# International Airline Alliance Network Design with Uncertainty
## 3 Airline Alliance Network
### Optimal Solution  
The robust optimization of the alliance route network design model with uncertainty is to find minimum \(\lambda\) satisfying the constraint condition and route network design plan (selection of gateway and route, arrangement of OD flow) of minimum \(\lambda\) avoiding risks to the maximum extent. The solution principle is to give a relatively small constant value of \(\lambda\) (\(\lambda\) can be solved by adding iterative calculation.), calculate the model (11) of each scenario with different gateway combination, and update \(\lambda\) to approach the minimum value of \(\lambda\) which the network design is continuously improved.  
The model of each scenario with different gateway combination can be formulated.  
\[Z(s)=min\sum_{\alpha\in A}\left[\sum_{i\in N^{\alpha}}\left(\sum_{k\in H^{\alpha}} \chi(s)C_{ik}Z_{ik}(s)+\sum_{k\in H^{\alpha}}\sum_{m\in H\backslash H^{\alpha}}\right.\right. \tag{11}\]  
which is subject to:  
\[\sum_{\alpha\in A}\sum_{k\in H^{\alpha}}h_{k}(s)=p \tag{12}\]  
\[\sum_{k\in H^{\alpha}}Z_{ik}(s)=O_{i}(s);\forall i\in N^{\alpha},\alpha\in A \tag{13}\]  
\[\sum_{m\in H^{(i)}}X^{i}_{mj}(s)=W_{ij}(s);\forall i\in N^{\alpha},j\in N \backslash N^{\alpha},\alpha\in A \tag{14}\]  
\[\sum_{m\in H\backslash H^{\alpha}}Y^{i}_{km}(s)=Z_{ik}(s);\forall i\in N^{ \alpha},k\in H^{\alpha},\alpha\in A \tag{15}\]  
\[\sum_{j\in N^{(m)}}X^{i}_{mj}(s)=\sum_{k\in H^{\alpha}}Y^{i}_{km}(s);\forall i \in N^{\alpha},m\in H\backslash H^{\alpha},\alpha\in A \tag{16}\]  
\[Z_{ik}(s)\leq O_{i}(s)h_{k}(s);\forall i\in N^{\alpha},k\in H^{\alpha},\alpha\in A \tag{17}\]  
\[X^{i}_{mj}(s)\leq W_{ij}(s)h_{m}(s);\forall i\in N^{\alpha},m\in H\backslash H ^{\alpha},j\in N^{(m)},\alpha\in A \tag{18}\]  
\[Z_{ik}(s),Y^{i}_{km}(s),X^{i}_{mj}(s)\geq 0; \tag{19}\]  
\[\forall i\in N^{\alpha},k\in H^{\alpha},m\in H\backslash H^{\alpha},j\in N^{( m)},\alpha\in A\]  
\[h_{k}(s)\in\{0,1\},\forall k\in H^{\alpha},\alpha\in A \tag{20}\]  
The proposed solver is presented as follows in Algorithm 1.  
```
1: Initialization \(Z^{*}(s)=+\infty\), \(s=1,\dots,S\), \(\lambda=T\), \(T\) is a small positive number.
2: Select \(p\) regional hubs as international gateways. The possible combinations corresponding to \(p\) international gateways are \(C^{i^{\prime}}_{H^{\alpha}}\times C^{b^{\prime}}_{H^{\beta}}\times C^{{}^{ \prime}}_{H^{\alpha}}\times\dots\), one of the combination is represented by \(H\), \(H^{\alpha}\) represents the set of optional hubs in area \(a\). \(\{a,b,c,\dots\}=A\), \(a^{{}^{\prime}}+b^{{}^{\prime}}+c^{{}^{\prime}}+\dots=p\), \(a^{{}^{\prime}},b^{{}^{\prime}},c^{{}^{\prime}}\geq 1\).
3: For the combination \(H\), use the model (11) to find the solution under scenario \(s\), \(s=1,\dots,S\), obtain the total transportation cost \(Z(H,s)\).
4: For scenario \(s\), if \(Z(H,s)\geq Z^{*}(s)\), turn to Step 5, else \(Z(H,s)<Z^{*}(s)\), renew \(Z^{*}(s)=Z(H,s)\), turn to Step 5.
5: If \(\frac{Z(H,s)-Z^{*}(s)}{Z^{*}(s)}\leq T\) for all \(s=1,\dots,S\), output \(\lambda=T\), \(H^{*}=H\), \(H^{*}\) is the optimal international gateway combination, else turn to Step 6.
6: if \(\frac{Z(H,s)-Z^{*}(s)}{Z^{*}(s)}\geq T\) for some s, turn to Step 2.
```  
**Algorithm 1** An Iterative Optimization Algorithm for the Alliance Route Network