Uncertainty in Data Envelopment Analysis

Corollary 5.10

Considering linear regular uncertain distribution such as ℒ(a, b) for the uncertain variables of Theorem 5.6, the following deterministic model is obtained instead of the model (5.29).

$\begin{array}{l}\underset{\gamma ,u_r,v_i}{max}\sum _{i=1}^m{\overline{p}}_i+\sum _{r=1}^s{\overline{w}}_r\\ \text{subject to}\\ \sum _{i=1}^m{p}_i+\sum _{r=1}^s{w}_r=1\\ \sum _{i=1}^m{\overline{p}}_i\left(b_{{\tilde{x}}_{\mathit{io}}}-a_{{\tilde{x}}_{\mathit{io}}}\right)-\sum _{r=1}^s{\overline{w}}_r\left(a_{{\tilde{y}}_{\mathit{ro}}}-b_{{\tilde{y}}_{\mathit{ro}}}\right)+\sum _{i=1}^m{p}_i{a}_{{\tilde{x}}_{\mathit{io}}}-\sum _{r=1}^s{w}_r{b}_{{\tilde{y}}_{\mathit{ro}}}\le 0\\ \begin{array}{l}\sum _{r=1}^s{w}_r\left(\left(1-\alpha \right)a_{{\tilde{y}}_{\mathit{rj}}}+\alpha b_{{\tilde{y}}_{\mathit{rj}}}\right)\le \sum _{i=1}^m{p}_i\left(\alpha a_{{\tilde{x}}_{\mathit{ij}}}+\left(1-\alpha \right)b_{{\tilde{x}}_{\mathit{ij}}}\right)j=1,2,\dots ,n\end{array}\\ \begin{array}{l}0<{\overline{p}}_i\le p_{i}i=1,2,\dots ,m\end{array}\\ \begin{array}{l}0<{\overline{w}}_r\le w_{r}r=1,2,\dots ,s\end{array}\end{array}$

  (5.31)

Proof. According to the inverse of the linear uncertain distribution, the following form of model (5.29) is obtained.

$\begin{array}{l}\underset{\gamma ,u_r,v_i}{\theta =max}\gamma \\ \text{subject to}\\ \sum _{i=1}^m{v}_i\left(\gamma \left(b_{{\tilde{x}}_{\mathit{io}}}-a_{{\tilde{x}}_{\mathit{io}}}\right)+a_{{\tilde{x}}_{\mathit{io}}}\right)-\sum _{r=1}^s{u}_r\left(\gamma \left(a_{{\tilde{y}}_{\mathit{ro}}}-b_{{\tilde{y}}_{\mathit{ro}}}\right)+b_{{\tilde{y}}_{\mathit{ro}}}\right)\le 0\\ \begin{array}{l}\sum _{r=1}^s{u}_r\left(\left(1-\alpha \right)a_{{\tilde{y}}_{\mathit{rj}}}+\alpha b_{{\tilde{y}}_{\mathit{rj}}}\right)\le \sum _{i=1}^m{v}_i\left(\alpha a_{}\right)\end{array}\end{array}$