# A Tailored Pricing Strategy for Different Types of Users in Hybrid Carsharing Systems
## 3 Model Formulation
### Bi-Objective Nonlinear Programming Model
#### 3.4.2 Model Formulation
}D^{wh}_{i_{t},t_{2}},\forall y\in \mathbf{Y},t\in\mathbf{T}\] (10) For both the CSCs' stations and SAVs' stations, the number of available sharing cars in the next time step \(A^{w}_{y_{t+1}}\) equals to the number of available sharing cars at the current time step \(A^{w}_{y_{t}}\), minus the number of cars rented by users at the same time step, then plus the number of users who arrived at this station at the current time step. The time steps used in Equation (10) are explained below. As shown in Figure 1, when users travel from node \(i\) to node \(j\), the nearest carsharing stations to node \(i\) and \(j\) are station \(y\) and station \(y^{\prime}\), respectively. The arriving time step at station \(y\) is \(t\). The departing time step at node \(i\) is \(t^{\prime}\), and the arriving time step at destination \(j\) is \(t^{\prime\prime}\). When users travel from node \(j\) to \(i\) and stop by carsharing station \(y^{\prime}\) and \(y\), the arriving time step at station \(y\) is still \(t\), \(t_{1}\) is the departing time step at node \(j\) and \(t_{2}\) corresponds to the arriving time step at node \(i\). Details about the time steps are shown in Table 3. * Constraints about demands Since the demand for each cluster is limited, users belonging to different clusters can choose the service of CSCs and the service of SAVs based on the utilities of each service mode. Then, there is an upper bound for the CSC users. \[D^{sh}_{i_{t},t^{\prime}}\leq Q^{h}_{i_{t},t^{\prime}}-D^{sh}_{i_{t},t^{\prime }},t^{\prime}=t+\delta^{th}_{ij},\forall i,j\in\mathbf{K},h\in\mathbf{H},t,t^{\prime} \in\mathbf{T}\] (11) Demands satisfied by CSCs are less than the potential CSCs' demands, which equals the total demands minus the demands satisfied by SAVs. Furthermore, under the TPS, the new demands equal to the initial demands \(D^{wh}_{i_{t},t^{\prime}}\) plus the demand variations \(\Delta D^{wh}_{i_{t},t^{\prime}}\) brought by price variations. The demand variation is  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & **CSCs** & **SAVs** \\ \hline \(t^{\prime}\) & \(max\left[0,t-\delta^{st}_{yy}h\right]\) & \(min\left[t+\delta^{sth}_{yy},T\right]\) \\ \(t^{\prime\prime}\) & \(min\left[t+\delta^{th}_{yy^{\prime}}+\delta^{(t+\delta^{th}_{yy})h}_{y^{\prime} },T\right]\) & \(min\left[t^{\prime}+\delta^{th}_{ij},T\right]\) \\ \(t_{1}\) & \(max\left[0,t-\delta^{(t+\delta^{th}_{yy})h}_{y^{\prime}}\right]\) & \(max\left[0,t_{2}-\delta^{(t_{1}+\delta^{th}_{yy})h}_{y^{\prime}}\right]\) \\ \(t_{2}\already given in Equation (3). The demand variation between the same nodes is different for different user clusters. \[D^{wh}_{i_{l}j_{l^{\prime}}}=D^{w0h}_{i_{l}j_{l^{\prime}}}+\Delta D^{wh}_{i_{l}j_{ l^{\prime}}},t^{\prime}=t+\delta_{ij}^{th},\forall i,j\in\mathbf{K},h\in\mathbf{H},w\in\mathbf{W},t,t^{ \prime}\in\mathbf{T}\] (12)
* Constraints about the fleet size The fleet size is restricted by the size of the station. \[A_{y_{t}}^{w}\leq Q_{y}^{w},\forall y_{t}\in\mathbf{X},y\in\mathbf{Y},w\in\mathbf{W}\] (13) Constraints (13) guarantee that the number of available sharing cars at each station is always no greater than the number of parking places at the same station. * Domain of decision variables Constraints (14)-(17) set the domain of all decision variables. Especially in constraints (14), the price is ensured to vary within the predefined scope, and it should be no greater than the highest price. \[0\leq P_{ij}^{vth}\leq P_{ij}^{wmaxh},\forall i,j\in\mathbf{K},h\in\mathbf{H},w\in\mathbf{W}\] (14) Demands satisfied by the CSCs and SAVs are restricted to be an integer in constraints (15) \[D^{wh}_{i_{l}j_{l^{\prime}}}\in\mathbf{Z}^{+},t^{\prime}=t+\delta_{ij}^{th},\forall i,j\in\mathbf{K},h\in\mathbf{H},w\in\mathbf{W}\] (15)  
The number of available sharing cars at each node should also take an integer as below. \[A_{y_{t}}^{w}\in\mathbf{Z}^{+},\forall y_{t}\in\mathbf{X},y\in\mathbf{Y},w\in\mathbf{W} \tag{16}\]  
The number of parking spaces at each carsharing station should be an integer for SAVs as shown in constraints (17).