# A Method to Optimize Routing Paths for City-Pair Airlines on Three-Layer Air Transport Networks
## 4 Experiments
### Optimization Results  
In order to demonstrate the effectiveness of the proposed routing path allocation method, we compare it with three other strategies: the \"\(Default\)\" routing path, the \"_Shortest_\" routing path, and the \"\(Random\)\" routing path.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameters** & **2012**[24] & **2015** & **2018** & **2019** \\ \hline Nodes & 147 & 198 & 227 & 233 \\ \hline Edges & 1055 & 2657 & 3919 & 4498 \\ \hline Average degree & 14.35 & 16.83 & 22.52 & 24.98 \\ \hline Average shortest path length & 2.20 & 2.29 & 2.29 & 2.25 \\ \hline Diameter & 4 & 3 & 3 & 3 \\ \hline Mixing coefficient & \(-0.15\) & \(-0.45\) & \(-0.38\) & \(-0.36\) \\ \hline Clustering coefficient & 0.79 & 0.62 & 0.55 & 0.58 \\ \hline \hline \end{tabular}
\end{table}
Table 2: A comparison between network parameters in airline networks in different years.  
1. _\"Default\"_, which takes the routing path used by air traffic managers in practice;
2. _\"Shortest\"_, which takes the shortest length routing path from the candidate set \(S_{crp}^{i,j}\) for airline \(l_{i,j}\);
3. _\"Random\"_, which takes a routing path for airline \(l_{i,j}\) from its candidate set \(S_{crp}^{i,j}\);
4. _\"Proposed\"_, which is the proposed simulated annealing-based method to select the optimal routing paths.  
Table 3 shows the traffic flows (the number of airlines) at the most heavily crowded air route segments in different time intervals using the method of different routing paths. Compared with the default routing paths, the shortest routing paths, and the random routing paths, the proposed routing-paths-optimization strategy can reduce the maximum traffic flows of the air route network by 2.4%, 4.6%, and 4.8%, respectively, which shows that the proposed optimization method has a particularly positive effect in alleviating the traffic congestion of air route network. This improvement is mainly due to the cost function 2, which takes both elements (traffic flow and flight distance) into consideration in a well-balanced way. However, other solutions (_\"Default\"_, _\"Shortest\"_ and _\"Random\"_) only cover some of these elements. _\"Default\"_ is derived from the flight schedule, which is the baseline. _\"Shortest\"_ only considers the flight distance of one aircraft, while _\"Random\"_ selects randomly from potential routes.  
To further demonstrate the effectiveness of the proposed optimization method, we also provide the first ten air route segment traffic flows in time interval 9:00-10:00 under different routing paths strategies, as shown in Figure 6. As shown in Figure 6, the proposed method could reduce the first ten traffic flows of air route segments on the whole, such that the congestion of some air route segments could be alleviated. In time interval 9:00-10:00, the maximum traffic flows of air route segment in the proposed method could be reduce the number of flights by two, four, and four, respectively, when compared with _\"Default\"_, _\"Shortest\"_ and _\"Random\"_.  
Figure 6: The first ten traffic flows of an air route network in the time interval 9:00–10:00 under different routing-path strategies.  
In addition, Figure 7 shows the distributions of traffic flow on air route network for different time intervals, where the routing paths for airlines were selected based on the proposed simulated-annealing method.  
\begin{table}
\begin{tabular}{c c c c c} \hline \multirow{2}{*}{**Time Interval**} & \multicolumn{4}{c}{**Maximum Traffic Flows**} \\ \cline{2-5}  & \(Default\) & \(Shortest\) & \(Random\) & \(Proposed\) \\ \hline  
\multirow{4}{*}{_Average_} & \multirow{2}{*}{33.08} & \multirow{2}{*}{33.88} & \multirow{2}{*}{33.96} & \multirow{2}{*}{32.33} \\ \cline{2-2}  & & & & \\ \cline{1-1} \cline{2-2}  & & & & \\ \cline{1-1} \cline{2-2}  & & & & \\ \cline{1-1} \cline{2-2}  & & & & \\ \hline \end{tabular}
\end{table}
Table 3: The maximum traffic flows of air route segments in different time intervals using different routing path methods.