# International Airline Alliance Network Design with Uncertainty
## 4 Case Study
### Computational Results  
In this section, we present a computational analysis of the robust optimization model of the alliance route network design problem in order to assess the effects of uncertainty in the unit transportation cost on the resulting solutions. We assumed that the number of international gateways selected from the two areas was three. The collection and distribution discount factors were taken to be equal to one; i.e., \(\chi=\delta=1\). For the alpha value, on the other hand, we let \(\alpha\in{0.2,0.4,0.6,0.8}\), as is customarily done in the literature. We solved the problem separately for each scenario with the goal of minimizing the total transportation cost and solving the robust model based on the nine scenarios. The results of the optimal international gateway locations and the total transportation costs are presented in Table 4.  
Note that in robust solutions, the total transportation costs are different in each scenario; hence, the values of the \"Trans. Costs\" are left empty. Observe in Table 4 that, for each scenario, the optimal international gateway locations are not always the same when the alpha value changes. For instance, the solution of the base case suggests locating tthe international gateways at Nodes 3, 12, and 14 for \(\alpha=0.2\), Nodes 1, 12, and 14 for \(\alpha=0.4\) \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{3}{c}{\(\alpha=0.2\)} & \multicolumn{2}{c}{\(\alpha=0.4\)} & \multicolumn{2}{c}{\(\alpha=0.6\)} & \multicolumn{2}{c}{\(\alpha=0.8\)} \\ \hline \multirow{3}{*}{**Intl. GL**} & \multicolumn{2}{c}{**Trans. Costs**} & \multirow{3}{*}{**Intl. GL**} & \multicolumn{2}{c}{**Trans. Costs**} & \multirow{3}{*}{**Intl. GL**} & \multicolumn{2}{c}{**Trans. Costs**} & \multirow{3}{*}{**Intl. GL**} & \multicolumn{2}{c}{**Trans. Costs**} \\ \cline{1-1} \cline{5-10}  & & & & & & & \\ \hline s1 & 3, 12, 14 & 223,129,610 & 1, 12, 14 & 365,549,030 & 1, 12, 14 & 506,512,065 & 1, 13, 14 & 647,009,456 \\ \hline s2 & 3, 12, 14 & 267,755,532 & 1, 12, 14 & 438,658,836 & 1, 12, 14 & 607,814,479 & 1, 13, 14 & 776,411,347 \\ \hline s3 & 3, 12, 14 & 178,503,688 & 1, 12, 14 & 292,439,224 & 1, 12, 14 & 405,209,652 & 1, 13, 14 & 517,607,565 \\ \hline s4 & 3, 12, 14 & 208,191,272 & 1, 12, 14 & 325,888,407 & 1, 12, 14 & 438,658,836 & 1, 12, 14 & 551,429,264 \\ \hline s5 & 1, 12, 14 & 241,491,590 & 1, 12, 14 & 410,647,234 & 1, 12, 14 & 579,766,838 & 1, 14, 15 & 743,667,894 \\ \hline s6 & 3, 13, 14 & 260,392,325 & 1, 13, 14 & 429,436,112 & 1, 13, 14 & 596,487,252 & 1, 13, 14 & 763,538,391 \\ \hline s7 & 3, 12, 14 & 183,698,164 & 3, 12, 14 & 297,876,805 & 1, 12, 14 & 410,647,234 & 1, 12, 14 & 523,417,662 \\ \hline s8 & 3, 13, 14 & 201,692,264 & 1, 13, 14 & 318,068,686 & 1, 13, 14 & 429,436,112 & 1, 13, 14 & 540,803,538 \\ \hline s9 & 1, 12, 14 & 236,054,009 & 1, 12, 14 & 405,209,652 & 1, 14, 15 & 572,475,719 & 1, 14, 15 & 736,129,225 \\ \hline RM & 3, 12, 14 & 1, 12, 14 & 1, 12, 14 & 1, 12, 14 & 1, 12, 14 & 1, 12, 14 & 1, 12, 14 \\ \hline \hline \end{tabular} Note: s1–s9: scenario number; RM: robust optimization model; Intl. GL: optimal international gateway locations; Trans. costs: total transportation costs in the respective optimal solutions.  
\end{table} 
Table 3: Scenarios. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Scenario** & **CASK-SH** & **CASK-MH** & **CASK-LH** \\
**Number** & **Deviation (\%)** & **Deviation (\%)** & **Deviation (\%)** \\ \hline
1 & 0 & 0 & 0 \\ \hline
2 & 20 & 20 & 20 \\ \hline
3 & \(-20\) & \(-20\) & \(-20\) \\ \hline
4 & 20 & 20 & \(-20\) \\ \hline
5 & 20 & \(-20\) & 20 \\ \hline
6 & \(-20\) & 20 & 20 \\ \hline
7 & 20 & \(-20\) & \(-20\) \\ \hline
8 & \(-20\) & 20 & \(-20\) \\ \hline
9 & \(-20\) & \(-20\) & 20 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Results with the uncertain unit transportation costs. and \(\alpha=0.6\), and Nodes 1, 13, and 14 for \(\alpha=0.8\). This proves that the optimal solutions are sensitive to the alpha value.