# Optimization of Multimodal Discrete Network Design Problems Based on Super Networks
## 3 Formulating the Multimodal Discrete Network Design Problem
### The Generalized Cost Formulas of Driving Links  
(1) Travel time  
In the actual transportation system, driving links \(A^{D,k}\), \(\forall k\in\{c\}\), stand for the road between adjacent intersections. In this paper, we set the travel time of car driving links as:  
\[T^{D,c}_{ij}=t^{D,c0}_{ij}\left(1+\varphi\left(\frac{x_{ij}+M^{D,c}_{ij}}{ \xi c^{D,c}_{ij}}\right)^{\sigma}\right),\forall(i,j)\in A^{D,c} \tag{8}\]  
where \(t^{D,c0}_{ij}\): the free-flow travel time, \(x_{ij}\): the newly assigned traffic flow, \(M^{D,c}_{ij}\): the original traffic flow, \(c^{D,c}_{ij}\): the capacity of driving links, \(\xi\): the average number of passengers and \(\varphi\), \(\sigma\): the undetermined coefficient.  
Meanwhile, in the actual transportation system, driving links \(A^{D,k}\), \(\forall k\in\{r,p\}\) indicate adjacent stations, and the travel time of the bus and rail transit driving links are set as:  
\[T^{D,K}_{ij}=t^{D,K}_{ij},\forall(i,j)\in A^{D,k},k\in\{p,r\} \tag{9}\]  
where \(t^{D,k}_{ij}\): the average travel time of the transportation mode \(k\).  
(2) Monetary cost  
Since the car is charged depending on mileages, in this paper, the monetary cost of the car driving links is described as:  
\[P^{D,c}_{ij}=p^{D,c0}_{ij}*L^{D,c}_{ij},\forall(i,j)\in A^{D,c} \tag{10}\]  
where \(p^{D,c0}_{a}\): the monetary cost per unit mileage,  
\(L^{D,c}_{ij}\): the length of the link \((i,j)\).  
Moreover, as the price of the ticket is up to mileages, the monetary cost of the bus and rail transit driving links are set as:  
\[P^{D,k}_{ij}=\left\{\begin{array}{cc}p^{D,k0}_{ij}&L^{D,k}_{ij}\leq L^{D,k0} \\ p^{D,k0}_{ij}+\left[L^{D,k}_{ij}-L^{D,k0}\right]*p^{D,k}_{ij}\ L^{D,k}_{ij}>L^{D,k0} \end{array}\right.,\forall(i,j)\in A^{D,k},k\in\{p,r\} \tag{11}\]  
where \(p^{D,k0}_{ij}\): the starting fare, \(L^{D,k0}\): the number of miles corresponding to the starting fare of the mode \(k\), \(L^{D,k}_{ij}\): the length of the link \((i,j)\) and \(p^{D,k}_{ij}\): the increasing ticket price per unit mileage.  
(3) Comfort loss As the comfort loss is proportional to the travel time on the car network, it can be formulated as:  
\[S_{ij}^{D,c}=\gamma^{D,c}.T_{ij}^{D,c},\forall(i,j)\in A^{D} \tag{12}\]  
where: \(\gamma^{D,c}\): the degree of comfort loss per unit time of the car mode.  
In addition, the comfort loss is mainly determined by travel time and congestion on the bus or rail transit network, so it can be described as:  
\[S_{ij}^{D,k}=\left(\eta_{0}^{D,k}+\eta_{1}^{D,k}\Big{(}max\Big{(}0,q_{ij}^{D,k }-f_{ij}^{D,k}.o_{ij}^{D,k}\Big{)}\Big{)}\right)T_{ij}^{D,k},\forall(i,j)\in A ^{D},k\in\{\,p,r\} \tag{13}\]  
where \(\eta_{0}^{D,k}\): the comfort loss parameter when the transportation mode \(k\) is empty, \(\eta_{1}^{D,k}\): the comfort loss parameter when congestion occurs, \(q_{ij}^{D,k}\): the number of passengers, \(f_{ij}^{D,k}\): the departure frequency and \(o_{ij}^{D,k}\): the capacity of vehicles.  
(4) Risk reserve time  
Similarly, we set the risk reservation time of the driving links as follows:  
\[R_{ij}^{D,k}=\left(\rho_{ij}^{D,k}-1\right)T_{ij}^{D,k},\;\forall(i,j)\in A^{ D,k},k\in K \tag{14}\]  
where \(\rho_{ij}^{D,k}\): the delay parameter of driving links and \(\rho_{ij}^{D,k}>1,\;\forall(i,j)\in A^{D,k},k\in K\).