## 3 A New Hybrid Technique for Finding DMU Targets in Non-Radial FHD Models
### Extended RBA
#### 3.2.3 Finding DMU Targets of Step 2 of STEM's Model  
Finding the optimum solutions of non-radial model 14 (step 2 of STEM), without solving any mathematical models, is described in two target finding stages below.  
The first target finding stage  
First, DMUs of feasible region in step 2 of the STEM model should be found. In DMUs of feasible region, three conditions should be regarded. The first and second conditions should be tested for \(\mathrm{DMU}_{j}\), \(j=1\),..., \(n\) as described in Section 3.2.2, but in the second condition, \(\beta_{i}\), \(1=1,\ldots,m+s\) should be determined again. The third condition relates to \(\mathrm{f}_{q}(\mathrm{u})\geq\mathrm{f}_{q}(\mathrm{u}^{P})-\Delta\mathrm{f}_{ q}\) and \(\mathrm{f}_{i}(\mathrm{u})\geq\mathrm{f}_{i}(\mathrm{u}^{P})\), \(1=1,\ldots,m+s\), \(1\
eq q\) constraints that \(\mathrm{f}_{q}(\mathrm{u})\), \(\Delta\mathrm{f}_{q}\), and \(\mathrm{f}_{i}(\mathrm{u}),1=1,\ldots,m+s\), \(1\
eq q\) are obtained from suggested target of DM. So in DMUs of feasible region, constraints \(\mathrm{f}_{q}(\mathrm{u})\geq-\mathrm{x}_{i0}\mathrm{x}_{i\mathrm{j}}-\Delta \mathrm{f}_{q}\), and \(\mathrm{f}_{i}(\mathrm{u})\geq-\mathrm{x}_{i0}\mathrm{x}_{i\mathrm{j}}\), \(1=1,\ldots,m\), \(1\
eq q\) and \(\mathrm{f}_{q}(\mathrm{u})\geq\mathrm{y}_{i\mathrm{j}}\mathrm{y}_{r0}\), \(1=m+1,\ldots,m+s\), \(1\
eq q\) for \(j=1\),..., \(n\) should be satisfied, too. So, the selected \(\mathrm{DMU}_{j}\) of \(\mathrm{S}_{\mathrm{step~{}2~{}of~{}STEM}}^{\mathrm{P}}\) set are defined as Equation (23).  
\[\mathrm{S}_{\mathrm{step~{}2~{}of~{}STEM}}^{\mathrm{P}}=\Big{\{}\mathrm{DMU}_{j}\,| \,\mathrm{DMU}_{j}\,\mathrm{satisfies~{}first~{}to~{}third~{}conditions},~{}j=1, \ldots,n\Big{\}}. \tag{23}\]  
The second target finding stage  
The optimum objective function value of step 2 of the STEM model should be calculated and DMU targets of mentioned model should be found in the second target finding stage. The optimum objective function value of step 1 of the STEM model is calculated by Equation (24).  
\[\mathrm{h}^{*}=\min_{\mathrm{DMU}_{j}\in\mathrm{S}_{\mathrm{step~{}2~{}of~{}STEM}}^{\mathrm{P}}}\Big{\{}\max_{i=1,\ldots,m,r=1,\ldots,s}\Big{\{}\big{(} \mathrm{f}_{i}^{*}+\mathrm{x}_{i0}\mathrm{x}_{i\mathrm{j}}\big{)}.\beta_{i},~{} ~{}\big{(}\mathrm{f}_{r}^{*}-\mathrm{y}_{i\mathrm{j}}\mathrm{y}_{r0}\big{)}. \beta_{r}\Big{\}}\Big{\}} \tag{24}\]  
After that, the DMUs of feasible region which have optimum objective function values show the targets of \(\mathrm{DMU}_{0}\) in the model (Equation (25)).  
\[\begin{array}{l}\mathrm{DMU}_{0}\mathrm{Targets~{}in~{}step~{}2~{}of~{}STEM}=\\ \Big{\{}\mathrm{DMU}_{j}\,|\,\mathrm{DMU}_{j}\in\mathrm{S}_{\mathrm{step~{}2~{}of~ {}STEM}^{\prime}}^{\mathrm{P}}\,\,j=1,\ldots,n,~{}\mathrm{objective~{} function~{}value}=\mathrm{h}^{*}\Big{\}}.\end{array} \tag{25}\]  
It is remarkable that \(\mathrm{DMU}_{0}\mathrm{Target}\) in step 2 of STEM exists, if the best compromise solution is found in step 2 of the STEM model (Section 3.1). The brief of applying extended RBA to find non-radial FDH models targets of the interactive algorithm is shown in Figure 2.