## 3 A New Hybrid Technique for Finding DMU Targets in Non-Radial FHD Models
### The Interactive Algorithm
#### 3.1.2 The Second Interactive Stage  
In the second interactive stage, non-efficient DMUs targets are determined by STEM. In this regard, a DMU\({}_{j}\in\text{M}\prime\) is considered as DMU\({}_{0}\) (a non-proceed DMU\({}_{j}\); \(j=1,\,\ldots,\,n\)) and \(x_{\text{io}}=x_{\text{ij}},\text{i}=1,\,\ldots,\,\text{m}\), \(j=1,\,\ldots,\,n\), and \(y_{\text{ro}}=y_{\text{rj}},\text{r}=1,\,\ldots,\,\text{s},\,j=1,\,\ldots,\,n\), are defined; then the pay-off table (as described in Section 2.2) is constructed by composing and solving the non-radial model 13 for \(f_{1}(u),\,1=1,\,\ldots,\,m+\text{s}\) (step 0 of STEM). \[\text{Max}\,\left\{f_{1}(u)=-x_{1o}\theta_{1},\ldots,f_{m}(u)=-x_{m0}\theta_{ m},f_{m+1}(u)=y_{1o}\varphi_{1},\ldots,f_{m+\text{s}}(u)=y_{so}\varphi_{\text{s}}\right\}\]  
\[\text{s.t}\]  
\[\text{u}\in\text{S}=\left\{\text{u}\left| \begin{array}{c}\sum_{j=1}^{n} \lambda_{j}=1,\,\lambda_{j}\in\{0,1\},\,j=1\ldots\text{n},\\ \sum_{j=1}^{n}\lambda_{j}x_{ij}\leq\theta_{i}x_{\text{io}},\,0\leq\theta_{i} \leq 1,\,i=1\ldots\text{m},\\ \sum_{j=1}^{n}\lambda_{j}y_{ij}\geq\varphi_{i}y_{\text{ro}},\varphi_{\text{r}} \geq 1,\,r=1\ldots\text{s}.\end{array} \right.\right\}.\]  
It is considerable that a DMU in S (in non-radial model 13) is efficient if all \(\theta_{i}^{*}=\ 1,\,i=1\ldots\text{m}\), and all \(\varphi_{\text{r}}^{*}=1,\,r=1\ldots\text{s}\). The DM is then asked to suggest a new target for DMU\({}_{0}\). If DM does not suggest a new target for DMU\({}_{0}\), \"There is no target for DMU\({}_{0}\) in FDH model\", then M\({}^{\prime}=\text{M}^{\prime}-\text{DMU}_{0}\) (omit DMU\({}_{0}\) from M\({}^{\prime}\) set), and go to the second interactive stage (to find targets of another non-proceed DMU). If DM suggest a new target for DMU\({}_{0}\), extract DM's opinions about the desired value of \(f_{1},\,1=1,\,\ldots,\,m+\text{s}\) from the suggested DMU\({}_{0}\) target. This means that the value of \(f_{1},\,1=1,\,\ldots,\,m+\text{s}\) should be as good as the suggested DMU\({}_{0}\) target by DM. So the proposed target of DM is considered as [PERSON]'s opinions for all \(m+\text{s}\) objective functions (\(f_{1},\,1=1,\,\ldots,\,m+\text{s}\)). Attempt are then made to find the best compromise solution considering [PERSON]'s opinions using the required iterations of step 1 and 2 of STEM (Section 2.2). After that, the first iteration of STEM begins (set p = 1). In step 1 of STEM (Section 2.2), first \(\pi_{1},\,1=1,\ldots,\,m+\text{s}\) is calculated by Equation (4) considering the results of the second interactive stage, then \(\beta_{1}=\frac{\pi_{1}}{\sum_{d=1}^{n}\pi_{d}},\,1=1,\ldots,\,m+\text{s}\) is calculated. If at least one \(\beta_{1},\,1=1,\,\ldots,\,m+\text{s}\) is not calculable, there is no compromise solution (in this situation, obtaining the suggested target of DM for DMU\({}_{0}\) is impossible) and the interactive algorithm continues from the start of the second interactive stage (to find another suitable target for DMU\({}_{0}\)). In the situation that the value of all \(\beta_{1},\,1=1,\,\ldots,\,m+\text{s}\), are calculable, model 14 is as follows:  
\[\text{Min}\,\,\,\text{h}\geq(f_{1}^{*}-f_{1}(u)).\beta_{1}\,1=1,\ldots,\,m+ \text{s},\]\[\text{u}\in\text{$\mathbb{S}^{\text{P}}$, }\text{h}\geq 0.\text{ Where $\mathbb{S}^{\text{P}}$}=\left\{\text{u}\left| \begin{array}{c}\sum_{j=1}^{\text{n}}\lambda_{j}=1,\text{ }\lambda_{j}\in\{0,1\},\text{ j}=1\dots\text{n},\\ \sum_{j=1}^{\text{n}}\lambda_{j}\text{x}_{j}\leq\theta_{i}\text{x}_{\text{i} \text{o}},\text{ }0\leq\theta_{i}\leq 1,\text{ i}=1\dots\text{m},\\ \sum_{j=1}^{\text{n}}\lambda_{j}\text{r}_{j}\geq\varphi_{\text{r}}\text{y}_{ \text{r}}\text{y}_{\text{r}}\geq 1,\text{ r}=1\dots\text{s}.\end{array} \right.\right\}\cup\text{$ \mathbb{S}^{\text{P}-1}$}\]  
It is considerable that \(\mathbb{S}^{\text{P}}\) denotes the feasible region in the \(\text{p}^{\text{th}}\) iteration.