# International Airline Alliance Network Design with Uncertainty
## 4 Case Study
### Computational Results
In addition, the magnitude of the total transportation costs increases with the increase in the alpha value. For each alpha value, the optimal international gateway locations may be different in each of the scenarios. For example, when \(\alpha=0.8\), although Node 1 and Node 14 were selected as international gateways in all of the scenarios, the third gateway was not always the same. Specifically, the optimal solution in Scenarios 1, 2, 3, 6, and 8 was \(\{1,13,14\}\), while in Scenarios 4 and 7, it was \(\{1,12,14\}\), and in Scenarios 5 and 9, it was \(\{1,14,15\}\). If an alliance route network is constructed based on the optimal solution in the base case, the international gateways are PEK, DTW, and LAX, which will be different from the optimal gateways PEK, JFK, and LAX in Scenarios 4 and 7 and the optimal gateways PEK, LAX, and MSP in Scenarios 5 and 9. This suggests that the international gateways selected for the base case will no longer be optimal in some other scenarios. In general, when only the cost data in a certain scenario are optimal, the solutions might not be optimal in other scenarios, and may even deviate significantly from the original optimal solutions. We can see from Table 4 that the optimal solutions of the robust model are different from the solutions obtained in some scenarios. In order to illustrate that the robust solution can adapt to many possible scenarios with different cost parameters, we took \(\alpha=0.8\) as an example and calculated the relative deviation of the robust solution in each scenario. The results are shown in Figure 2.  
The calculation results show that in each scenario, the relative deviation of the robust solution was small. Even in the worst case, such as Scenario 8 (\(-20\%\), \(20\%\), \(-20\%\)), the relative deviation was \(1.68\%\). In other words, even if the robust solution \(\{1,12,14\}\) differed from the optimal solution in Scenario 8, the relative deviation from the total cost of the optimal solution\(\{1,13,14\}\) did not exceed \(2\%\). In addition, we found that even though the robust solution was the same as the optimal solutions of Scenarios 4 and 7, there was a certain deviation in the total transportation cost. This is due to the differences in transport paths and flow volume on each path. We conclude that, since the optimal solutions are sensitive to the unit transportation costs, with high amounts of uncertainty in the unit transportation costs, it is better to adopt the solution obtained with the robust model instead of adopting the solution of a particular scenario. This proves that the model presented in Section 3.2 can be used to realize transportation through the alliance routes and reduce the impact of cost uncertainty on the design of an alliance route network. Taking \(\alpha=0.8\) as an example, the international gateways chosen according to the robust solution are PEK, JFK, and LAX. Connections between the city airports can be seen in Figure 3.  
Figure 2: Relative deviations. The dotted lines indicate the connections between \"spokes\" and international gateways, and the solid lines indicate the connections between the international gateways. PEK was selected as the gateway of China, and JFK and LAX were selected as the gateways of the US. Thus, passengers across the two areas can be transported through the links between the international gateways. To evaluate the impact of the number of international gateways on the robust solutions, we calculated the optimal solutions for different \(p\) values according to different alpha values, as shown in Table 5.  
As can be seen from Table 5, for different alpha values, the frequency of each gateway selection is not the same. In our case, for example, when \(\alpha=0.6\) or \(\alpha=0.8\), the frequency of gateway selection is \(1,14>12>3>15\). With the increase in the number of international gateways, the total transportation cost of the alliance network continues to decrease.