Uncertainty in Data Envelopment Analysis

Theorem 6.19

(Mohammad Nejad & Ghaffari-Hadigheh, 2018) The nonlinear model (6.45) is linearized as the below model.

$\begin{array}{l}{q}^{\ast }=max\sum _{r=1}^s{u}_r{c}_{\mathit{ro}}-{\theta }_0\sum _{i=1}^m{v}_i{a}_{\mathit{io}}^{\prime }\\ \text{subject to}\\ \begin{array}{l}\sum _{r=1}^s{u}_r\left(c_{\mathit{ro}}-b_{\mathit{ro}}\right)+{\theta }_o\sum _{i=1}^m{v}_i\left(b_{\mathit{io}}^{\prime }-a_{\mathit{io}}^{\prime }\right)\le 1\\ \sum _{r=1}^s{u}_r{\phi }_{\mathit{rj}}^{-1}\left({\alpha }_j\right)-\sum _{i=1}^m{v}_i{\varphi }_{\mathit{ij}}^{-1}\left(1-{\alpha }_j\right)\le 0& j=1,2,\dots ,n\\ \sum _{r=1}^s{u}_r{c}_{\mathit{ro}}-{\theta }_0\sum _{i=1}^m{v}_i{a}_{\mathit{io}}^{\prime }\ge 0\\ v_i\ge \varepsilon & i=1,2,\dots ,m\\ u_r\ge \varepsilon & r=1,2,\dots ,s\end{array}\end{array}$

  (6.47)

Proof. Assume (β^∗, u^∗, v^∗) as the optimal solution of model (6.45). It is obvious that ${\beta }^{\ast }=min\left\{\frac{1}{2},h\left(u^{\ast },v^{\ast }\right)\right\}$. Similar to Case 1, the below bi-level model is defined.

$\begin{array}{l}maxmin\left\{\frac{1}{2},h\left(u,v\right)\right\}\\ \text{subject to}\\ \begin{array}{ll}\sum _{r=1}^s{u}_r{\phi }_{\mathit{rj}}^{-1}\left({\alpha }_j\right)-\sum _{i=1}^m{v}_i{\varphi }_{\mathit{ij}}^{-1}\left(1-{\alpha }_j\right)\le 0& j=1,2,\dots ,n\\ \sum _{r=1}^s{u}_r{c}_{\mathit{ro}}-{\theta }_0\sum _{i=1}^m{v}_i{a}_{\mathit{io}}^{\prime }\ge 0\\ v_i\ge \end{array}\end{array}$