# Optimization of Multimodal Discrete Network Design Problems Based on Super Networks
## 3 Formulating the Multimodal Discrete Network Design Problem
### Bi-Objective Programming Model  
The objective of multimodal DNDP is to decide on an optimal scheme \(U^{*}\) from the candidate schemes \(U=\{U^{c},U^{t},U^{p}\}\) to minimize the total cost, which includes the network operation cost and the construction cost of the optimization scheme. \(U\) is the set of candidate links; \(U^{c}\) is the set of added or expanded links for the car network; \(U^{r}\) is the set of added or expanded links for the rail transit network, and \(U^{p}\) is the set of added or expanded links for the bus network. (1) Network operation cost  
According to the generalized cost formulas mentioned above, the network operation cost of the multimodal transportation network can be formulated as follows:  
\[ \begin{split}& W=\sum_{(i,j)\in A\cup U\&\in K} \hskip-14.226378 ptw_{ij}^{k}\cdot x_{ij}^{k}\\ &=\sum_{(i,j)\in A\cup U\&\in K}(\pi T_{ij}^{k}+ \beta P_{ij}^{k}+\gamma S_{ij}^{k}+\delta R_{ij}^{k})\cdot x_{ij}^{k}\\ &=\sum_{(i,j)\in A\cup U\&\in K}(\pi T_{ij}^{E,k}+ \beta P_{ij}^{E,k}+\gamma S_{ij}^{E,k}+\delta R_{ij}^{E,k})\cdot x_{ij}^{E,k} \\ &\hskip 14.226378 pt+(\pi T_{ij}^{L,k}+\beta P_{ij}^{L,k}+\gamma S_{ ij}^{L,k}+\delta R_{ij}^{L,k})\cdot x_{ij}^{L,k}\\ &\hskip 14.226378 pt+(\pi T_{ij}^{D,k}+\beta P_{ij}^{D,k}+\gamma S_{ ij}^{D,k}+\delta R_{ij}^{D,k})\cdot x_{ij}^{D,k}\\ &\hskip 14.226378 pt+(\pi T_{ij}^{T,k}+\beta P_{ij}^{T,k}+\gamma S_{ ij}^{T,k}+\delta R_{ij}^{T,k})\cdot x_{ij}^{T,k}\end{split}  \tag{21}\]  
Here, \(x_{ij}^{k}\) is the traffic flow of the link \((i,j)\). (2) Construction cost  
For DNDP, there are two methods to optimize the network: one is to add a new link; the other one is to expand the original link. In the multimodal super network topology, an added link is to connect two unconnected nodes, while an expanded link is to add a link to the originally connected nodes. As shown in Figure 10: for the car network, since node 3 and node 14 were not connected originally, the added link (3,14) (the blue dotted link) represents the added optimization. Meanwhile, as node 7 and node 10 were connected originally, so link (7,10) (the blue dotted link) illustrates the expanded optimization. Therefore, we use the 0-1 variable \(y_{ij}^{c}\) and \(z_{ij}^{c}\) to respectively express whether to add or expand links. If the link \((i,j)\) is added/expanded, \(y_{ij}^{c}/z_{ij}^{c}\) is equal to 1, otherwise \(y_{ij}^{c}/z_{ij}^{c}\) is equal to 0, \(\forall(i,j)eU^{c}\)  
For the bus network, the added method indicates adding a new link on the existed route, like link (16,17) (the green dotted link). The expanded method means to increase the frequency of departure, so it will change the whole route, such as link (1,2), (2,3), (3,4), (4,5),(5,6) (the green dotted link). Hence, if the link \((i,j)\) is added/expanded the 0-1 variable \(y_{ij}^{p}/z_{ij}^{p}\) is equal to 1, otherwise \(y_{ij}^{p}/z_{ij}^{p}\) is equal to 0, \(\forall(i,j)eU^{p}\) in the bus network. Similarly, for the rail transit network, the added method expresses to add a new link on the existed route, like link (1,3) (the red dotted link). And the expanded method represents to increase the frequency of departure, so it will change the whole route, such as links (10,12), (12,13), (13,14) (the red dotted link). Thus, if the link \((i,j)\) is added/expanded the 0-1 variable \(y_{ij}^{r}/z_{ij}^{r}\) is equal to 1, otherwise \(y_{ij}^{r}/z_{ij}^{r}\) is equal to 0, \(\forall(i,j)eU^{r}\) in the rail transit network. Consequently, the construction cost of the optimization scheme is formulated as:  
\[ \begin{split} B=\sum_{(i,j)\in U}\sum_{Uk\in K}& B_{ij}^{k1}y_{ij}^{k}+B_{ij}^{k2}z_{ij}^{k}\\ =\sum_{(i,j)\in U^{r}}(B_{ij}^{r1}\cdot& y_{ij}^{r }+B_{ij}^{r2}\cdot z_{ij}^{c})+\sum_{(i,j)\in U^{p}}(B_{ij}^{p1}\cdot y_{ij}^ {p}+B_{ij}^{p2}\cdot z_{ij}^{p})\\ &+\sum_{(i,j)\in U^{r}}(B_{ij}^{r1}\cdot y_{ij}^{r}++B_{ij}^{r2} \cdot z_{ij}^{r})\end{split}