Deriving the Efficiency Frontier for Two-Stage Structures: Input – Output Oriented Approach of Radial and Non-Radial

Early models of data envelopment analysis are not suitable for evaluating two-stage structures due to the black box view and lack of attention to internal processes. In these structures, Deriving the efficiency frontier and fairly determining the optimal value of variables is the most important challenge. In many existing two-stage models, the efficiency frontier is not plotted or the optimal value of intermediate variables is determined by one of two steps. This leads to incorrect calculation of the efficiency of the next stage and the total efficiency. In fact, in these models, poor performance of one stage leads to reduced efficiency of the other stage. In this study, by keeping the intermediate variables constant at the current level and with an input-output oriented approach, radial and non-radial models were developed on a constant and variable returns to scale in terms of efficiency. Using mathematical relations, the validity of the models was proved and shown that in the proposed models, the performance of the units in steps is compared with a unit on the efficiency frontier, and the models make the whole structure efficient by bringing the steps to the efficiency frontier. The proposed models were used in an applied study to evaluate the sustainability of nine supply chains of tomato producers. Their performance results were expressed by four models as well as the optimal value of inefficient unit variables in each of these models.