## 3 A New Hybrid Technique for Finding DMU Targets in Non-Radial FHD Models
### The Interactive Algorithm
#### 3.1.2 The Second Interactive Stage
By solving model 14, alternative targets of \(\text{DMU}_{\text{o}}\)\(\left(\text{DMU}_{\text{o}}\text{ Target}=\left\{\text{DMU}_{\text{j}}\left|\lambda_{j}^{*}=1,\text{ }\forall j\right\}\right\}\right)\) are found. The required parameters for Step 2 of STEM (Section 2.2) are then calculated. In this phase, the objective function values of the first step in the \(\text{p}^{\text{th}}\) iteration are considered as \(\text{F}^{\text{P}}=\left(\text{f}_{1}(\text{u}^{\text{P}}),\dots,\text{f}_{ \text{m+s}}(\text{u}^{\text{P}})\right)\) where \(\text{u}^{\text{P}}\) denotes the feasible solution in the \(\text{p}^{\text{th}}\) iteration. If all objective function values are satisfied considering the DM's opinion (the situation that the best compromise solution is found), \"consider suggested target as the target of \(\text{DMU}_{\text{o}}\)\", \(\text{M}^{\prime}=\text{M}^{\prime}-\text{DMU}_{\text{o}}\), go to the second interactive stage. If \"DM is not satisfied with all objective function values\", or \"no feasible integer solution is found\" (no compromise solution exists in these two situations), algorithm is continued from the start of the second interactive stage. Otherwise \(\text{f}_{1}(\text{u}^{\text{P}})=\text{f}_{1}^{\text{P}}\) considering [PERSON]'s opinions, which means the values of \(\text{f}_{1}\) in the \(\text{p}^{\text{th}}\) iteration are satisfied. Consequently, in order to improve the other objective functions, the \(\text{f}_{1}\) values are modified by the amounts of \(\Delta\text{f}_{\text{q}}\) considering [PERSON]'s opinions. In this case, DMUs in the feasible region are distinguished in the next iteration. As mentioned, finding the best compromise solution in the second interactive stage is desired through STEM. If the best compromise solution is obtained using required iterations of step 1 and 2 of STEM, suggested targets are considered as targets of \(\text{DMU}_{\text{o}}\). In the situation that no compromise solution is found (in two situations a compromise solution does not exist: 1- at least one \(\beta_{\text{l}}\), 1 = 1,..., m + s is not calculable, 2- there is only one similar unsatisfied objective function in two consecutive steps of STEM), the second interactive stage repeats. The brief of the interactive algorithm for finding DMU targets is shown in Figure 1. In the interactive stage 1, efficient DMUs and their targets are determined. It is remarkable that by applying the first interactive stage of the interactive algorithm, at least one \(\text{DMU}_{\text{j}}\), j = 1,..., n is efficient (at least, one member belongs to M set). The number of \(\text{M}^{\prime}\) set members are determined by composing and solving of non-radial model 12 for all DMUs. The required number of iterations for the interactive algorithm depends on the number of \(\text{M}^{\prime}\) set members. Non-radial model 12 should be constructed and solved n (number of DMUs) times. If the ratio of \"the number of \(\text{M}^{\prime}\) set members\" to \"all DMUs\" is near to 0, the interactive algorithm needs lesser iterations. If this ratio is near to 1, the interactive algorithm needs more iterations. In the interactive stage 2, non-efficient DMUs targets are determined using STEM. The required times to construct and find targets of non-radial model 13 (step 0 of STEM) are equal to the number of \(\text{M}^{\prime}\) set members. The required times to construct and find targets of non-radial model 14 (steps 1 and 2 of STEM) is dependent on DM's opinions  
Figure 1: The interactive algorithm for finding DMU targets in non-radial FDH models. about the suitable target for each considered DMU\({}_{\rm o}\). The required iteration number to apply the second interactive stage is equal to the number of M\({}^{\prime}\) set's members, which is defined the first time.