# International Airline Alliance Network Design with Uncertainty
## 3 Airline Alliance Network
### Alliance Route Network Model  
Sets, parameters, and decision variables are introduced before describing the mathematical formulation. Sets and parameters:  
\(A\) is a set of international areas that need to establish alliances with airlines; \(a\in A\). \(N\) is a set of all nodes in the network; \(i\in N\), \(j\in N\). \(N^{a}\) is a set of nodes in the network of area \(a\in A\); \(N^{a}\subseteq N\). \(H\) is a set of all regional hubs in the network; \(r\in H\), \(k\in H\), \(m\in H\), \(t\in H\), \(H\subseteq N\). \(H^{a}\) is a set of regional hubs within area \(a\in A\); \(H^{a}\subseteq H\). \(S\) is a set of scenarios for the uncertain transportation cost; \(s\in S\). \(W_{ij}\) is the passenger demand from origin \(i\in N^{a}\) to destination \(j\in N\backslash N^{a}\). \(O_{i}\) is the total traffic flow starting from node \(i\in N^{a}\); \(O_{i}=\sum_{j\in N\backslash N^{a}}W_{ij}\). \(C_{ik}(s)\) is the unit transportation cost from \"spoke\" \(i\in N^{a}\) to international gateway \(k\in H^{a}\) under scenario \(s\in S\). \(C_{km}(s)\) is the unit transportation cost from international gateway \(k\in H^{a}\) to international gateway \(m\in H\backslash H^{a}\) under scenario \(s\in S\). \(C_{mj}(s)\) is the unit transportation cost from international gateway \(m\in H\backslash H^{a}\) to \"spoke\" \(j\in N^{(m)}\) under scenario \(s\in S\), where \(N^{(m)}\) denotes the set of nodes in the area containing gateway \(m\in H\backslash H^{a}\). Decision variables:  
\(h_{k}\) is 1 if an international gateway is located at node \(k\in H^{a}\), and 0 otherwise. \(Z_{ik}\) is the total amount of flow from regional hub \(i\in N^{a}\) to international gateway \(k\in H^{a}\). \(Y_{km}^{i}\) is the amount of flow from regional hub \(i\in N^{a}\) to international gateway \(k\in H^{a}\) that arrives at the gateway \(m\in H\backslash H^{a}\). \(X_{mj}^{i}\) is the amount of flow from regional hub \(i\in N^{a}\) to node \(j\in N\backslash N^{a}\) that goes through the international gateway \(m\in H\backslash H^{a}\). \(Z^{*}(s)\) is the minimum total transportation cost of an alliance network constructed under scenario \(s\). Given the uncertain transportation effects, an alliance route network model is proposed as:  
\(min\;\lambda\),  
which is subject to:\[\sum_{\alpha\in A}\left[\sum_{i\in N^{\alpha}}\left(\sum_{k\in H^{ \alpha}}\chi\mathcal{C}_{ik}(s)Z_{ik}+\sum_{k\in H^{\alpha}}\sum_{m\in H\backslash H ^{\alpha}}\alpha\mathcal{C}_{km}(s)Y^{i}_{km}+\right.\right. \tag{1}\] \[\left.\sum_{m\in H\backslash H^{\alpha}}\sum_{j\in N^{(m)}}\delta \mathcal{C}_{mj}(s)X^{i}_{mj}\right)\right]\leq(1+\lambda)Z^{*}(s),\forall s\in S\] \[\sum_{\alpha\in A}\sum_{k\in H^{\alpha}}h_{k}=p\] (2) \[\sum_{k\in H^{\alpha}}Z_{ik}=\mathcal{O}_{i};\forall i\in N^{ \alpha},\alpha\in A\] (3) \[\sum_{m\in H^{(i)}}X^{i}_{mj}=W_{ij};\forall i\in N^{\alpha},j\in N \backslash N^{\alpha},\alpha\in A\] (4) \[\sum_{m\in H\backslash H^{\alpha}}Y^{i}_{km}=Z_{ik};\forall i\in N ^{\alpha},k\in H^{\alpha},\alpha\in A\] (5) \[\sum_{j\in N^{(m)}}X^{i}_{mj}=\sum_{k\in H^{\alpha}}Y^{i}_{km}; \forall i\in N^{\alpha},m\in H\backslash H^{\alpha},\alpha\in A\] (6) \[Z_{ik}\leq\mathcal{O}_{i}h_{k};\forall i\in N^{\alpha},k\in H^{ \alpha},\alpha\in A\] (7) \[X^{i}_{mj}\leq W_{ij}h_{mi};\forall i\in N^{\alpha},m\in H \backslash H^{\alpha},j\in N^{(m)},\alpha\in A\] (8) \[Z_{ik},Y^{i}_{km},X^{i}_{mj}\geq 0;\] (9) \[\left.\sum_{m\in H\backslash H^{\alpha}}\sum_{j\in N^{(m)}}\delta \mathcal{C}_{mj}(s)X^{i}_{mj}\right)\right]\leq(1+\lambda)Z^{*}(s),\forall s\in S\]0 \[\left.\sum_{m\in H\backslash H^{\alpha}}\sum_{j\in N^{(m)}}\delta \mathcal{C}_{mj}(s)X^{i}_{mj}\right)\right]\leq(1+\lambda)Z^{*}(s),\forall s\in S\]1  
Formula (1) is the requirement of relatively robust optimization, that is, for each design of the alliance route network, the relative deviation between the total transportation cost and the optimal transportation cost under each scenario is calculated, and the maximum relative deviation must be minimized.