## Appendix A
### The Detail Results of Applying Extended RBA for the Second Case Study  
Applying extended RBA (Section 3.2) is described to find targets of the non-radial FDH model 12-14 for DMU\({}_{07}\) of the second case study (related non-radial FDH models in the interactive algorithm). Each target of the non-radial FDH model is found using two target finding stages. A similar process should be carried out to find the target of the mentioned non-radial FDH models for other DMUs. First \(\mathrm{x_{io}}=\mathrm{x_{ij}}\), \(\mathrm{i}=1,2\), and \(\mathrm{y_{ro}}=\mathrm{y_{\bar{\eta}}}\), \(\mathrm{r}=1,2,3\) (\(\mathrm{j}=1,\ldots,13\)) for DMU\({}_{0}=\mathrm{DMU}_{07}\) should be defined. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multicolumn{2}{c}{**DMU\({}_{\mathbf{o}}\)**} & \multicolumn{1}{c}{**Target\({}_{1}\)**} & \multicolumn{1}{c}{**Target\({}_{2}\)**} & \multicolumn{1}{c}{**Target\({}_{3}\)**} & \multicolumn{1}{c}{**Target\({}_{4}\)**} & \multicolumn{1}{c}{**Target\({}_{5}\)**} & \multicolumn{1}{c}{**Target**} \\ \hline \multirow{4}{*}{DMU\({}_{04}\)} & Model 14 & 1 & 1 & 8 & 7 & 8 & DMU\({}_{01}\) \\ \cline{2-8}  & DM’s opinions & 2 & 3 & 6 & 8 & 7 & DMU\({}_{06}\) \\ \cline{2-8}  & \(\Delta_{\mathrm{f}}\) & \(-1\) & \(-2\) & 2 & \(-1\) & 1 & - \\ \cline{2-8}  & Unsatisfied & 0 & 0 & 0 & 1 & 0 & - \\ \hline \multirow{4}{*}{DMU\({}_{07}\)} & Model 14 & 1 & 1 & 8 & 7 & 8 & DMU\({}_{06}\) \\ \cline{2-8}  & DM’s opinions & 2 & 3 & 6 & 8 & 7 & DMU\({}_{06}\) \\ \cline{2-8}  & \(\Delta_{\mathrm{f}}\) & \(-1\) & \(-2\) & 2 & \(-1\) & 1 & - \\ \cline{2-8}  & Unsatisfied & \(\underline{0}\) & \(\underline{0}\) & \(\underline{0}\) & \(\underline{0}\) & \(\underline{0}\) & - \\ \hline \hline \end{tabular}
\end{table} 
Table 11: The results of solving model 14 (Step 2 of STEM for p = 2) and DM’s opinions about them for two DMUs in the second case study. Target finding of non-radial FDH models 12 and 13 for DMU\({}_{07}\) in the second case study. By applying the first target finding stage, DMUs of feasible region were found. As described in Section 3.2.1, firstly, DMUs of feasible region (\(\mathbb{S}_{\text{Step 0 of STEM}}\)) should be found using Equation (15) (Table A8). As is observed in the last column of Table A8, only five DMUs (DMU\({}_{01}\), DMU\({}_{02}\), DMU\({}_{05}\), DMU\({}_{06}\) and DMU\({}_{07}\)) satisfied all five constraints. So, these DMUs belonged to the feasible region (\(\mathbb{S}_{\text{step 0 of STEM}}=\{\text{DMU}_{01},\text{DMU}_{02},\text{DMU}_{05}, \text{DMU}_{06},\text{DMU}_{07}\}\)). By applying the second target finding stage, the optimum objective function values were then calculated, and DMU targets were found. In this regard, firstly, the optimum objective function values of models should be calculated by Equations (16) and (17). The objective function value of DMU\({}_{01}\), DMU\({}_{02}\), DMU\({}_{05}\), DMU\({}_{06}\), and DMU\({}_{07}\) (in non-radial FDH model 12, \(Z_{1}^{0}=\frac{1}{3}\sum_{r=1}^{3}\frac{Y_{1}}{Y_{\text{ro}}}-\frac{1}{2} \sum_{i=1}^{2}\frac{x_{i}}{x_{\text{ro}}},j=1,\ldots,13\), and in non-radial FHD model 13, \(\mathbb{f}_{1}(\mathbf{u})=-x_{1\text{ro}}x_{\text{ro}}x_{\text{ro}},\mathbb{f }_{2}(\mathbf{u})=-x_{2\text{ro}}x_{\text{ro}}x_{\text{ro}},\mathbb{f}_{3}( \mathbf{u})=y_{1\text{ro}}y_{1\text{ro}},\mathbb{f}_{4}(\mathbf{u})=y_{2\text{ro }}y_{2\text{ro}},\mathbb{f}_{5}(\mathbf{u})=y_{3\text{ro}}y_{3\text{ro}}\)) are shown in Table A9. The last row of Table A9 shows the maximum value of \(Z_{1}^{0}\) and \(\mathbb{f}_{r},1=1\ldots 5\), respectively. These values show the optimum objective function value (optimum value of \(Z_{1}^{0}\) and \(\mathbb{f}_{r},1=1,\ldots,5\)) of each non-radial FDH model. DMU\({}_{0}\) targets are found by comparing the optimum objective function value (optimum value of \(Z^{*0}\) and \(\mathbb{f}_{r}^{*},1=1,\ldots,5\)) with the objective function value of each DMU (\(Z_{1}^{0}\) and \(\mathbb{f}_{r},1=1,\ldots,5\)) by Equations (15) and (16) (Table A9).