# Optimizing a Reverse Supply Chain Network for Electronic Waste under Risk and Uncertain Factors
## 4 Proposed Approach
### Converting the FMILP Model to the Auxiliary Crisp Model  
[PERSON] et al. 's approach can be effectively applied several kinds of fuzzy numbers such as trapezoidal, triangular, and either linear or nonlinear problems. In addition, this method is generally superior to address fuzzy linear problems because it still maintains the linearity of the original model. As the first step, it is assumed that a triangular fuzzy number \(\widetilde{a}\ =(a_{1},a_{2},a_{3})\) is given, where \(a_{1}\) represents the pessimistic value, \(a_{2}\) is the most likely value, and \(a_{3}\) presents the optimistic value of the triangular fuzzy number. The triangular membership function of \(\widetilde{a}\) can be described as below:  
\[\mu_{\widetilde{a}}(x)=\left\{ \begin{array}{ll}f_{a}(x)=\frac{x-a_{1}}{a_{2 }-a_{1}},&\text{if }a_{1}\leq x\leq a_{2}\\ 1&\text{if }x=a_{2}\\ g_{a}(x)=\frac{a_{3}-x}{a_{3}-a_{2}}&\text{if }a_{2}\leq x\leq a_{3}\\ 0&\text{otherwise}\end{array} \right. \tag{26}\]  
Based on [PERSON]'s [61] approach, \(EI(\widetilde{a})\) is considered as the expected interval of a triangular number \(\widetilde{a}\), and the expected value of a triangular number \(\widetilde{a}\) is denoted as \(EV(\widetilde{a})\) can be computed in Equations (27) and (28). \[EI(\widetilde{a})=[E_{1}^{a},E_{2}^{a}]=\left[\int_{0}^{1}f_{a}^{-1}(x)dx\int_ {0}^{1}g_{a}^{-1}(x)dx\right]=\left[\frac{1}{2}(a_{1}+a_{2}),\frac{1}{2}(a_{2 }+a_{3})\right] \tag{27}\]  
\[EV(\widetilde{a})=\frac{E_{1}^{a}+E_{2}^{a}}{2}=\frac{a_{1}+2a_{2}+a_{3}}{4} \tag{28}\]  
According to [60], if there are two fuzzy numbers \(\widetilde{a}\) and \(\widetilde{b}\), and \(\widetilde{a}\) is bigger than \(\widetilde{b}\), the membership function \(\mu_{M}\) can be defined by:  
\[\mu_{M}\Big{(}\widetilde{a},\widetilde{b}\Big{)}=\left\{ \begin{array}{ll}0& \text{if }E_{2}^{a}\ -E_{1}^{b}\ <0\\ \frac{E_{2}^{a}-E_{1}^{b}-\left(E_{1}^{a}-E_{2}^{b}\right)}{E_{2}^{a}-E_{1}^{b} -\left(E_{1}^{a}-E_{2}^{b}\right)}&\text{if }0\in\left[E_{1}^{a}-E_{2}^{b},E_{2}^{a}-E_{1}^{b} \right]\\ 1&\text{if }E_{1}^{a}-E_{2}^{b}>0\end{array} \right. \tag{29}\]In this case, \(\mu_{M}\left(\widetilde{a},\widetilde{b}\right)\ \geq\ \alpha\) which indicates that \(\widetilde{b}\) is less than \(\widetilde{a}\) at least in a degree \(\alpha\). When \(\widetilde{a}\) and \(\widetilde{b}\) are indifferent in a degree \(\alpha\), it is noted \(\frac{\alpha}{2}\leq\mu_{M}\left(\widetilde{a}\,\ \widetilde{b}\ \right)\leq 1-\frac{ \alpha}{2}\)[62]. Hence, a fuzzy model can be presented as follows:  
\[\text{Min}\ w=\widetilde{c}^{T}x\]  
Subject to  
\[\widetilde{a}_{i}x\geq\widetilde{b}_{i},\ i=1,\ldots,k\] \[\widetilde{a}_{i}x=\widetilde{b}_{i},\ i=k+1,\ldots n \tag{30}\] \[x\geq 0\]  
where \(\widetilde{c}^{T}\) is a fuzzy vector. According to Equations (29) and (30), the two above constraints \(\widetilde{a}_{i}x\geq\widetilde{b}_{i}\) and \(\widetilde{a}_{i}x=\widetilde{b}_{i}\) can be converted into the equivalent constraints, respectively, in Equations (31) and (32). \[\frac{E_{2}^{a_{i}x}-E_{1}^{b_{i}}}{E_{2}^{a_{i}x}-E_{1}^{a_{i}x}+E_{2}^{b_{i} }-E_{1}^{b_{i}}}\geq\alpha,\ i=1,\ldots,k \tag{31}\]  
\[\frac{\alpha}{2}\leq\frac{E_{2}^{a_{i}x}}{E_{2}^{a_{i}x}-E_{1}^{a_{i}x}+E_{2}^ {b_{i}}-E_{1}^{b_{i}}}\leq 1-\frac{\alpha}{2},\ i=k+1,\ldots,n \tag{32}\]  
According to [60], with various feasible solutions of x, the feasible solution \(x_{0}\) is selected based on an \(\alpha\)-acceptable optimal solution of the model of Equation (30) in the case that Equation (33) is achieved:  
\[\widetilde{c}^{T}x\ \geq\ \frac{1}{2}\widetilde{c}^{T}x_{0} \tag{33}\]  
Finally, the fuzzy model in Equation (30) can be converted into a crisp \(\alpha\)-parametric model as follows:  
\[