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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.

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