## 4 Two Real Case Studies
### The First Case Study: University Departments
#### 4.1.1 Applying the Interactive Algorithm for the First Case Study
As M\({}^{\prime}=\{\text{DMU}_{05},\text{ DMU}_{06},\text{DMU}_{16}\}\
eq\varnothing\) (m\({}^{\prime}\) has 3 members), the interactive algorithm (Figure 1) should be continued to find targets of M\({}^{\prime}\) set considering \(\text{DMU}_{05},\text{ DMU}_{06}\), and \(\text{DMU}_{16}\), one by one. But here, the process of applying the interactive algorithm for \(\text{DMU}_{05},\text{ DMU}_{06}\) and \(\text{DMU}_{16}\) are described simultaneously. So, each of these three DMUs (\(\text{DMU}_{0}=\text{DMU}_{05}\), \(\text{DMU}_{06}\) and \(\text{DMU}_{16}\)) was considered as a non-proceed DMU that their targets should be defined. Firstly, \(\text{x}_{\text{io}}=\text{x}_{\text{ij}},\text{i}=1,\text{ 2},\text{ j}=1,\text{ \ldots},\text{ 17}\), and \(\text{y}_{\text{ro}}=\text{y}_{\text{rj}},\text{r}=1,\text{ 2},\text{ j}=1,\text{ \ldots},\text{ 17}\) were defined for \(\text{DMU}_{\text{o}}\), o = 5,6, and 16. Then the pay-off tables (Table 4) were constructed by composing and solving of model 13 for f\({}_{1}\)(u), l = 1, \(\ldots,\text{ 4}\). In model 13, n = 17 and m = s = 2 are considered for each DMU\({}_{\text{o}}\), o = 5, 6, and 16. It is mentionable that the required times to construct and solve model 13 in the interactive algorithm were equal to 3 (the number of M\({}^{\prime}\) set members). In this interactive stage, finding a suitable DMU in feasible region as the target of \(\text{DMU}_{\text{o}}\), o = 5, 6, and 16 was desired which the value of f\({}_{1},1=1,\text{ \ldots},\text{ 4}\) was as good as the suggested DMU\({}_{\text{o}}\) target, o = 5, 6, and 16 by [PERSON]. Supposed that the targets of \(\text{DMU}_{05},\text{ DMU}_{06}\) and \(\text{DMU}_{16}\) according to [PERSON]'s opinions were \(\text{DMU}_{03},\text{ DMU}_{07}\) and \(\text{DMU}_{13}\), respectively. So, suggested targets of [PERSON] were considered as [PERSON]'s opinions for f\({}_{1}\), l = 1, \(\ldots,\text{ 4}\). After, the first iteration of STEM (p = 1) was started. In step 1 of STEM (calculation phase), first \(\pi_{1},1=1,\text{ \ldots},\text{ 4}\) was calculated by Equation (4) considering Table 4, then \(\beta_{1}=\frac{\pi_{1}}{\sum_{\text{d=1}}^{4}\pi_{\text{d}}},\text{ 1}=1,\text{ \ldots},\text{ 4}\) was calculated as shown in Table 5.  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \(\text{DMU}_{\text{o}}\) & \multicolumn{1}{c}{**f\  
Then model 14 for DMU\({}_{05}\), DMU\({}_{06}\) and DMU\({}_{16}\) (Step 1 of STEM for p = 1) were composed. The results of solving these models and [PERSON]'s opinions about them are shown in Table 6. As seen in the last column of Table 6, the targets of DMU\({}_{05}\), DMU\({}_{06}\) and DMU\({}_{16}\) via solving of model 14 (Step 1 of STEM for p = 1) were DMU\({}_{03}\), DMU\({}_{07}\) and DMU\({}_{13}\), respectively. As described previously, these targets were acceptable according to [PERSON]'s opinion, too. As the obtained solutions for DMU\({}_{05}\) and DMU\({}_{06}\) were satisfied via [PERSON]'s opinions, the best compromise solutions were obtained for these DMUs. Then, DMU\({}_{0}\) (DMU\({}_{05}\) and DMU\({}_{06}\)) from M\({}^{\prime}\) set removed (M\({}^{\prime}=\{\)DMU\({}_{16}\}\
eq\varnothing\) (M\({}^{\prime}\) has 1 member). As the value of the fourth target of DMU\({}_{16}\) was not satisfied via [PERSON]'s opinions, the obtained solution was not the best compromise solution. So, composing model 14 related to Step 2 of STEM (p = 1) was required for DMU\({}_{16}\). The required parameters were calculated as shown in Table 7.