## 3 Methodology
### Composited Group Decision-Making Approach
#### 3.2.1 Super-Efficiency Slack-Based Model (Super-SBM)
As a result, the individual SF decision matrices \(\left(\widehat{S}^{k}\right)\) are constructed, as shown in Equation (36). Based on the decision makers' weights (\(\Psi_{k}\)), the SF decision matrix \(\left(\widehat{S}\right)\) is aggregated using the spherical weight arithmetic mean, as shown in Equation (37). \[\widehat{S}^{k}= \begin{bmatrix}\widehat{s}_{11}^{k}&\widehat{s}_{12}&\cdots& \widehat{s}_{1J}^{k}\\ \widehat{s}_{21}^{k}&\widehat{s}_{22}^{k}&\cdots&\widehat{s}_{2J}^{k}\\ \vdots&\vdots&\ddots&\vdots\\ \widehat{s}_{I1}^{k}&\widehat{s}_{I2}&\cdots&\widehat{s}_{IJ}^{k}\end{bmatrix}  \;with\;\widehat{s}_{ij}^{k}=\left(\theta_{\widehat{s}_{ij}^{k}},\,\mu_{ \widehat{s}_{ij}^{k}},\;\;\pi_{\widehat{s}_{ij}^{k}}\right);i=1\ldots I,j=1 \ldots J \tag{36}\]  
\[\widehat{S}= \begin{bmatrix}\widehat{s}_{11}&\widehat{s}_{12}&\cdots&\widehat {s}_{1J}\\ \widehat{s}_{21}&\widehat{s}_{22}&\cdots&\widehat{s}_{2J}\\ \vdots&\vdots&\ddots&\vdots\\ \widehat{s}_{I1}&\widehat{s}_{I2}&\cdots&\widehat{s}_{IJ}\end{bmatrix} \;with \;\widehat{s}_{ij}=\left(\theta_{\widehat{s}_{ij}^{\prime}},\mu_{\widehat{s }_{ij}^{\prime}},\;\pi_{\widehat{s}_{ij}}\right);i=1\ldots I,j=1\ldots J \tag{37}\]  
Step 8. The SF decision matrix \((S)\) is then defuzzied, as represented in Equation (38). Applying regret theory, the utility matrix \((U)\) is constructed with the risk-aversion coefficient \((\varphi)\) as Equation (39). \[S= \begin{bmatrix}s_{11}&s_{12}&\cdots&s_{1I}\\ s_{21}&s_{22}&\cdots&s_{2J}\\ \vdots&\vdots&\ddots&\vdots\\ s_{I1}&s_{I2}&\cdots&s_{IJ}\end{bmatrix}  \tag{38}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Linguistic Term** & **Spherical Fuzzy Number** & **Linguistic Term** & **Spherical Fuzzy Number** \\  & (\(\theta\),\(\mu\),\(\pi\)) & & (\(\theta\),\(\mu\),\(\pi\)) \\ \hline Absolutely Low & (0.1, 0.9, 0.1) & Slightly High & (0.6, 0.4, 0.4) \\ Very Low & (0.2, 0.8, 0.2) & High & (0.7, 0.3, 0.3) \\ Low & (0.3, 0.7, 0.3) & Very High & (0.8, 0.2, 0.2) \\ Slightly Low & (0.4, 0.6, 0.4) & Absolutely High & (0.9, 0.1, 0.1) \\ Neutral & (0.5, 0.5, 0.5) & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Linguistic terms for decision matrix [42]. \[U= \begin{bmatrix}u_{11}&u_{12}&\cdots&u_{1J}\\ u_{21}&u_{22}&\cdots&u_{2J}\\ \vdots&\vdots&\ddots&\vdots\\ u_{I1}&u_{I2}&\cdots&u_{IJ}\end{bmatrix} \text{with }u_{ij}=\big{(}s_{ij}\big{)}^{\varphi};i=1\ldots I,j=1\ldots J,0<\varphi<1 \tag{39}\]  
Step 9. Based on the utility matrix, the vector of ideal points \((U^{*})\) is defined as Equations (40) and (41). Hence, the regret matrix \((T)\) is determined with the regret-aversion coefficient \((\lambda)\), according to Equation (42). \[U^{*}= \begin{bmatrix}u_{1}^{*}&u_{2}^{*}&\cdots&u_{J}^{*}\end{bmatrix}  \tag{40}\]  
where  
\[u_{j}^{*}=\max_{1\leq i\leq I}(u_{ij});\,j=1\ldots J \tag{41}\]  
\[T= \begin{bmatrix}t_{11}&t_{12}&\cdots&t_{1J}\\ t_{21}&t_{22}&\cdots&t_{2J}\\ \vdots&\vdots&\ddots&\vdots\\ t_{I1}&t_{I2}&\cdots&t_{IJ}\end{bmatrix} \text{with }t_{ij}=1-e^{-\lambda(u_{ij}-u_{I}^{*})};i=1 \ldots I,j=1\ldots J,\lambda>0 \tag{42}\]  
Step 10. The overall utility matrix \((V)\) is defined according to Equation (43). \[V= \begin{bmatrix}v_{11}&v_{12}&\cdots&v_{1J}\\ v_{21}&v_{22}&\cdots&v_{2J}\\ \vdots&\vdots&\ddots&\vdots\\ v_{I1}&v_{I2}&\cdots&v_{IJ}\end{bmatrix} \text{with }v_{ij}=u_{ij}+t_{ij};\,i=1 \ldots I,j=1\ldots J \tag{43}\]  
Step 11. The vector of average solution \((\overline{V})\) is defined as Equations (44) and (45). \[\overline{V}= \begin{bmatrix}\overline{v}_{1}&\overline{v}_{2}&\cdots&\overline {v}_{J}\end{bmatrix}  \tag{44}\]  
where  
\[\overline{v}_{j}=\frac{1}{I}\sum_{i=1}^{I}v_{ij} \tag{45}\]  
Step 12. The positive \((V^{+})\) and negative distance \((V^{-})\) from average solution matrices are determined according to Equations (46) and (47). \[V^{+}= \begin{bmatrix}v_{11}^{+}&v_{12}^{+}&\cdots&v_{1J}^{+}\\ v_{21}^{-}&v_{22}^{-}&\cdots&v_{2J}^{+}\\ \vdots&\vdots&\ddots&\vdots\\ v_{I1}^{+}&v_{I2}^{+}&\cdots&v_{IJ}^{+}\end{bmatrix}