# Optimization of Multimodal Discrete Network Design Problems Based on Super Networks
## 3 Formulating the Multimodal Discrete Network Design Problem
### Bi-Objective Programming Model
 \tag{22}\]  
where \(B_{ij}^{k1}\): the cost of the added link \((i,j)\) for the mode \(k\), \(\forall k\in K\) and \(B_{ij}^{k2}\): the cost of the expanded link \((i,j)\) for the mode \(k\), \(\forall k\in K\). (3) Bi-objective programming model  
In summary, a bi-objective programming model for multimodal DNDP is proposed based on Equations (21) and (22) to minimize the network operation cost and construction cost as follows:  
\[\min Z(x,y,z)=\min[\theta W+\tau B]=min[\theta\sum_{(i,j)\in A\cup Uk\in K}& w_{ij}^{k}\cdot x_{ij}^{k}+\tau\sum_{(i,j)\in Uk\in K}& (B_{ij}^{k1}\cdot y_{ij}^{k}++B_{ij}^{k2}\cdot z_{ij}^{k})] \tag{23}\]  
\[s.t.\cdot\cdot\cdot\cdot\cdot\cdot 0\leq\sum_{(i,j)\in U}B_{ij}^{k1}\cdot y_{ij}^{r }+B_{ij}^{k2}\cdot z_{ij}^{c}\leq BG^{c}\ \ \forall(i,j)\in U^{c} \tag{24}\]  
\[0\leq\sum_{(i,j)\in U}B_{ij}^{p1}\cdot y_{ij}^{p}+B_{ij}^{p2}\cdot z_{ij}^{c} \leq BG^{p}\ \ \ \forall(i,j)\in U^{p} \tag{25}\]  
\[0\leq\sum_{(i,j)\in U}B_{ij}^{p1}\cdot y_{ij}^{r}+B_{ij}^{p2}\cdot z_{ij}^{r} \leq BG^{r}\ \ \ \forall(i,j)\in U^{r} \tag{26}\]  
Figure 10: The diagram of optimization schemes. \[0\leq x_{ij}^{k}+M_{ij}^{k}\leq c_{ij}^{k}+y_{ij}^{k}c_{ij}^{kI^{\prime}}+z_{ij}^{ k}c_{ij}^{k2^{\prime}}\quad\forall(i,j)\in A\cup U,\,k\in K \tag{27}\] \[\sum_{i}x_{ri}^{E}-\sum_{i}x_{ir}^{E}=q_{rs}\quad\forall(i,j)\in A \cup U,k\in K\] (28) \[\sum_{i}x_{si}^{k}-\sum_{i}x_{is}^{k}=-q_{rs}\quad\forall(i,j)\in A \cup U,k\in K\] (29) \[\sum_{i}x_{ij}^{k}-\sum_{i}x_{ji}^{k}=0\quad\forall(i,j)\in A\cup U,k\in K\] (30) \[x_{ij}^{k}\geq 0\quad\forall(i,j)eA\cup U,k\in K\] (31) \[s.t.\cdot\cdot\cdot\cdot\cdot\cdot 0\leq\sum_{(i,j)\in U}B_{ij}^{k1}\cdot y_{ij}^{r }+B_{ij}^{k2}\cdot z_{ij}^{c}\leq BG^{c}\ \ \forall(i,j)\in U^{c} \tag{24}\]0  
where \(\theta\), \(\tau\): the weight coefficients and we have \(\theta+\tau=1\), \(BG^{k}\): the budget cost of the transportation mode \(k\), \(M_{ij}^{k}c_{ij}^{k}\): the original traffic flow and the original capacity of the link \((i,j)\) respectively, \(q_{rs}\): the traffic demand of OD pair \(rs\), \(c_{ij}^{k1^{\prime}}/c_{ij}^{k2^{\prime}}\): the capacity of the added/expanded link \((i,j)\) for the mode \(k\), \(\omega\): the binary variable, if the link \((i,j)\) is expanded, \(\omega\) is equal to 1, otherwise, \(\omega\) is equal to 0 on the bus or rail transit route and \(N\): the number of expanded links on the bus or rail transit route. Constraints (24)-(26) indicate that the cost of optimization schemes is less than the budget cost for car, bus, and rail transit network. Constraint (27) shows that the sum of the assigned traffic flow and the original traffic flow is less than the sum of the original capacity and the increased capacity of the added or expanded optimization. Constraints (28)-(31) are traffic flow equilibrium constraints, where constraint (28) means that for the starting point, the outflow minus the inflow is equal to \(q_{rs}\). Constraint (29) shows that for the destination point, the outflow minus the inflow is equal to \(-q_{rs}\). Constraint (30) expresses that for each node, the outflow and inflow must be equal. Constraint (31) indicates that the traffic flow cannot be less than zero. Constraint (32) ensures that the expanded method will change the whole existed route for the bus and rail transit network. Specifically, if one link of the route is expanded, then the other links must be expanded.