On the dual formulation of Russell measure model

The Russell Measure Model (RM), a nonlinear data envelopment analysis (DEA) model for evaluating decision-making units, allows for independent and disproportionate inputs and outputs, which makes it superior and more accurate than the radial models. The model is formulated as a second-order cone programming (SOCP) problem, and its dual is derived using SOCP duality. Previous studies have noted the complexity and limited interpretability of this dual formulation and have proposed an alternative using semidefinite programming (SDP) problem. This paper demonstrates the equivalence of these dual formulations through variable transformations. In addition, a new SOCP formulation of the dual RM model is introduced, which is in the usual form of multiplier models without any variable transformations. It is shown that this new formulation is equivalent to the SDP model. Moreover, using the conic model, a new approach is proposed to identify the unique maximal reference set and projection by solving one model, thereby improving upon the existing two-stage approach. Two examples demonstrate the advantages of the proposed models.