Solving data envelopment analysis models with sum-of-fractional objectives: a global optimal approach based on the multiparametric disaggregation technique

The majority of data envelopment analysis (DEA) models can be linearized via the classical Charnes–Cooper transformation. Nevertheless, this transformation does not apply to sum-of-fractional DEA efficiencies models, such as the secondary goal I (SG-I) cross efficiency model and the arithmetic mean two-stage network DEA model. To solve a sum-of-fractional DEA efficiencies model, we convert it into bilinear programming. Then, the obtained bilinear programming is relaxed to mixed-integer linear programming (MILP) by using a multiparametric disaggregation technique. We reveal the hidden mathematical structures of sum-of-fractional DEA efficiencies models, and propose corresponding discretization strategies to make the models more easily to be solved. Discretization of the multipliers of inputs or the DEA efficiencies in the objective function depends on the number of multipliers and decision-making units. The obtained MILP provides an upper bound for the solution and can be tightened as desired by adding binary variables. Finally, an algorithm based on MILP is developed to search for the global optimal solution. The effectiveness of the proposed method is verified by using it to solve the SG-I cross efficiency model and the arithmetic mean two-stage network DEA model. Results of the numerical applications show that the proposed approach can solve the SG-I cross efficiency model with 100 decision-making units, 3 inputs, and 3 outputs in 329.6 s. Moreover, the proposed approach obtains more accurate solutions in less time than the heuristic search procedure when solving the arithmetic mean two-stage network DEA model.