum \(\left(\widetilde{\tau}_{j}\right)\) and the SF column sum \(\left(\widetilde{c}_{j}\right)\) are defuzzied according to Equation (22). Based on the results of defuzzification, the prominence, the relation, and the weight \(\left(w_{j}\right)\) of the \(jth\) criteria are determined as follows:  
\[prominence_{j}=r_{j}+c_{j} j=1\dots J \tag{32}\] \[relation_{j}=r_{j}-c_{j} j=1\dots J\] (33) \[w_{j}=\frac{r_{j}+c_{j}}{\sum_{j=1}^{I}\left(r_{j}+c_{j}\right)} j=1\dots J \tag{34}\]  
_Step 9: Construct the spherical fuzzy decision matrix_  
In this step, each expert provides the linguistic evaluation of alternatives \((i=1,2,\dots,I)\) according to the criteria. Based on the relationships presented in Table 3, the linguistic evaluations are transformed into the corresponding SFNs. Using the SWAM, the SF decision matrix \(\left(\widetilde{X}=\left[\widetilde{x}_{ij}\right]_{I\times I}\right)\) is constructed as Equation (35). \[\widetilde{x}_{ij}=SWAM_{\theta}\left(\widetilde{x}_{ij}^{1},\widetilde{x}_{ ij}^{2},\dots,\widetilde{x}_{ij}^{k}\right)=\theta_{1}\widetilde{x}_{ij}^{1}+ \theta_{2}\widetilde{x}_{ij}^{2}+\dots+\theta_{k}\widetilde{x}_{ij}^{k} i=1\dots I;j=1\dots J \tag{35}\] \begin{table}
\begin{tabular}{c c c} \hline
**Linguistics Term** & **Notation** & **Spherical Fuzzy Numbers** \\ \hline Absolutely Low & AL & (0.1, 0.9, 0.1) \\ Very Low & VL & (0.2, 0.8, 0.2) \\ Low & L & (0.3, 0.7, 0.3) \\ Slightly Low & SL & (0.4, 0.6, 0.4) \\ Medium & M & (0.5, 0.5, 0.5) \\ Slightly High & SH & (0.6, 0.4, 0.4) \\ High & H & (0.7, 0.3, 0.3) \\ Very High & VH & (0.8, 0.2, 0.2) \\ Absolutely High & AH & (0.9, 0.1, 0.1) \\ \hline \end{tabular}
\end{table} 
Table 3: Linguistics term and respective SFNs for the SF EDAS method. Step 10: Identify the spherical fuzzy average solution  
Based on the SF decision matrix, the SF average solution is determined by SWAM as Equation (34). The average solution represented by \(\widetilde{AV}=\big{[}\widehat{av}_{j}\big{]}_{I}\). \[\widehat{av}_{j}=SWAM\big{(}\widetilde{x}_{1j},\widetilde{x}_{2j},\ldots, \widetilde{x}_{lj}\big{)}=\frac{1}{I}\widetilde{x}_{1j}+\frac{1}{I}\widetilde{ x}_{2j}+\ldots+\frac{1}{I}\widetilde{x}_{lj}\qquad\qquad\qquad j=1\ldots J \tag{36}\]  
Step 11: Identify the crisp decision matrix and the crisp average solution  
The crisp decision matrix and crisp average solution are constructed by defuzzification process according to Equation (22). Step 12: Identify the positive distance from average matrix and the negative distance from average matrix  
Based on the crisp decision matrix and the crisp average solution, the positive distance from average matrix \(\Big{(}PDA=\left[d_{ij}^{+}\right]_{IxI}\Big{)}\) and the negative distance from average matrix \(\Big{(}NDA=\left[d_{ij}^{-}\right]_{IxI}\Big{)}\) are defined as Equations (37) and (38). \[d_{ij}^{+} =\frac{\max\big{(}0,x_{ij}-av_{j}\big{)}}{av_{j}} \tag{37}\] \[d_{ij}^{-} =\frac{\max\big{(}0,av_{j}-x_{ij}\big{)}}{av_{j}} \tag{38}\]  
Step 13: Calculate the weighted sum positive distance and the weighted sum negative distance  
In this step, the weighted sum positive distance \(\big{(}s_{i}^{+}\big{)}\) and the weighted sum negative distance \(\big{(}s_{i}^{-}\big{)}\) are computed as Equations (39) and (40), respectively, in which the criteria weights (\(w_{j}\)) are determined in Step 8. \[s_{i}^{+} =\sum_{j=1}^{J}w_{j}d_{ij}^{+} \tag{39}\] \[s_{i}^{-} =\sum_{j=1}^{J}w_{j}d_{ij}^{-} \tag{40}\]  
Next, the normalized weighted sum of positive distance \(\big{(}ns_{i}^{+}\big{)}\) and the normalized weighted sum of negative distance \(\big{(}ns_{i}^{-}\big{)}\) are determined as follows.