Second order cone programming approach to two-stage network data envelopment analysis

Efficiency aggregation and efficiency decomposition are two techniques used in modeling decision making units (DMUs) with two-stage network structures under network data envelopment analysis (DEA). Multiplicative efficiency decomposition (MED) is usually used in a very specialized two-stage structure when constant returns to scale (CRS) is assumed. MED-based network DEA retains the property of the conventional DEA in the sense that input- and output-oriented models yield the same efficiency scores. Compared with the additive efficiency decomposition (AED), MED does not require predetermined weights to combine individual stage efficiencies. However, if there are external inputs to the second stage, and/or some outputs leave the first stage and do not become inputs to the second stage, or if we assume variable returns to scale (VRS), MED has limited capability to address these extensions. Alternatively, multiplicative efficiency aggregation (MEA), which is highly nonlinear and is impossible to be transformed into a linear programming problem, defines the overall efficiency as a product of stage efficiency scores and can be easily applied to general two-stage network structures. The current study discovers that MEA DEA model for general two-stage networks corresponds to a cone structure in disguise, and can be transformed into the form of second order cone programming (SOCP). Therefore, MEA in two-stage network DEA can be effectively and efficiently solved, regardless of the network structures. We show that AED can also be solved using SOCP and demonstrate that input and output-oriented AED models may not yield the same efficiency scores under CRS. The current research enables us to solve both MEA and AED using SOCP which is considered as effective as linear programming.