## 4 Two Real Case Studies
### The First Case Study: University Departments
#### 4.1.1 Applying the Interactive Algorithm for the First Case Study
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{\(\text{DMU}_{\text{o}}\)} & \multirow{2}{*}{**1**} & \multirow{2}{*}{**i** or **r**} & \multirow{2}{*}{\(\theta_{\text{i}}^{*}\)**or \(\varphi_{\text{r}}^{*}\)**} & \multirow{2}{*}{**f1**} & \multirow{2}{*}{**Unsatisfied**} & \multirow{2}{*}{**Target**} & \multirow{2}{*}{**(DM)**} & \multirow{2}{*}{\(\Delta_{\text{f}_{\text{i}}}\)} & \multirow{2}{*}{\(\beta_{\text{1}}\)} \\  & & & & & & & & \\ \hline \multirow{4}{*}{\(\text{DMU}_{16}\)} & 1 & i = 1 & 0.827 & \(-\)186 & 0 & 186 & 0.000 & 0.000 \\ \cline{2-11}  & 2 & i = 2 & 1.000 & 20 & 0 & 20 & 0.000 & 0.000 \\ \cline{2-11}  & 3 & r = 1 & 1.063 & 85 & 0 & 85 & 0.000 & 0.000 \\ \cline{2-11}  & 4 & r = 2 & 2.000 & 0 & \(\underline{1}\) & 10 & 1.000 & 0.000 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Calculated parameters through solving of step 2 of STEM (p = 1) models for DMU\({}_{16}\) (the first case study). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{\(\text{DMU}_{\text{o}}\)} & \multirow{2}{*}{**Member of M Set**} & \multirow{2}{*}{**Obtained Target by Solving of Step 1 of STEM Model**} & \multirow{2}{*}{**Obtaining DM’s Target**} & \multirow{2}{*}{**Related Interactive Stages of the DM’s Target**} & \multirow{2}{*}{**Case**} \\  & & & & & & \\ \hline DMU\({}_{01}\) & & & & & & \\ \hline DMU\({}_{02}\) & & & & & & \\ \hline DMU\({}_{03}\) & & & & & & \\ \hline DMU\({}_{04}\) & & & & & & \\ \hline DMU\({}_{07}\) & & & & & & \\ \hline DMU\({}_{08}\) & & & & & & \\ \hline DMU\({}_{09}\) & \(\surd\) & - & - & - & - & - \\ \hline DMU\({}_{10}\) & & & & & & \\ \hline DMU\({}_{11}\) & & & & & & \\ \hline DMU\({}_{12}\) & & & & & & \\ \hline DMU\({}_{13}\) & & & & & & \\ \hline DMU\({}_{14}\) & & & & & & \\ \hline DMU\({}_{15}\) & & & & & & \\ \hline DMU\({}_{17}\) & & & & & & \\ \hline DMU\({}_{05}\) & \(\times\) & DMU\({}_{03}\) & DMU\({}_{03}\) & \(\surd\) & The best compromise solution via DM’s opinions was found in first iteration of step 1 of STEM & 2 \\ \hline DMU\({}_{06}\) & & & DMU\({}_{07}\) & DMU\({}_{07}\) & & The best compromise solution was not found because there was only one similar unsatisfied objective function in the step 1 and first iteration of step 2 of STEM & 3 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: The brief of applying the interactive algorithm for finding DMU targets (the first case study). As shown in the last column of Table 8, applying the interactive algorithm for university departments are considered as three cases. DMU\({}_{01}\), DMU\({}_{02}\), DMU\({}_{03}\), DMU\({}_{04}\), DMU\({}_{07}\), DMU\({}_{08}\), DMU\({}_{09}\), DMU\({}_{10}\), DMU\({}_{11}\), DMU\({}_{12}\), DMU\({}_{13}\), DMU\({}_{14}\), DMU\({}_{15}\), and DMU\({}_{17}\) were members of M set and the targets of these DMUus were themselves (case 1). As best compromise solutions were found through step 1 of STEM for DMU\({}_{05}\), DMU\({}_{06}\), these DMUs belonged to case 2. As there was only one similar unsatisfied objective function in two consecutive steps of STEM for DMU\({}_{16}\), this DMU belonged to case 3. It should be mentioned that in the first case study, 14 DMUs from 17 DMUs (about 82%) lay on the efficient frontier. Moreover, non-radial FDH models 13 and 14 were composed and solved, three and four times through the second interactive stage, respectively. Here, the process of target determination for all DMUs was carried out, and the interactive algorithm finished. In fact, the required repetition times for composing and solving non-radial model 14 depend on DM's opinions about suitable targets for each considered DMU\({}_{o}\).