## 4 Two Real Case Studies
### The First Case Study: University Departments
#### 4.1.1 Applying the Interactive Algorithm for the First Case Study  
To find DMU targets, the interactive algorithm (Figure 1), including two interactive stages, was applied for universities departments. Applying the first interactive stage  
In this stage, efficient DMUs and their targets were determined. For o = 1,..., 17, parameters xio = xij, i = 1, 2, j = 1,..., 17, and yro = yij, r = 1, 2, j = 1,..., 17 were defined. Non-radial model 12 was then composed for DMUj, j = 1,..., 17. The results of solving of model 12 for DMUj, j = \(\cdots\),..., 17 are shown in Table 3.  
Figure 2: Extended RBA for finding non-radial FDH models targets of the interactive algorithm. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\mathbf{DMU_{j}}\) & \(\mathbf{I_{1}}\) & \(\mathbf{I_{2}}\) & \(\mathbf{O_{1}}\) & \(\mathbf{O_{2}}\) \\ \hline \(\mathrm{DMU_{01}}\) & 26 & 7 & 12 & 3 \\ \hline \(\mathrm{DMU_{02}}\) & 29 & 6 & 10 & 7 \\ \hline \(\mathrm{DMU_{03}}\) & 40 & 8 & 20 & 6 \\ \hline \(\mathrm{DMU_{04}}\) & 42 & 7 & 12 & 6 \\ \hline \(\mathrm{DMU_{05}}\) & 45 & 9 & 18 & 6 \\ \hline \(\mathrm{DMU_{06}}\) & 92 & 12 & 40 & 2 \\ \hline \(\mathrm{DMU_{07}}\) & 83 & 11 & 58 & 3 \\ \hline \(\mathrm{DMU_{08}}\) & 87 & 14 & 52 & 7 \\ \hline \(\mathrm{DMU_{09}}\) & 149 & 16 & 61 & 4 \\ \hline \(\mathrm{DMU_{10}}\) & 177 & 17 & 54 & 12 \\ \hline \(\mathrm{DMU_{11}}\) & 191 & 19 & 61 & 11 \\ \hline \(\mathrm{DMU_{12}}\) & 185 & 14 & 73 & 4 \\ \hline \(\mathrm{DMU_{13}}\) & 186 & 20 & 85 & 10 \\ \hline \(\mathrm{DMU_{14}}\) & 74 & 12 & 36 & 5 \\ \hline \(\mathrm{DMU_{15}}\) & 164 & 22 & 69 & 8 \\ \hline \(\mathrm{DMU_{16}}\) & 225 & 20 & 80 & 5 \\ \hline \(\mathrm{DMU_{17}}\) & 108 & 10 & 27 & 3 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Data of the first case study. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(\mathbf{DMU_{j}}\) & \(\mathbf{Z_{j}^{9}}\) & \(\mathbf{\phi_{1}^{*}}\) & \(\mathbf{\theta_{2}^{*}}\) & \(\mathbf{\varphi_{1}^{*}}\) & \(\mathbf{\varphi_{2}^{*}}\) & \(\mathbf{DMU_{j}Target}\) \\ \hline \(\mathrm{DMU_{01}}\) & 0 & 1 & 1 & 1 & 1 & \(\mathrm{DMU_{01}}\) \\ \hline \(\mathrm{DMU_{02}}\) & 0 & 1 & 1 & 1 & 1 & \(\mathrm{DMU_{02}}\) \\ \hline \(\mathrm{DMU_{03}}\) & 0 & 1 & 1 & 1 & 1 & \(\mathrm{DMU_{03}}\) \\ \hline \(\mathrm{DMU_{04}}\) & 0 & 1 & 1 & 1 & 1 & \(\mathrm{DMU_{04}}\) \\ \hline \(\mathrm{DMU_{05}}\) & 0.166 & 0.89 & 0.89 & 1.11 & 1 & \(\mathrm{DMU_{03}}\) \\ \hline \(\mathrm{DMU_{06}}\) & 0.565 & 0.9 & 0.92 & 1.45 & 1.5 & \(\mathrm{DMU_{07}}\) \\ \hline \(\mathrm{DMU_{07}}\) & 0 & 1 & 1 & 1 & 1 & \(\mathrm{DMU_{07}}\) \\ \hline \(\mathrm{DMU_{08}}\) & 0 & 1 & 1 & 1 & 1 & \(\mathrm{DMU_{08}}\) \\ \hline \(\mathrm{DMU_{09}}\) & 0 & 1 & 1 & 1 & 1 & \(\mathrm{DMU_{09}}\) \\ \hline \(\mathrm{DMU_{10}}\) & 0 & 1 & 1 & 1 & 1 & \(\mathrm{DMU_{10}}\) \\ \hline \(\mathrm{DMU_{11}}\) & 0 & 1 & 1 & 1 & 1 & \(\mathrm{DMU_{11}}\) \\ \hline \(\mathrm{DMU_{12}}\) & 0 & 1 & 1 & 1 & 1 & \(\mathrm{DMU_{12}}\) \\ \hline \(\mathrm{DMU_{13}}\) & 0 & 1 & 1 & 1 & 1 & \(\mathrm{DMU_{13}}\) \\ \hline \(\mathrm{DMU_{14}}\) & 0 & 1 & 1 & 1 & 1 & \(\mathrm{DMU_{14}}\) \\ \hline \(\mathrm{DMU_{15}}\) & 0 & 1 & 1 & 1 & 1 & \(\mathrm{DMU_{15}}\) \\ \hline \(\mathrm{DMU_{16}}\) & 0.617 & 0.83 & 1 & 1.06 & 2 & \(\mathrm{DMU_{13}}\) \\ \hline \(\mathrm{DMU_{17}}\) & 0 & 1 & 1 & 1 & 1 & \(\mathrm{DMU_{17}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The results of solving model 12 (related to the first interactive stage) in the first case study. model 12 (the first interactive stage) showed that 14 DMUs lay on the efficient frontier, and 3 DMUs did not. By applying the second interactive stage, the DMU targets of these three DMUs were distinguished. Applying the second interactive stage  
In the second interactive stage, the targets of DMUs belonging to set M\({}^{\prime}\) were found by STEM. First, the pay-off tables for non-proceed DMUs were constructed.