Uncertainty in Data Envelopment Analysis

6.5: Analysis and stability of uncertain model (5.20)

To study the stability of uncertain model (5.20), this study is done in two directions of efficient and inefficient DMUs (Lio & Liu, 2018).

6.5.1: Analysis and stability of inefficient DMUs

Assume DMU_o as an inefficient DMU. Lio and Liu (2018) introduced the model for the obtain the maximum value for η such that the new form of this DMU with data of $\left({\tilde{x}}_o+\eta e_m,{\tilde{y}}_o-\eta e_s\right)$ remains efficient, that e_j = (1, 1, …, 1) is a j-dimensional vector with values of 1.

$\begin{array}{l}min\eta \\ \text{subject to}\\ \begin{array}{l}E\left[\frac{\sum _{r=1}^s{u}_r\left({\tilde{y}}_{\mathit{ro}}+\eta \right)}{\sum _{i=1}^m{v}_i\left({\tilde{x}}_{\mathit{io}}-\eta \right)}\right]=1\\ E\left[\frac{\sum _{r=1}^s{u}_r{\tilde{y}}_{\mathit{rj}}}{\sum _{i=1}^m{v}_i{\tilde{x}}_{\mathit{ij}}}\right]\le 1& j=1,2,\dots ,n\\ v_i\ge 0& i=1,2,\dots ,m\\ u_r\ge 0& r=1,2,\dots ,s\end{array}\end{array}$

  (6.26)

Theorem 6.8

(Lio & Liu, 2018 ...