## 3 A New Hybrid Technique for Finding DMU Targets in Non-Radial FHD Models
### Extended RBA
#### 3.2.2 Finding DMU Targets of Step 1 of STEM's Model  
The process of applying extended RBA to find DMU targets of step 1 of STEM's model is described in two target finding stages as follows:  
The first target finding stage  
In the first target finding stage, finding DMUs of feasible region (in step 1 of STEM model) is desired. In DMUs of feasible region (DMU\({}_{j}\), j = 1,..., n), two conditions should be satisfied. Firstly, in these DMUs, similar to non-radial FDH models 12 and 13, constrains \(\frac{x_{\text{ij}}}{x_{\text{io}}}\leq 1\), i = 1,..., m, and \(\frac{y_{\text{fi}}}{y_{\text{ro}}}\geq 1\), r = 1,..., s should be satisfied for DMU\({}_{j}\), j = 1,..., n. The method of checking this condition for DMUs of feasible region (DMU\({}_{j}\), j = 1,..., n) has been described in Section 3.2.1. Secondly, in DMUs of feasible region, the constraint \(\max_{\text{i}}\{(f^{\text{*}}_{1}-f_{1}(u)).\beta_{1}\}\geq 0\), \(1=1,\ldots,m+s\) should be satisfied for DMU\({}_{j}\), j = 1,..., n. In constraints of the second condition, the values of \(f^{\text{*}}_{i},1=1,\ldots,m+s\) and \(\beta_{1},1=1,\ldots,m+s\) are obtained from solving non-radial model 13 and calculated using Equation (4), respectively. For checking the second condition, \((f^{\text{*}}_{i}+x_{\text{io}}x_{\text{ij}}).\beta_{i},i=1,\ldots,m\) and \((f^{\text{*}}_{r}-y_{\text{rj}}y_{\text{ro}}).\beta_{r},r=1,\ldots,s\) are calculated for DMU\({}_{j}\), j = 1,..., n that were compatible in the first condition. Then constraints \(\max_{\text{i}}\{(f^{\text{*}}_{1}-f_{1}(u)).\beta_{1}\}\geq 0\), \(1=1,\ldots,m+s\) are checked for mentioned DMUs. In DMUs belonging to the feasible region of non-radial model 14 (step 1 of STEM), in addition to the first condition, constraints \(\max_{\text{i=1,\ldots,m,r=1,\ldots,s}}\left\{(f^{\text{*}}_{i}+x_{\text{io}}x_ {\text{ij}}).\beta_{i},\ \left(f^{\text{*}}_{r}-y_{\text{rj}}y_{\text{ro}}\right).\beta_{r}\right\}\geq 0\) should be satisfied for j = 1,..., n (the second condition), too. In DMUs of feasible region, the first and second conditions should be satisfied in DMU\({}_{j}\), j = 1,..., n, simultaneously. Therefore, the DMUs of S\({}_{\text{step 1 of STEM}}^{\text{P}}\) are determined using Equation (20).  
\[\text{S}_{\text{step 1 of STEM}}^{\text{P}}=\left\{\text{DMU}_{j}\left|\text{ DMU}_{j}\text{ satisfies first and }\sec\text{ond conditions},\ j=1,\ldots,n\right\}. \tag{20}\]The second target finding stage  
The optimum objective function values of step 1 of the STEM model should be calculated and DMU targets of the model should be found in the second target finding stage. The optimum objective function value of step 1 of STEM model is calculated by Equation (21).  
\[\mathrm{h}^{*}=\min_{\mathrm{DMU}_{j}\in\mathrm{S}_{\mathrm{step~{}1~{}of~{}STEM} }^{\mathrm{P}}}\Big{\{}\max_{i=1,\ldots,m,r=1,\ldots,s}\Big{\{}\big{(}\mathrm{ f}_{i}^{*}+\mathrm{x}_{i0}\mathrm{x}_{i\mathrm{j}}\big{)}.\beta_{i},~{}~{} \big{(}\mathrm{f}_{r}^{*}-\mathrm{y}_{i\mathrm{j}}\mathrm{y}_{r0}\big{)}. \beta_{r}\Big{\}}\Big{\}} \tag{21}\]  
After that, the DMUs of \(\mathrm{S}_{\mathrm{step~{}1~{}of~{}STEM}}^{\mathrm{P}}\) which have optimum objective function value shows the targets of \(\mathrm{DMU}_{0}\) in the model. Therefore, the \(\mathrm{DMU}_{0}\) targets in step 1 of STEM are determined using Equation (22).  
\[\begin{array}{l}\mathrm{DMU}_{0}\mathrm{Targets~{}in~{}step~{}1~{}of~{}STEM}=\\ \Big{\{}\mathrm{DMU}_{j}\,|\,\mathrm{DMU}_{j}\in\mathrm{S}_{\mathrm{step~{}1~{}of~ {}STEM}^{\prime}}^{\mathrm{P}}\,\,j=1,\ldots,n,~{}\mathrm{objective~{} function~{}value}=\mathrm{h}^{*}\Big{\}}.\end{array} \tag{22}\]