# Optimization of Multimodal Discrete Network Design Problems Based on Super Networks
## 5 Numerical Examples
### Actual Network  
The actual transportation network in this case is an area from Xizhimen to Dongdan in Beijing, China. It is one of the busiest traffic areas in Beijing. The network of this area includes multiple bus, rail transit lines and integrates various travel modes, such as car, rail transit, bus, bicycle, and walking. It has many important transfer hub stations. So it is a typical multimodal transportation network. Moreover, there are many facilities for sharing bicycles near each transfer station, which facilitates passengers' transfer behavior. That is consistent with the model proposed in this paper, which takes bike-sharing as a transfer mode. In addition, due to the huge passenger demand from Xizhimen to Dongdan and the bottleneck of the transportation network, the network capacity is insufficient. Thus, it is necessary to expand or add some links for the car, bus, and rail networks respectively. This is also an important reason why this regional network is selected as an actual case. The multimodal transportation network diagram is as follows, which combined car, bus, and rail transit modes. As shown in Figure 15: the origin is Xizhimen while the destination is Dongdan. The transfer relationship among modes is as follows, among them, the pink nodes are the transfer stations:  
* Firstly, inter-modal transfers between bus and rail transit include 3 routes: (1) bus line 1 \(\rightarrow\) rail line 3; (2) bus line 2 \(\rightarrow\) rail line 3; (3) bus line 3\(\rightarrow\) bus line 4 \(\rightarrow\) rail line 2. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{\(\alpha\)} & \multirow{2}{*}{\(\beta\)} & \multirow{2}{*}{\(\gamma\)} & \multirow{2}{*}{\(\delta\)} & \multirow{2}{*}{\(\theta\)} & \multirow{2}{*}{\(\tau\)} & \multicolumn{3}{c}{**Exact Algorithm**} & \multicolumn{3}{c}{**Heuristic Algorithm**} & \multirow{2}{*}{\(GAP\)} \\ \cline{5-8} \cline{8-10}  & & & & & & & **Optimization** & & & **Scheme** & \\ \hline
0.25 & 0.25 & 0.25 & 0.25 & 0.9 & 0.1 & (1,3), (7,10) & 3962.2 & (1,3), (7,10) & 3962.2 & 0.00\% \\
0.25 & 0.25 & 0.25 & 0.25 & 0.1 & 0.9 & (1,3) & 3111.4 & (1,3) & 3111.4 & (1,3) & 3111.4 & 0.00\% \\
0.4 & 0.4 & 0.1 & 0.1 & 0.9 & 0.1 & (1,3), (7,10) & 4532.3 & (1,3), (7,10) & 4532.3 & 0.00\% \\
0.4 & 0.4 & 0.1 & 0.1 & 0.1 & 0.9 & (1,3) & 3281.2 & (1,3) & 3281.2 & 0.00\% \\
0.6 & 0.2 & 0.15 & 0.05 & 0.5 & 0.5 & (1,3), (3,14) & 4927.3 & (1,3), (3,14) & 4927.3 & 0.00\% \\
0.6 & 0.2 & 0.15 & 0.05 & 0.9 & 0.1 & (1,3),(7,10),(16,17) & 4028.6 & (1,3),(7,10),(3,14) & 4079.2 & 1.26\% \\
0.6 & 0.2 & 0.15 & 0.05 & 0.1 & 0.9 & (1,3) & 3356.4 & (1,3) & 3356.4 & 0.00\% \\
0.2 & 0.6 & 0.05 & 0.15 & 0.5 & 0.5 & (1,3), (16,17) & 3241.7 & (1,3), (16,17) & 3241.7 & 0.00\% \\
0.2 & 0.6 & 0.05 & 0.15 & 0.9 & 0.1 & (1,3),(3,14),(16,17) & 4239.6 & (1,3),(3,14) & 4245.7 & 0.14\% \\
0.2 & 0.6 & 0.05 & 0.15 & 0.1 & 0.9 & (1,3) & 3027.3 & (1,3) & 3027.3 & 0.00\% \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Comparison between heuristic and exact algorithm under different weight coefficients. * Secondly, the inter-route transfers in the bus network include 2 routes: (1) bus line 3 \(\rightarrow\) bus line 5; (2) bus line 3 \(\rightarrow\) bus line 4 \(\rightarrow\) bus line 5. * Moreover, the inter-route transfer in the rail transit network only includes 1 route: rail line 2 \(\rightarrow\) rail line 3. * Besides, the inter-modal transfer between car and rail transit includes 1 route: passengers drive from the origin Xizhimen to node 13 and transfer to rail line 3 by node 13. Before optimization, the maximum feasible flow of the network is 1524, and the traffic demand \(q_{rs}\) is 2000, so it is necessary to improve the network.