The role of unobserved units in two-stage network data envelopment analysis

In a typical two-stage network (2SN) data envelopment analysis (DEA), all the outputs from the first stage become the inputs to the second stage and there are no additional inputs directly into the second stage. These outputs from the first stage are called intermediates. In this study, we show that the existing 2SN DEA assumes that the 2SN production possibility sets (PPSs) of inputs-intermediates and intermediates-outputs are known, but the 2SN PPS of inputs-outputs is unknown and defined from the two known PPSs. In order to construct the 2SN PPS of inputs-outputs, the existing 2SN DEA has to generate unobserved units which are infeasible to the standard PPS of inputs-outputs. This approach can be interpreted as connecting two systems by a transitivity property. In the current study, we assume that the unobserved (infeasible) units are not allowed in constructing the 2SN PPS and the 2SN PPS of inputs-outputs is a subset of the standard PPS. We then define the corresponding 2SN PPSs of inputs-intermediates and intermediates-outputs. From this point of view, we consider the 2SN system as one system including two sections where the performance of each section cannot be measured without considering the performance of the entire system. We develop a linear programming model to radially measure the new-overall efficiency as the product of two stages efficiencies.