# Optimization of Multimodal Discrete Network Design Problems Based on Super Networks
## 5 Numerical Examples
### Actual Network
According to the passenger demand, the candidate links are given, as shown by the dotted line in Figure 15: for the car network, the blue dotted links (1,2), (2,3) represent the expansion optimization, and the blue dotted link (3,7) is the candidate link for the adding optimization, namely \(U^{c}\)=((1,2), (2,3), (3,7)); for the bus network, the green dotted links (12,13), (15,16), (17,20) are the candidate links for adding optimization, namely \(U^{p}\)=((12,13), (15,16), (17,20)); for the rail transit network, the red dotted link (6,7) is the candidate link for adding optimization, namely \(U^{r}=\) ((6,7)). Therefore, it can be concluded that the set of candidate schemes \(U=\{U^{c},U^{r},U^{p}\}=\) {(1,2), (2,3), (3,7), (12,13), (15,16), (17,20), (6,7)}, so there are totally \(2^{7}\) schemes. Next, we use the heuristic algorithm to find the near-optimal scheme from the \(2^{7}\) schemes. The multimodal transportation super network topology is built and the link information of each network is given to calculate the generalized cost of each type of links, as shown in Figure 16: in brackets, the first term is the generalized cost of links and the second term is capacity of links. Finally, the proposed heuristic algorithm will be coded by PYTHON for the numerical test based on Section 4, which combines the minimum cost flow algorithm and simulated annealing algorithm. And we set the initial temperature \(t_{max}=1000\), the end temperature \(t_{end}=100\), the maximum number of iterations \(M=200\). To analyze the optimal schemes under different weight coefficients, 12 groups of numerical experiments are carried out. The near-optimal schemes under different weight coefficients are shown in the table below. Figure 15: Multimodal transportation network from Xizhimen to Dongdan. It can be concluded from Table 9 that the choice of the near-optimal scheme is affected by the weight coefficients:  
1. When the value of _ff_ is larger than _fi_ in Equation (1), the travel time is the main priority, and passengers are more willing to choose the car mode that can reach the destination quickly. In this case, when _ff_ = 0.6 (group 8,9,10), the car's candidate links (1,2), (2,3) are all adopted in various groups, which are consistent with the choice of passengers. On the contrary, the monetary cost is focused on, and passengers prefer to choose the transportation mode that can save a lot of money. While _fi_ = 0.6 (group 11,12,13), the bus's candidate links (12,13), (15,16), (17,20) are all adopted in various groups, which are also tally with the actual situation
2. When the value of'is larger than \(o\) in Equation (23), the decision-maker pays more attention to the network operation cost. In this case, when'= 0.9 (group 2,5,8,11), the car's candidate links (1,2), (2,3) are the principal part of the near-optimal scheme under different groups. Conversely, the construction cost is mainly considered. While \(o\) = 0.9 (group 3,6,9,12), the bus's candidate links (12,13), (15,16), (17,20) are the main component of the near-optimal scheme in various groups. These are consistent with the actual situation, which proves that the developed heuristic algorithm can efficiently find the near-optimal solution with good stability and reliability. Therefore, in an actual operation situation, the value of each parameter can be set according to the attention of the decision-maker, for example, when the travel time is more important, the value of \(a\) is larger than \(b\), otherwise, the monetary cost is mainly considered; when decision-maker pays more attention to the operation cost, the value of \(o\) is larger than \(t\), conversely, the construction cost is principal considered. Consequently, the good results of the two numerical tests are obtained in this section, which shows that the proposed model and developed algorithm in this paper are feasible and efficient. Figure 16: The generalized costs and capacity of each link in the super network topology.