Uncertainty in Data Envelopment Analysis

5.12: Uncertain SBM model

In this section, the uncertain form of the SBM model is considered. To evaluate DMUs by this uncertain model, two crisp versions are proposed in the rest of this section.

5.12.1: Expected value SBM model

The following expected value crisp form of the uncertain SBM model was proposed by Jamshidi et al. (2021a).

$\begin{array}{l}min\rho =E\left[\frac{1-\frac{1}{m}\sum _{i=1}^m\frac{s_i^{-}}{{\tilde{x}}_{\mathit{io}}}}{1+\frac{1}{s}\sum _{r=1}^s\frac{s_r^{+}}{{\tilde{y}}_{\mathit{ro}}}}\right]\\ \text{subject to}\\ \begin{array}{ll}E\left[\sum _{j=1}^n{\lambda }_j{\tilde{x}}_{\mathit{ij}}-{\tilde{x}}_{\mathit{io}}+s_i^{-}\approx 0\right]& i=1,2,\dots ,m\end{array}\\ \begin{array}{ll}E\left[\sum _{j=1}^n{\lambda }_j{\tilde{y}}_{\mathit{rj}}-{\tilde{y}}_{\mathit{ro}}-s_r^{+}\approx 0\right]& r=1,2,\dots ,s\end{array}\\ \sum _{j=1}^n{\lambda }_j=1\\ s_i^{-}\ge 0i=1,2,\dots ,m\\ s_r^{+}\ge 0r=1,2,\dots ,s\\ {\lambda }_j\ge 0j=1,2,\dots ,n\end{array}$

  (5.58)

Definition 5.20

The vector (s_i⁻, s_r⁺, λ_j) ≥ 0 for i = 1, 2, …, m, r = 1, 2, …, s, and j = 1, 2, …, n is a feasible solution of model