## 4 Two Real Case Studies
### The First Case Study: University Departments
#### 4.1.2 Applying Extended RBA for the First Case Study  
As described in Section 4.1.1, for applying the interactive algorithm for the first case study, solving of non-radial FDH models 12, 13, model 14 of step 1 of STEM (p = 1), and model 14 of step 2 of STEM (p = 1), 17, 3, 3, and 1 times were needed, respectively. So, solving 24 mixed 0-1 LP models were required to find all DMU targets. The DMU targets of mentioned non-radial FDH models could be found using extended RBA through two target finding stages without solving any mathematical programming models. In this section, applying extended RBA for finding DMU targets of these non-radial FDH models (models 12-14 for DMU05 of university departments) are described. First \(\text{x}_{\text{io}}=\text{x}_{\text{ij}},\text{i}=1,\ 2\), and \(\text{y}_{\text{ro}}=\text{y}_{\text{ij}},\text{s}=1,\ 2\) (j = 1, \ldots,\ 17) were defined for DMU\({}_{o}=\text{DMU}_{05}\) (Figure 2). Target finding of non-radial FDH models 12 and 13 for DMU\  
The optimum objective function values and DMU targets should then be found in applying the second target finding stage. At first, the optimum objective function values of models should be calculated by Equations (16) and (17) (Figure 2). The objective function values of DMU\({}_{03}\) and DMU\({}_{05}\) (in model 12, \(Z_{1}^{\rm o}=\frac{1}{2}\sum_{r=1}^{2}\frac{Y_{\rm i}}{Y_{\rm no}}-\frac{1}{2} \sum_{i=1}^{2}\frac{x_{\rm i}}{x_{\rm i0}}\), \(j=1,\ldots,\ 17\), and in model 13, \(f_{1}(u)=-x_{10}x_{1j},f_{2}(u)=-x_{20}x_{2j},f_{3}(u)=Y_{10}Y_{1j},f_{4}(u)= y_{20}Y_{2j}\)) are shown in Table 10.  
The last row of Table 10 shows the maximum values of \(Z_{1}^{\rm o}\), \(\rm DMU_{j}\in S_{step\ 0\ \rm\ of\ STEM}\) and \(f_{1},1=1,\ \ldots,\ 4\), respectively. These values show the optimum objective function value of each model (\(Z^{\rm o},\mathsf{f}_{1}^{\rm s},1=1,\ldots,4\)). The \(\rm DMU_{o}\) targets were then found by comparing the optimum objective function values of each model with the objective function value of each DMU by Equations (18) and (19) (Figure 2). It is considerable that non-radial model 13 with \(\mathsf{f}_{4}\) objective function had alternative solutions. Target finding of non-radial model 14 for \(\rm DMU_{05}\) in the first case study  
First, step 1 of STEM (p = 1) should be considered. DMUs of feasible region (in step 1 of STEM model) should be found through applying the first target finding stage. First, DMUs belonging to the feasible region should be found. In DMUs of feasible region, two conditions should be satisfied. Firstly, constraints \(\frac{x_{\rm i}}{x_{\rm i0}}\leq 1,i=1,\ 2\), and \(\frac{Y_{\rm i}}{Y_{\rm no}}\geq 1,r=1,\ 2\), (\(j=1,\ldots,17\)) should be satisfied (the first condition). As it is seen in Table 9, only \(\rm DMU_{03}\) and \(\rm DMU_{05}\) satisfied these constraints. Moreover, the constraints \(\max_{\rm i}\{(\mathsf{f}_{1}^{\rm s}-f_{1}(u)).\beta_{1}\}\geq 0,\ 1=1,\ldots,\ 4\) should be satisfied for each DMU of feasible region (the second condition). Checking the second condition for \(\rm DMU_{03}\) and \(\rm DMU_{05}\) are shown in Table 11.  
As seen in Table 11, the constraints of the second condition were also satisfied for \(\rm DMU_{03}\) and \(\rm DMU_{05}\). So, \(\rm S_{step\ 1\ \rm\ of\ STEM}=\{DMU_{03},DMU_{05}\}\). The optimum objective function value of step 1 of STEM model was then calculated, and DMU targets were found through applying the second target finding stage. With respect to Equation (21), the minimum values of the left-hand side of the second condition for \(\rm DMU_{03}\) and \(\rm DMU_{05}\) showed the optimum objective function value of non-radial model 14 (last row of Table 11). As is seen in Equation (22), each DMU with an objective function value equal to the optimum objective function value, showed the targets  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{ \begin{tabular}{c} \(\mathbf{DMU}_{\rm j}\) \\ \end{tabular}