DEA cross-efficiency evaluation and ranking method based on interval data

Abstract

Data envelopment analysis (DEA) is an important method of efficiency evaluation. Cross-efficiency evaluation is one of the main aspects of research in the field of DEA that has been applied in various fields. In the traditional cross-efficiency evaluation model, the variable data of decision-making units is exact. Dynamic information is frequently unable to reflect the whole characteristic when determining the exact data. In this study, we select interval data to represent the dynamic information of some variables in the evaluation process. We then build a solution method based on interval efficiency and DEA cross-efficiency. This method retains the reflection of interval data on uncertain variable properties. Finally, the stochastic multi-criteria acceptability analysis 2 (SMAA2) is introduced to solve the whole sequence problem of interval efficiency. We present a case study from a set of 19 reservoir dams suffered from Wenchuan earthquake in Luojiang County, Sichuan Province to demonstrate the applicability of the proposed model.

Appendix

Models (7) and (8) have feasible solutions.

Proof

Here, we use Model (7) as an example. In Model (7), constraint $\sum \nolimits _{r=1}^s u_{ro} \underline{y_{rj} }\le \sum \nolimits _{i=1}^m v_{io} \overline{x_{ij} } j=1,2,\ldots ,n,j\ne d,j\ne o$ is smaller than the constraint set used to solve the optimal efficiency value of $DMU_o$. Thus, the solution set is expanded. $DMU_o$ will surely reach its optimal efficiency value. Thus, Model (7) will achieve the solution.

Secondly, we need to prove that the solution set of nonlinear constraint $\overline{\theta _o } *\sum \nolimits _{i=1}^m v_{io} \widetilde{x_{io}} -\sum \nolimits _{r=1}^s u_{ro} \widetilde{y_{ro}} =0$ is a convex set.

Suppose that the vector group $\left( {\vec {v_1 },\vec {u_1 }\hbox {}} \right)$, $\left( {\vec {v_2 },\vec {u_2 }\hbox {}} \right)$ are two solutions that satisfy this constraint. We then prove that vector $\left( {\lambda \vec {v_1 }+\left( {1-\lambda } \right) \vec {v_2 },\lambda \vec {u_1 }+\left( {1-\lambda } \right) \vec {u_2 }} \right)$, is also a solution that satisfy this constraint.

$$\begin{aligned}&\overline{\theta _o } *\left( {\lambda \vec {v_1 }+\left( {1-\lambda } \right) \vec {v_2 }} \right) \vec {x_o }-\left( {\lambda \vec {u_1 }+\left( {1-\lambda } \right) \vec {u_2 }} \right) \vec {y_o } \\&\quad =\lambda *\left( {\overline{\theta _o } *\vec {v_1 }\vec {x_o }-\vec {u_1 }\vec {y_o }} \right) +\left( {1-\lambda } \right) (\overline{\theta _o } *\vec {v_2 }\vec {x_o }-\vec {u_2 }\vec {y_o } \\&\quad =0 \\ \end{aligned}$$

The solution set of this constraint is a convex set, whereas other constraints and object functions are linear. Thus, the partial solution for the model will be the optimal solution. $\square$