## 3 Problem Modeling
### Notations  
* Sets and Parameters: \(N\), set of ground-underground network nodes, \(N=\{1,2,3,\ldots\ldots\ldots,n,n+1,\ldots,2n\}\). \(N_{1}\), set of ground network nodes, \(N_{1}=\{1,2,3,\ldots\ldots,n\}\). \(N_{2}\), set of underground candidate nodes, \(N_{2}=\{n+1,n+2,\ldots\ldots,2n\}\). \(B\), budget for underground network construction. \(W\), set of OD pairs. \(M\), total number of OD pairs. \(A\), set of ground-underground network arcs. \(S\), set of Monte Carlo disaster scenarios. \(D_{kl}\), potential demand of the OD pair \(kl\), where \(k\) is the origin node and \(l\) is the destination node. \(cap_{i}\), capacity of the node \(i\). \(cap_{ij}\), capacity of an arc \((i,j)\). \(b_{1}\), fixed cost of the underground arc construction per kilometer. \(b_{2}\), fixed cost of underground node construction. \(d_{ij}\), distance of an arc \((i,j)\). \(a_{ij}\), network-arc indicator (1, if arc \((i,j)\) is in the ground network; 0, otherwise). \(e_{ij}\), network-arc indicator (1, if arc \((i,j)\) is the arc between the ground node and the corresponding underground node; 0, otherwise). * Decision variables: \(X_{i}\), binary variable indicating whether the underground node \  
\[\sum_{i\in N_{2}}b_{1}X_{i}+\sum_{i\in N_{2}}\sum \begin{array}{c}j\in N_{2}\\ i\
eq j\end{array} \;\;b_{2}d_{ij}Y_{ij}\leq B \tag{2}\]  
\[Y_{ij}\leq X_{i},\;\forall i,j\in N_{2},i\
eq j \tag{3}\]  
\[Y_{ij}\leq X_{j},\;\forall i,j\in N_{2},i\
eq j \tag{4}\]  
\[X_{i},\;Y_{ij}\in\{0,1\},\;\forall i,j\in N_{2},i\
eq j \tag{5}\]  
Stage II:  
\[maxZ(s)=\frac{1}{M}\sum \begin{array}{c}j\in N\\ j\
eq k\end{array} \;\;\sum \begin{array}{c}k\in N_{1}\\ k\
eq l\end{array} \;\;\sum_{l\in N_{1}}\;\frac{F_{kj}^{kl}}{D_{kl}} \tag{6}\]  
\[\sum \begin{array}{c}j\in N\\ j\
eq k\end{array} \;\;Z_{kj}^{kl}\leq 1,\;\forall k,l\in N_{1},k\
eq l \tag{7}\]  
\[\sum \begin{array}{c}i\in N\\ i\
eq l\end{array} \;\;Z_{il}^{kl}\leq 1,\;\forall k,l\in N_{1},k\
eq l \tag{8}\]  
\[\sum \begin{array}{c}i\in N\\ i\
eq l\end{array} \;\;Z_{ij}^{kl}\leq 1,\;\forall k,l\in N_{1},k\
eq l, \forall j\in N,j\
eq k,l \tag{9}\]  
\[\sum \begin{array}{c}i\in N\\ i\
eq j,l\end{array} \;\;Z_{ij}^{kl}=\sum \begin{array}{c}i\in N\\ i\
eq j,k\end{array} \;\;Z_{ij}^{kl},\;\forall k,l\in N_{1},k\
eq l,\forall j \in N,j\
eq k,l \tag{10}\]  
\[Z_{ij}^{kl}\leq a_{ij},\;\forall k,l,i,j\in N_{1},k\
eq l,i\
eq j,i\
eq l,j\
eq k \tag{11}\]  
\[Z_{ij}^{kl}\leq Y_{ij},\;\forall\;i,j\in N_{2},i\
eq j,\forall k,l\in N_{1},k \
eq l,i\
eq l+n,j\
eq k+n \tag{12}\]  
\[Z_{ij}^{kl}\leq e_{ij}X_{j},\;\forall i,k,l\in N_{1},\forall j\in N_{2},k\
eq l,i=j-n,i\
eq l \tag{13}\]  
\[Z_{ij}^{kl}\leq e_{ij}X_{i},\;\forall j,k,l\in N_{1},\forall i\in N_{2},k\
eq l,j=i-n,j\
eq k \tag{14}\]  
\[F_{ij}^{kl}\leq cap_{ij}Z_{ij}^{kl},\;\forall k,l\in N_{1},\;k\
eq l \tag{15}\]  
\[\sum \begin{array}{c}i\in N\\ i\
eq j,l\end{array} \;\;F_{ij}^{kl}=\sum \begin{array}{c}i\in N\\ i\
eq j,k\end{array} \;\;F_{ij}^{kl},\;\forall k,l\in N_{1},\forall j\in N,k \
eq l,j\
eq k,j\
eq l \tag{16}\]  
\[\sum \begin{array}{c}j\in N\\ j\
eq k\end{array} \;\;F_{ij}^{kl}\leq D_{kl},\;\forall k,l\in N_{1},k\
eq l \tag{17}\]  
\[\sum \begin{array}{c}k\in N_{1}\\ k\
eq j\end{array} \;\;\sum \begin{array}{c}l\in N_{1}\\ l\
eq k,i\end{array} \;\;F_{ij}^{kl}\leq cap_{i},\;\forall i,j\in N,i\
eq j \tag{18}\]  
\[\sum \begin{array}{c}k\in N_{1}\\ k\
eq j\end{array} \;\;\sum \begin{array}{c}l\in N_{1}\\ l\
eq k,i\end{array} \;\;F_{ij}^{kl}\leq cap_{j},\;\forall i,j\in N,i\
eq j \tag{19}\]  
\[\sum \begin{array}{c}k\in N_{1}\\ k\
eq j\end{array} \;\;\sum \begin{array}{c}l\in N_{1}\\ l\
eq k,i\end{array} \;\;F_{ij}^{kl}\leq cap_{i},\;\forall i,j\in N,i\
eq j\
eq k,i\
eq j\
eq k,i\
eq j\
eq k,i\
eq j\
eq k,i\
eq j\
eq k,i\
eq j\
eq k,i\
eq j \
eq k,i\
eq j \[\sum_{ \begin{array}{c}k\in N_{1}\\ k\
eq l,j\end{array} }\sum_{ \begin{array}{c}l\in N_{1}\\ l\
eq i\end{array}