# 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  
A bi-objective nonlinear programming model is constructed by simultaneously considering the operator's profit and all users' costs. The TPS is applied to help the operator and users achieve the goal. The price for each user cluster, fleet size, size of SAV stations, and the demands satisfied by CSCs and SAVs are all determined by the model. 1. Objective Functions  
The objective functions contain two objectives; one is the operator's profit maximization, and another is to minimize all users' costs. Since the carsharing operator can provide two kinds of service, including the service of CSCs and the service of SAVs, then the profit also comes from the two modes. The first objective function is given in Equation (8), and the operator's profit is maximized. \[ \begin{array}{ll}Max\;\boldsymbol{\pi}=&\sum\limits_{\begin{subarray}{c}i\in \mathbf{K},y,y^{\prime}\in\mathbf{Y}\in\mathbf{H}\in\mathbf{H}\\ r^{\prime}=t+\delta_{yy}^{ath}\end{subarray}}D_{i_{i},j_{t+\delta_{ij}^{th}} ^{sh}}^{sh}\times\left(P_{yy^{\prime}}^{st^{\prime}h}-C_{y}^{s}\right)\times \delta_{yy^{\prime}}^{st^{\prime}h}+&\sum\limits_{\begin{subarray}{c}i,j\in \mathbf{K},h\in\mathbf{H}\\ t^{\prime}=t+\delta_{yi}^{ath},y\in\mathbf{Y}\end{subarray}}D_{i_{i},j_{t+ \delta_{ij}^{th}}}^{ah}\times P_{ij}^{st^{\prime}h}\times\delta_{ij}^{st^{ \prime}h}\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad t^{\prime }=t+\delta_{yi}^{ath},y\in\mathbf{Y}\end{array}  \tag{8}\]  
The first and second terms show the profit gained by the service of CSCs and the service of SAVs, respectively. For the service of CSCs, the operator's profit equals the rental fees paid by users between stations minus the driving costs during the trips. For the service of SAVs, the profit is equal to the rental fees between origins and destinations paid by users. The third term means the extra driving cost paid by the operator during the trips of SAVs, i.e., the driving cost of SAVs from the carsharing stations to users' origins and from users' destinations to carsharing stations. The fourth term represents the depreciation cost of all sharing cars, and the fifth part refers to the maintenance cost of all parking spaces. The last term gives the penalty cost of the unsatisfied carsharing demands. The second objective is given in Equation (9), and all users' costs are minimized. \[\begin{array}{ll}\mathit{Min\boldsymbol{\varphi}=}&\sum\limits_{ \begin{subarray}{c}i,j\in \mathbf{K},\;y,y^{\prime}\in\mathbf{Y},h\in\mathbf{H}\\ t^{\prime}=t+\delta_{iy}^{ath}\end{subarray} }D_{i_{t},j_{t+\delta_{ij}^{th}} }^{sh}\times\left(P_{ij}^{st^{\prime}h}\times\delta_{yy^{\prime}}^{st^{\prime }h}+\alpha^{h}\times\left(\delta_{yi}  
All users' costs are composed of CSC users and SAV users. CSC users need to pay the rental fees of CSCs between carsharing stations and the walking costs. Combined with the service provided by CSCs (Figure 1a), the walking costs include two components; one is the walking costs from origins to carsharing stations, and another is the walking costs from drop-off stations to destinations. In Figure 1b, SAV users need to pay the rental fees from origins to destinations and the waiting costs when SAVs drive from the nearest carsharing stations to origins. (2) Constraints Constraints in the programming model include the flow conservation, constraints about the fleet size and demands, and restrictions about decision variables. * Flow conservation The fleet size in consecutive time steps at each station follows the flow conservation rules, as shown in Equation (10). \[A^{w}_{y_{t+1}}=A^{w}_{y_{t}}-\sum_{ \begin{array}{c}h\in\mathbf{H},t^{\prime}\in \mathbf{T}\\ i,j\in\mathbf{K},y^{\prime}\in\mathbf{Y}\end{array} }D^{wh}_{i_{t},t_{1}}+\sum_{ \begin{array}{c}h\in\mathbf{H},t_{1}\in\mathbf{T}\\ i,j\in\mathbf{K},y^{\prime}\in\mathbf{Y}\end{array}