Hybrid efficiency measurement and target setting based on identifying defining hyperplanes of the PPS with negative data

The traditional data envelopment analysis (DEA) models can be used for efficiency evaluation of decision making units (DMUs) with nonnegative data. However, in the real world there are DMUs which have negative inputs and/or outputs. Consequently, in the literature of DEA various approaches have been offered in order to deal with negative data. The current research proposes an algorithm concerned with a pure mathematical optimization method to measure hybrid efficiency of DMUs in the presence of negative data, and also a pure mathematical procedure for target setting. In doing so, explicit form equations of strong and weak defining hyperplanes of production possibility set (PPS) based on the multiple criteria decision making methodology are obtained. In characterizing these hyperplanes, a new multiple objective linear programming (MOLP) problem is presented whose feasible region of criterion space is similar to the PPS with variable returns to scale technology on nonnegative and negative data. The MOLP problem is solved using the multicriteria simplex method where its process of solving leads to construct all strong and weak defining hyperplanes. Then, using the strong and weak hyperplanes obtained and also without using any DEA optimization model, a hybrid measure of efficiency and a strong efficient target unit for each inefficient DMU with negative data are obtained. Finally, the results are discussed using two examples, where the first one, in detail, explains the proposed methods and algorithm, and the second one briefly shows the contributions are applicable for the real data.