A novel median-based optimization model for eco-efficiency assessment in data envelopment analysis

Abstract

Economic-environmental performance or eco-efficiency is a topic of great interest due to the “green movement.” Data Envelopment Analysis (DEA) is a non-parametric method for measuring the eco-efficiencies in comparable Decision-Making Units (DMUs) under various technology assumptions, e.g., constant or variable returns to scale. In the case of variable returns to scale, the returns to scale (RTS) values show whether the DMUs under consideration have the correct scale size or can be improved by upsizing or downsizing. However, sometimes the RTS values for some DMUs are unusually high or low and hence useless in practice. The RTS-mavericks test is devised to propose RTS bounds to fix this flaw. However, these bounds can be ineffective in practice. Even if this flaw is rectified, it needs to be clarified how the concept of RTS-mavericks influences eco-efficiency analysis. For the case of a single technology and a combination of two technologies (a so-called pollution-generating technology), we derive RTS equations and develop new median-based optimization problems to correct this flaw and show that the new concept can lead to non-convex technologies. We also demonstrate the applicability and exhibit the efficacy of the proposed model in the context of eco-efficiency analysis in the European Union.

Appendices

Appendix 1

For the case of a pollution-generating technology, we immediately show the linearized Model (1A) because the linearization follows the same logic as in the single technology case.

$$\begin{array}{ll}\mathrm{max} {\mathit{eff}}_{k}={\mathbf{U}}_{k}^{g\mathrm{T}}{\mathbf{y}}_{k}^{g} - {\widetilde{\mathbf{U}}}_{k}^{b\mathrm{T}}{\mathbf{y}}_{k}^{b} +{ u}_{k} \\ \mathrm{s}.\mathrm{t}. \\ {\mathbf{V}}_{k}^{g\mathrm{T}}{\mathbf{x}}_{k}^{g} {+\mathbf{V}}_{k}^{b\mathrm{T}}{\mathbf{x}}_{k}^{b} {-\widetilde{\mathbf{V}}}_{k}^{b\mathrm{T}}{\mathbf{x}}_{k}^{b}=1 \\ {\mathbf{U}}_{k}^{g\mathrm{T}}{\mathbf{y}}_{j}^{g} - {\mathbf{V}}_{k}^{g\mathrm{T}}{\mathbf{x}}_{j}^{g} {-\mathbf{V}}_{k}^{b\mathrm{T}}{\mathbf{x}}_{j}^{b}+ {u}_{k}\le 0& \forall j\in \mathcal{J}\\ {\widetilde{\mathbf{U}}}_{k}^{b\mathrm{T}}{\mathbf{y}}_{j}^{b}-{\widetilde{\mathbf{V}}}_{k}^{b\mathrm{T}}{\mathbf{x}}_{j}^{b}\ge 0& \forall j\in \mathcal{J}\\ { u}_{k}+\left({-\mathbf{U}}_{k}^{g\mathrm{T}}{\mathbf{y}}_{k}^{g}\right)\cdot \left[-1+\mathrm{CV}\cdot \widehat{\sigma }+median\left({\varvec{\rho}}\right)\right]\le 0 \\ -{ u}_{k}+\left({-\mathbf{U}}_{k}^{g\mathrm{T}}{\mathbf{y}}_{k}^{g}\right)\cdot \left[1+\mathrm{CV}\cdot \widehat{\sigma }-median\left({\varvec{\rho}}\right)\right]\le 0& \\ {\mathbf{U}}_{k}^{g},{\mathbf{V}}_{k}^{g},{\mathbf{V}}_{k}^{b},{\widetilde{\mathbf{U}}}_{k}^{b},{\widetilde{\mathbf{V}}}_{k}^{b}\ge 0 \,\mathrm{and}\,{ u}_{k} \mathrm{free} \end{array}$$

(21)

Proposition 10

Again, let $\mathcal{L}$ be the index set of RTS-mavericks. Then, problem (1A) leads to modified eco-efficiency scores ${eff}_{k}^{mod**}<{eff}_{k}^{**} \forall k\in \mathcal{L}$, with ${eff}_{k}^{mod**}and {eff}_{k}^{**}$ being the optimal eco-efficiencies of models (1A) and (9).

Proof

To prove this proposition, one can use the same reasoning as in Proposition 7. □

For the sake of completeness, we give Algorithm 2, which can be used in the case of a pollution-generating technology.

Appendix 2

See Tables

Table 8 Pollution-generating technology results for the E.U. member states (Algorithm 2)

8 and

Table 9 Normal vs. half-normal distribution-based estimates

9