Extended secondary goal approach for common equilibrium efficient frontier selection in DEA with fixed-sum outputs

As a non-parametric programming approach, Data envelopment analysis (DEA) has been extended to consider the situation of fixed-sum outputs, which causes competition among the evaluated decision making units (DMUs). Minimum reduction strategy of the fixed-sum output has been proposed to form a common equilibrium efficient frontier to solve the problem. However, the non-uniqueness of the common equilibrium efficient frontier problem has reduced the usefulness of this extended method. Aiming at solving the problem, we propose an extended secondary goal approach to further narrow the scope of the common equilibrium efficient frontier. Compared with traditional secondary goal approaches, the new approach has considered each DMU's minimum and maximum inefficiency value. Specially, a Max-min model based on satisfaction degree is proposed to reflect each DMU's satisfaction on achieving its final efficiency value. In addition, two effective algorithms are given to solve the non-linear Max-min model and further guarantee the uniqueness of common equilibrium efficient frontier. Last, we use a numerical example to illustrate our proposed models.