## 3 Problem Description and Modeling
### Bilevel Chance Constrained Programming Model  
According to the fuzzy chance constrained programming model [30], given confidence levels \(\alpha^{u}\) and \(\alpha^{l}\), the above type-II fuzzy double-objective optimization model can be transformed into a bilevel chance constrained programming model.  
According to the upper level membership function \((r_{ij1}^{u},r_{ij2}^{u},r_{ij3}^{u},r_{ij4}^{u},\omega_{ij}^{u})\) of the allocation risk, the upper level chance constrained programming model can be obtained as follows:  
\[\left\{\begin{array}{l}\text{ming},\text{min}f^{u}\\ s.t.\\ \\ \text{C}r\left\{\sum\limits_{t=1}^{T}\sum\limits_{i=1}^{N}\sum\limits_{j=1}^{N }\widetilde{f}_{{t}{i}{j}}\leq f^{u}\right\}\geq\alpha^{u}\\ \\ \text{Equations (\ref{eq:1})--}\eqref{eq:1}\end{array}\right. \tag{13}\]  
Similarly, according to the lower level membership function \((r_{ij1}^{u},r_{ij2}^{u},r_{ij3}^{u},r_{ij4}^{u},\omega_{ij}^{u})\), the lower level chance constrained programming model can be obtained as follows:  
\[\left\{\begin{array}{l}\text{ming},\text{min}f^{u}\\ s.t.\\ \\ \text{C}r\left\{\sum\limits_{t=1}^{T}\sum\limits_{i=1}^{N}\sum\limits_{j=1}^{N }\widetilde{f}_{{t}{i}{j}}\leq f^{u}\right\}\geq\alpha^{u}\\ \\ \text{Equations (\ref{eq:1})--}\eqref{eq:1}\end{array}\right. \tag{14}\]Next, according to the confidence equivalent formula of the interval type-II fuzzy model [31], the above opportunity constraints can be converted as follows:  
\[\begin{array}{l}Cr\left\{\sum\limits_{t=1}^{T}\sum\limits_{i=1}^{N}\sum\limits_{ j=1}^{N}\left(r_{ij}{}^{u}\times\sum\limits_{k=1}^{K}x_{ij}{}^{k}\right)\leq f ^{u}\right\}\geq\alpha^{u}\Rightarrow\sum\limits_{t=1}^{T}\sum\limits_{i=1}^{N }\sum\limits_{j=1}^{N}\left(h_{ij}{}^{u}\times\sum\limits_{k=1}^{K}x_{ij}{}^{ k}\right)\leq f^{u}\\ Cr\left\{\sum\limits_{t=1}^{T}\sum\limits_{i=1}^{N}\sum\limits_{j=1}^{N} \left(r_{ij}{}^{l}\times\sum\limits_{k=1}^{K}x_{ij}{}^{k}\right)\leq f^{l} \right\}\geq\alpha^{l}\Rightarrow\sum\limits_{t=1}^{T}\sum\limits_{i=1}^{N} \sum\limits_{j=1}^{N}\left(h_{ij}{}^{l}\times\sum\limits_{k=1}^{K}x_{ttj}{}^{ k}\right)\leq f^{l}\end{array} \tag{15}\]  
where, \(Cr\{\)A\(\}\) represents the degree of confidence of event A, \(h_{ij}{}^{u}\) and \(h_{ij}{}^{l}\) are defined as follows.  
\[\begin{array}{l}h_{ij}^{u}=\left\{\begin{array}{l}\frac{(\omega_{ij}^{u}-2 \alpha^{u})r_{ij1}^{u}+2\alpha^{u}r_{ij2}^{u}}{\omega_{ij}^{u}},if\;\alpha^{u} \leq\frac{\omega_{ii}^{u}}{2}\\ \frac{2(\omega_{ij}^{u}-2\alpha^{u})r_{ij1}^{u}+(2\alpha^{u}-\omega_{ij}^{u})r_ {ij4}^{u}}{\omega_{ij}^{u}},if\;\alpha^{u}>\frac{\omega_{ii}^{u}}{2}\end{array}\right. \end{array} \tag{16}\]  
\[\begin{array}{l}h_{ij}^{l}=\left\{\begin{array}{l}\frac{(\omega_{ij}^{l}-2 \alpha^{l})r_{ij1}^{l}+2\alpha^{l}r_{ij2}^{l}}{\omega_{ij}^{l}},if\;\alpha^{l} \leq\frac{\omega_{ij}^{l}}{2}\\ \frac{2(\omega_{ij}^{l}-2\alpha^{l})r_{ij1}^{l}+(2\alpha^{l}-\omega_{ij}^{l})r_ {ij4}^{l}}{\omega_{ij}^{l}},if\;\alpha^{l}>\frac{\omega_{ij}^{l}}{2}\end{array}\right. \end{array} \tag{17}\]  
Combining Equations (13)-(17), the above bilevel chance constrained programming model can be transformed as follows with the weighted method.  
\[\left\{\begin{array}{l}\text{ming}\\ \text{min}f_{r}=\frac{1}{2}\sum\limits_{t=1}^{T}\sum\limits_{i=1}^{N}\sum \limits_{j=1}^{N}\left((h_{ij}{}^{u}+h_{ij}{}^{l})\times\sum\limits_{k=1}^{K}x _{tij}{}^{k}\right)\\ s.t.\\ \text{Equations (\ref{eq:11})--(\ref{eq:12})}\end{array}\right. \tag{18}\]