Integrated one-stage models considering undesirable outputs and weighting preference in slacks-based measure of efficiency and superefficiency

In the evaluation scenario with undesirable outputs, the SBM-Undesirable model is usually used together with the SuperSBM-Undesirable model to fully differentiate efficient and inefficient decision-making units (DMUs). The existing SuperSBM-Undesirable model has two main forms, but both have problems in implementation. One form may be infeasible under the variable returns to scale (VRS); Another form has an inappropriate definition for undesirable output efficiency. These two problems are seriously ignored in the existing applications. In this paper, we first propose an improved SuperSBM-Undesirable model with strongly efficient projections, which is feasible under the constant returns to scale (CRS) or VRS technology. Then, by integrating the SBM-Undesirable and the improved SuperSBM-Undesirable models, we focus on proposing a concise, precise and practical one-stage model considering undesirable outputs to differentiate all DMUs and determine their strongly efficient projections simultaneously. The proposed one-stage model only contains essential decision variables and constraints, thereby effectively conserving computational time for large-scale practical applications. Further, by introducing the weighting preference of inputs and outputs, we construct the one-stage model with weighting preference to differentiate the importance of different indicators. Finally, through several data experiments, the superiority of our models in computational results and computational scale are verified.