Transformation

Resulting deterministic problem ( $P^{\prime }$ )

Approach

references

$T_1$

$T_2$

$D{F}_2$

$D{R}_2$

$\max cx$

$Prob\left[{\displaystyle {\sum }_{j=1}^n}{\overline{a}}_{ij\omega \alpha r}{x}_j\le {\underset{\_}{b}}_{i\omega \alpha r}\right]\ge \beta \mathrm{,}$

$r=1,\cdots ,p;i=1,\cdots ,m;x_j\ge 0,$

$j=1,\cdots ,n$

1

[36]

$D{F}_4$

$D{R}_5$

Find the fuzzy probability

distribution of W and its

expectation given respectively by:

${\tilde{F}}_W\left(x\right)={\displaystyle {\cup }_{\alpha \in \left(\mathrm{0,1}\right]}}\alpha \left[F_{H_{{\alpha }^{+}}}\left(x\right)\mathrm{,}{F}_{H_{{\alpha }^{-}}}\left(x\right)\right]$

${\tilde{E}}_W\left(x\right)={\displaystyle {\cup }_{\alpha \in \left(\mathrm{0,1}\right]}}\alpha \left[E\left(H_{\alpha }^{-}\right)\mathrm{,}E\left(H_{\alpha }^{+}\right)\right]$

2

[37]

$D{F}_2$

$D{R}_3$

$\max \left[cx-E\left({\displaystyle {\sum }_{r=1}^p}{\displaystyle {\sum }_{i=1}^m}{g}_i^{\alpha r}{y}_i^{\alpha r}\left(\omega \right)\right)\right]$

$y_i^{\alpha r}\left(\omega \right)={\displaystyle {\sum }_{j=1}^n}{\overline{a}}_{ij\omega \alpha r}{x}_j+{\underset{\_}{b}}_{i\omega \alpha r}$

3

[38]

$D{F}_4$

$D{R}_4$

Use the random simplex method to

find $x_{\alpha }$ solution of $\max cx$

$\underset{\_}{a}\alpha \left(\omega \right)x\le \underset{\_}{b}\alpha \left(\omega \right)\mathrm{<mo>\; ;\; </mo>}{\overline{a}}_{\alpha }\left(\omega \right)x\le {\overline{b}}_{\alpha }$

$\alpha \in \left(\mathrm{0,1}\right]$ and construct $x^{\ast }={\displaystyle {\cup }_{\alpha \in \left(0,1\right]}}\alpha x_{\alpha }$

4

[39]