Two-Stage Network DEA with Bad Outputs

Abstract

Conventional black-box DEA models allow producer performance to be measured for technologies where undesirable outputs are jointly produced by-products of desirable output production. These models allow for non-radial scaling of desirable outputs, undesirable outputs, and inputs and can account for slacks in the constraints that define the technology. We review some of these black-box performance measures and show how to measure performance in two-stage network models. In these kinds of network models inputs are used to produce intermediate outputs in a first stage and then, those intermediate outputs become inputs to a second stage where final desirable outputs and undesirable outputs are produced. The bias from using a black-box model when a network technology exists is examined as well as the bias from ignoring slacks in the constraints defining the network technology.

Appendix

The synthetic data set in Table A.1 is used to estimate the black-box and two-stage network performance indicators presented in the paper. In the first stage N = 3inputs are used to produce Q = 2 intermediate outputs. In the second stage the Q = 2 intermediate outputs become inputs to produce M = 3 desirable outputs and L = 1 undesirable output. We choose a directional vector of g = (g ^x₁ ,g ^x₂ ,g ^x₃ ,g ^y₁ ,g ^y₂ ,g ^y₃ ,g ^b₁ ) = (1,1,1,1,1,1,1) to estimate each of the black-box and network performance indicators given in (19.8), (19.11), (19.16), and (19.19). We choose g = (g ^y₁ ,g ^y₂ ,g ^y₃ ,g ^b₁ ) = (1,1,1,1) for the black-box and network directional output distance functions given by (19.44), (19.45), (19.48), and (19.50). Estimates were obtained using GAMS (Generalized Algebraic Modeling System) with the Minos solver.

Table A.1 Synthetic data

Table A.2 Performance estimates