# A Tailored Pricing Strategy for Different Types of Users in Hybrid Carsharing Systems
## 3 Model Formulation
### Notations
\begin{tabular}{p{142.3 pt} p{142.3 pt}} \hline \hline Set & \\ \(\mathbf{W}=\{s,a\}\) & Set of service modes, \(\forall w\in\mathbf{W}\). \(s\) and \(a\) separately represent the service of CSCs and the service of SAVs. \\ \(\mathbf{H}=\{1,\ldots,h,\ldots H\}\) & Set of user clusters, \(\forall h\in\mathbf{H}\). \(H\) is the maximum cluster. \\ \(\mathbf{T}=\{1,\ldots,t,\ldots,T\}\) & Time steps in the operational period. \(T\) is the maximum time step of the studied period. \\ \(\mathbf{K}=\{1,\ldots,k,\ldots,K\}\) & Nodes in the hybrid carsharing system, \(i,j,k\) are commonly used indices. \\ \(\mathbf{Y}=\{1,\ldots,y,\ldots,Y\}\) & Carsharing stations for the service of both CSCs and SAVs, \(\mathbf{Y}\subseteq\mathbf{K}\). Commonly used indices are \(y\) and \(y^{\prime}\). \\ \(\mathbf{X}=\{1,\ldots,k_{t},\ldots,k_{T}\}\) & Time-space nodes set combines the nodes and time steps, and \(k_{t}\) means node \(k\) at time step \(t\). \\ Parameters & Travel demands of user cluster \(h\) between nodes \(i\) and node \(j\), the departing time step is \(t\), and the arriving time step is \(t^{\prime}\). The demands can be satisfied either by CSCs or SAVs, \(t^{\prime}\) is dependent on the service mode the user cluster \(h\) chooses, \(\forall(i_{t},j_{t^{\prime}})\in\mathbf{A}\), \(h\in\mathbf{H}\). \\ \(\delta_{iy}^{sth}\) & Walking time for user cluster \(h\) who choose service \(s\), between origin \(i\) and station \(y\), departing from time step \(t\), \(\forall y\in\mathbf{Y}\), \(i\in\mathbf{K}\), \(t\in\mathbf{T}\), \(h\in\mathbf{H}\). \\ \(\delta_{y^{\prime}}^{sth}\) & Walking time for user cluster \(h\) choosing service \(s\), between station \(y^{\prime}\) and destination \(j\), and the departing time step is \(t\), \(\forall y^{\prime}\in\mathbf{Y}\), \(j\in\mathbf{K}\), \(t\in\mathbf{T}\), \(h\in\mathbf{H}\). \\ \(\delta_{yi}^{sth}\) & Waiting time for user cluster \(h\) choosing service \(s\), between station \(y\) and origin \(i\), departing from time step \(t\), \(\forall y\in\mathbf{Y}\), \(i\in\mathbf{K}\), \(h\in\mathbf{H}\). \\ \(\alpha^{h}\) & Walking cost per minute for user cluster \(h\) who choose service \(s\), \(\forall h\in\mathbf{H}\). \\ \(\mu^{h}\) & Waiting cost per minute for user cluster \(h\) who choose service \(a\), \(\forall h\in\mathbf{H}\). \\ \(D^{u0h_{i}\hat{t}}\) & The initial number of trips for user cluster \(h\) who choose service \(w\), departing from node \(i\) at time step \(t\) and arriving at node \(j\) at time step \(t^{\prime}\), \(\forall(i_{t},j_{t^{\prime}})\in\mathbf{A}\), \(h\in\mathbf{H}\), \(w\in\mathbf{W}\). \(t^{\prime}\) is different when users choose different service modes. \\ \(p_{ij^{\prime}}^{v0h},p_{ij^{\prime}}^{p^{\prime}}\) & The initial or the highest price can be set between node \(i\) and \(j\) for user cluster \(h\) who choose service \(w\), \(\forall y\), \(y^{\prime}\in\mathbf{Y}\), \(h\in\mathbf{H}\), \(w\in\mathbf{W}\). \\ \(Q_{iy}^{s}\) & The number of parking places of CSC station \(y\), \(\forall y\in\mathbf{Y}\). \\ \(C_{m}^{v}\) & The maintenance cost for a parking place per day under service \(w\), \(\forall w\in\mathbf{W}\). \\ \(C_{z}^{v}\) & The depreciation cost per car per day under service \(w\), \(\forall w\in\mathbf{W}\). \\ \(C_{v}^{v}\) & The driving cost, the operator, should pay for a car per time step under service \(w\), \(\forall w\in\mathbf{W}\). \\ Travel cost for user cluster \(h\) who chooses service \(w\), departing from node \(i\) at time step \(t\) and arriving at node \(j\) at \(t^{\prime}\), \(\forall i\), \(j\in\mathbf{K}\), \(h\in\mathbf{H}\). \\ Decision variables & The number of trips for user cluster \(h\) who choose service \(s\), departing from node \(i\) at time step \(t\) and arriving at node \(j\) at \(t^{\prime}\), \(t^{\prime}=t+\delta_{ij}^{th}\), \(\forall(i_{t},j_{t^{\prime}})\in\mathbf{A}\), \(h\in\mathbf{H}\), \(w\in\mathbf{W}\). \(t^{\prime}\) is dependent on the service mode users choose. \\ \(p_{ij}^{path}\) & Rental price per time step between nodes \(i\) and \(j\) for user cluster \(h\) choosing service \(w\), and the departing time step is \(t\), \(\forall i,j\in\mathbf{K}\), \(h\in\mathbf{H}\), \(w\in\mathbf{W}\). \\ \(Q_{q}^{s}\) & The number of parking places for SAV station \(y\), \(\forall y\in\mathbf{Y}\). \\ Auxiliary variables & The probability of carsharing users who would potentially choose SAVs between nodes \(i\) and \(j\), and the departing time step is \(t\), \(\forall i,j\in\mathbf{K}\), \(h\in\mathbf{H}\). \\ \(\Delta D^{uph}i_{i}\), & Demand variation between nodes \(i\) and \(j\) for user cluster \(h\) who choose service \(w\), \(\forall(i_{t},j_{t^{\prime}})\in\mathbf{A}\), \(h\in\mathbf{H}\), \(w\in\mathbf{W}\). \\ \(\Delta P_{ij}^{truth}\) & Price variation between nodes \(i\) and \(j\) for each user cluster \(h\) under service \(w\), and the departing time step is \(t\), \(\forall i,j\in\mathbf{K}\), \(h\in\mathbf{H}\), \(w\in\mathbf{W}\). \\ \hline \hline \end{tabular}