# Optimization of Multimodal Discrete Network Design Problems Based on Super Networks
## 5 Numerical Examples
### Simple Network  
Before optimization, the maximum feasible flow of the network is 78, and the traffic demand \(q_{rs}\) is 100, so it is necessary to improve the network. According to the passenger demand, some candidate links are given, as shown in Figure 12: the blue dotted links (3,14), (7,10) are the candidate links for the car network, namely \(U^{c}\)={(3,14), (7,10)}; the green dotted link (16,17) is the candidate link for the bus network, namely \(U^{p}\)={(16,17)}; the red dotted link (1,3) is the candidate link for the rail transit network, namely \(U^{r}\)={(1,3)}. Therefore, it can be concluded that the set of candidate schemes \(U=\{U^{c},U^{r},U^{p}\}\)= {(3,14), (7,10), (16,17), (1,3)}, so there are in total \(2^{4}\) schemes. Next, we  
Figure 11: The flow chart of the multimodal DNDP solution algorithm. will use the exact algorithm and heuristic algorithm respectively to find the near-optimal scheme \(U^{*}\) from the \(2^{4}\) schemes. (1) Exact algorithm  
To test the feasibility of the developed heuristic algorithm, an exact algorithm is first used to solve the model. The specific steps are as follows:  
_Step 1:_ Calculate the generalized cost of each type of links  
Firstly, give the information of links for each network in the multimodal transportation, among which, the car network mainly includes length (km), zero traffic flow driving time (min), existing vehicle (pcu/h), capacity (pcu/h); the bus network and rail transit network mainly include average driving time (min), frequency (number/ hour); transfer network mainly includes length (km), walking time (min) and cycling time (min). Then, calculate the generalized cost of each type of links according to the formulas in Section 3, and the results are shown in Figure 13: in brackets, the first term is the generalized cost of links and the second term is the capacity of links. As shown in Figure 13: in the multimodal transportation super network topology, the blue dotted links (26,27) and (23,29) respectively represent the candidate links (3,14), (7,10) (in Figure 12) for the car network; the red dotted link (14,15) stands for the candidate link (1,3) (in Figure 12) for the rail transit network, and the green dotted link (12,13) expresses the candidate link (16,17) (in Figure 12) for the bus network. _Step 2:_ Traffic flow assignment  
The minimum cost flow algorithm is coded by PYTHON based on the steps in Section 4.1 to assign the traffic flow. The traffic flow assignment and the composition of transportation mode for each shortest path are obtained, as shown in Table 5, in which all candidate links are taken. Figure 12: The optimization schemes on a simple multimodal network. _Step 3_: Calculate the value of the objective function  
The optimal result of traffic flow assignment in step 2 is used to calculate the value of the network operation cost by Equation (21); then based on the optimization cost and the budget cost of each candidate link to calculate the value of construction cost by Equation (22); finally, the weighted sum of the operation cost and the construction cost by Equation (23) is the value of the objective function. When the generalized cost formula coefficients \(\alpha=0.25\), \(\beta=0.25\), \(\gamma=0.25\), \(\delta=\)0.25 in Equation (1), and the objective function weight coefficients \(\theta=0.5\), \(\tau=0.5\) in Equation (23), the calculation results are illustrated in Table 6 and Figure 14.  
It can be concluded from Table 6 and Figure 14 that the optimal solution of the exact algorithm is \(U^{*}=(1,3)\), and the corresponding operation cost is 4114, construction cost is 3000, and total cost is 3557. Some schemes are without values of construction cost \(B\) and objective function \(Z\) in Table 6. Because the maximum feasible flow of the schemes does not reach \(q_{rs}=100\), so the optimization schemes are not adopted, and there is no need to calculate the objective function.