## 3 A New Hybrid Technique for Finding DMU Targets in Non-Radial FHD Models
### Extended RBA
#### 3.2.1 Finding DMU Targets of Non-Radial FDH Models 12 and 13  
As the constraints of non-radial FDH models 12 and 13 are exactly the same, finding DMU targets of these models are described simultaneously.  
The first target finding stage  
First, DMUs of feasible region should be found. Similar to RBA, if the constraints \(\frac{\text{x}_{\rm ij}}{\text{x}_{\rm io}}\ \leq 1,\text{i}=1,\ldots,\text{m}\) and \(\frac{\text{y}_{\rm ij}}{\text{y}_{\rm no}}\geq 1,\ \text{r}=1,\ldots,\text{s}\) are satisfied for DMU\({}_{\rm j}\), j = 1,..., n (the first condition), DMU j j = 1,..., n belongs to S. So, the DMUs of feasible region of non-radial FDH models 12 and 13 are determined using Equation (15).  
\[\text{S}_{\text{step 0 of STEM}}=\Big{\{}\text{DMU}_{\rm j}|\text{DMU}_{\rm j} \text{ satisfies first condition, j = 1,\ldots,n}\Big{\}}. \tag{15}\]  
As mentioned above, the first condition of extended RBA is similar to RBA.  
The second target finding stage  
The optimum objective function values should be calculated and DMU targets should be found in the second target finding stage. To achieve this, the objective function value of each DMU\({}_{j}\), j = 1,..., n in S\({}_{\text{Step 0 of STEM}}\) (in non-radial model 12, \(Z^{\text{o}}_{j}=\frac{1}{s}\sum_{t=1}^{s}\frac{y_{j_{0}}}{y_{\text{ro}}}- \frac{1}{m}\sum_{i=1}^{m}\frac{x_{i}}{x_{\text{io}}}\) and in non-radial model 13, \(f_{1}(u)=-x_{\text{lo}}x_{\text{ij}},\ldots,f_{m}(u)=-x_{\text{mo}}x_{\text{mj} },f_{m+1}(u)=y_{1}y_{1}o,\ldots,f_{m+s}(u)=y_{\text{sj}}y_{\text{so}}\)) is calculated. The maximum obtained value for DMUs belonging to S\({}_{\text{Step 0 of STEM}}\), shows optimum objective function value (the optimum objective function value of non-radial FDH models 12 and 13 are \(Z^{\text{o}}_{j}\) and \(f_{1}(u)\), \(1=1,\ldots,m+s\), respectively). Therefore, the optimum objective function values of non-radial FDH models 12 and 13 are calculated by Equations (16) and (17), respectively.  
\[Z^{\text{*o}}=\max_{\text{DMU}_{j}\in\text{S}_{\text{Step 0 of STEM}}}\left\{Z^{\text{o}}_{j}\right\},\ Z^{\text{o}}_{j}= \frac{1}{s}\sum_{t=1}^{s}\frac{y_{\text{rj}}}{y_{\text{ro}}}-\frac{1}{m}\sum_ {i=1}^{m}\frac{x_{\text{ij}}}{x_{\text{io}}} \tag{16}\]  
\[f^{\text{*}}_{1}=\max_{\text{DMU}_{j}\in\text{S}_{\text{Step 0 of STEM}}}\left\{f_{\text{ij}} \right\},f_{\text{ij}}(u)=-x_{\text{lo}}x_{\text{ij}},\ 1=1,\ldots,m,f_{\text{ij}}(u)=y_{(1-m)}y_{(1-m)o^{ \prime}},\ 1=m+1,\ldots,m+s \tag{17}\]  
Then, the DMUs of S\({}_{\text{Step 0 of STEM}}\) which have optimum objective function value show the targets of DMU\({}_{\text{o}}\) in each model. Therefore, DMU\({}_{\text{o}}\) targets in non-radial FDH models 12 and 13 are determined by Equations (18) and (19), respectively.  
\[\begin{split}&\text{DMU}_{\text{o}}\text{Targets in model 12}=\\ &\left\{\text{DMU}_{j}\left|\text{DMU}_{j}\right.\in\text{S}_{ \text{Step 0 of STEM}},\ j=1,\ldots,n,\text{ objective function value}=Z^{\text{*o}}\right\}\end{split} \tag{18}\]  
DMU\({}_{\text{o}}\)Targets in model 13 \(=\\ &\left\{\text{DMU}_{j}\left|\text{DMU}_{j}\right.\in\text{S}_{ \text{Step 0 of STEM}},\ j=1,\ldots,n,\text{ objective function value}=f^{\text{*}}_{1},\ 1=1,\ldots,m+s\right\}\end{split}\] (19)  
As described in Section 3.1, both steps 1 and 2 of STEM's model are shown by non-radial model 14. Finding DMU targets of related non-radial FDH models by extended RBA are described in Sections 3.2.2 and 3.2.3.