## 3 Model
### Passenger Order Cancellation Model
#### 3.2.2 Passenger Order Cancellation with Penalty  
For cases where ride-hailing platforms assign orders to taxis, passengers need to consider the penalty fee, \(F_{p}\), when canceling orders. In such situations, the cost for passengers to cancel the order and take a cruising taxi, \(C_{Q}\), and the cost for passengers to retain the online taxi order, \(C_{KT}\), are given by Equations (12) and (13), respectively:  
\[C_{Q}=M_{T}L+\delta_{i}RT_{X}+F_{P} \tag{12}\]  
\[C_{KT}=M_{T}L+\delta_{i}R \tag{13}\]  
In this case, the equilibrium point is \(T-\frac{F_{p}}{\delta_{i}R}\). When \(T-\frac{F_{p}}{\delta_{i}R}\) falls within the range of \((T_{P},T)\), passengers will cancel their online taxi order and switch to a cruising taxi if the taxi's arrival time, \(T_{X}\), is within \([\ T_{P},T-\frac{F_{p}}{\delta_{i}R}\ )\). When a penalty is incurred for passenger order cancellation, the probability, \(P_{T}\), of passengers canceling their online taxi orders is given by Equation (14).  
\[P_{T}=\int_{0}^{T-\frac{F_{p}}{\delta_{i}R}}f(T_{X})dT_{X}=1-e^{-\frac{[N-(1- \theta_{T})D][\pi]}{2S}\left(T-\frac{F_{p}}{\delta_{i}R}\right)} \tag{14}\]  
In other intervals, the conditions for imposing a penalty for order cancellation are not met, meaning passengers will not cancel their orders, and thus, \(P_{T}=0\).  
For cases where ride-hailing platforms assign orders to ride-hailing cars, the equilibrium point for order cancellation after the penalty time, \(T_{P}\), is \(T-\frac{(M_{T}-M_{C})L+\delta_{i}P}{\delta_{i}R}\)In scenarios where passengers need to pay a penalty for canceling their orders, when \(T-\frac{(M_{T}-M_{C})L+F_{P}}{\delta,k}\) falls within the range of \(\left(T_{P},T-\frac{(M_{T}-M_{C})L}{\delta,k}\right)\), passengers will cancel their ride-hailing orders and switch to a cruising taxi if the arrival time of the cruising taxi is within \(0\leq T_{X}<T-\frac{(M_{T}-M_{C})L+F_{P}}{\delta_{i}R}\). Therefore, when a penalty is incurred for order cancellation, the probability of passengers canceling their ride-hailing orders is given by Equation (15).  
\[P_{\text{C}}=\int_{0}^{T-\frac{(M_{T}-M_{C})L+F_{P}}{\delta_{i}R}}f(T_{X})dT_{X} =1-e^{-\frac{[N-(1-\theta_{\text{H}})D]\varpi}{2S}\left[T-\frac{(M_{T}-M_{C})L+ F_{P}}{\delta_{i}R}\right]} \tag{15}\]  
In other cases, the conditions for imposing a penalty for order cancellation are not met, meaning passengers will not cancel their orders. At this point, \(P_{\text{C}}=0\).  
In summary, for cases where ride-hailing platforms assign orders to taxis, the probability, \(P_{T}\), of passengers canceling their orders can be expressed as Equation (16).  
\[P_{T}=\left\{\begin{array}{cc}1-e^{-\frac{[N-(1-\theta_{\text{H}})D]\varpi} {2S}T_{P}}&0\leq T_{X}<T_{P}\text{ and }T_{X}<T\\ 1-e^{-\frac{[N-(1-\theta_{\text{H}})D]\varpi}{2S}\left(T-\frac{F_{P}}{\delta_ {i}R}\right)}&T_{P}\leq T_{X}<T-\frac{F_{P}}{\delta_{i}R}\\ 0&\text{other}\end{array}\right. \tag{16}\]  
For cases where ride-hailing platforms assign orders to their own ride-hailing vehicles, the probability, \(P_{C}\), of passengers canceling their orders can be expressed as Equation (17).  
\[P_{\text{C}}=\left\{\begin{array}{cc}1-e^{-\frac{[N-(1-\theta_{\text{H}})D] \varpi}{2S}\left[T-\frac{(M_{T}-M_{C})L}{\delta_{i}R}\right]}&0\leq T_{X}<T- \frac{(M_{T}-M_{C})L}{\delta_{i}R}<T_{P}\\ 1-e^{-\frac{[N-(1-\theta_{\text{H}})D]\varpi}{2S}\left[T-\frac{(M_{T}-M_{C}) L+F_{P}}{\delta_{i}R}\right]}&T_{P}\leq T_{X}<T-\frac{(M_{T}-M_{C})L+F_{P}}{ \delta_{i}R}\\ 0&\text{other}\end{array}\right. \tag{17}\]