## 3 Model
### Passenger Order Cancellation Model
#### 3.2.1 Passengers Incur No Penalty for Order Cancellation  
For ride-hailing platforms dispatching orders to taxis, the cost, \(C_{Q}\), for a passenger to cancel an order and take a cruising taxi and the cost, \(C_{KT}\), for a passenger to keep an online taxi order are shown in Equations (6) and (7), respectively. These costs consist of the fare paid by the passenger and the waiting time cost, where \(\delta_{i}\) represents the time preference coefficient of passenger \(i\). Different passengers have varying degrees of travel urgency, which affects their waiting time cost accordingly. Passengers with lower travel urgency have lower time preference coefficients and thus lower actual waiting time costs. Conversely, for passengers with higher travel urgency, their time preference coefficient \(\delta_{i}>1\), leading to a significant increase in waiting time costs. According to loss aversion theory, the psychological pain of a loss is 2 to 2.5 times that of a gain of the same amount. With passengers' waiting time costs considered a \"loss\", \(\delta_{i}\) is set within the range of \(\delta_{i}\in(0,2.5)\).  
\[C_{Q}=M_{T}L+\delta_{i}RT_{X} \tag{6}\]  
\[C_{KT}=M_{T}L+\delta_{i}RT \tag{7}\]  
The issue of passenger order cancellation considered in this paper arises when a passenger encounters a cruising taxi before the arrival of an online-hailed taxi, i.e., \(T_{X}<T\). Within this timeframe, \(C_{KT}>C_{Q}\), passengers who have placed online orders assigned to taxis by the platform will cancel their platform orders upon encountering an empty cruising taxi while waiting. Therefore, when no penalty is incurred for order cancellation,the probability that a passenger will cancel a taxi order assigned by the platform, \(P_{T}\), can be expressed as Equation (8).  
\[P_{T}=\int_{0}^{T_{p}}f(T_{X})dT_{X}=1-e^{-\frac{[N-(1-\theta_{T})][\pi]}{2S}T_{p}} \tag{8}\]  
For cases where ride-hailing platforms assign orders to private vehicles, the cost for passengers to retain the platform's order consists of two parts: the fare paid and the cost of passenger waiting time, as shown in Equation (9).  
\[C_{KC}=M_{C}L+\delta_{i}RT \tag{9}\]  
The equilibrium point for order cancellation before the penalty time \(T_{P}\) is \(T-\frac{(M_{T}-M_{C})L}{\delta_{i}R}\). When \(0\leq T-\frac{(M_{T}-M_{C})L}{\delta_{i}R}<T_{p}\), passengers will cancel their online order allocated to ride-hailing cars and switch to a cruising taxi, when the arrival time of the cruising taxi is within \(0\leq T_{X}<T-\frac{(M_{T}-M_{C})L}{\delta_{i}R}\). Therefore, when no penalty is incurred for order cancellation, and the cost advantage of ride-hailing cars is significant, the probability of passengers canceling their orders allocated to ride-hailing cars is given by Equation (10).  
\[P_{C}=\int_{0}^{T-\frac{(M_{T}-M_{C})L}{\delta_{i}R}}f(T_{X})dT_{X}=1-e^{- \frac{[N-(1-\theta_{T})D][\pi]}{2S}\left[T-\frac{(M_{T}-M_{C})L]}{\delta_{i}R }\right]} \tag{10}\]  
When \(0\leq T_{X}<T_{P}\leq T-\frac{(M_{T}-M_{C})L}{\delta_{i}R}\), passengers will cancel their orders and switch to a cruising taxi if \(0\leq T_{X}<T_{P}\). In this scenario, where no penalty is incurred for order cancellation and ride-hailing cars do not have a significant cost advantage, the probability of passengers canceling their ride-hailing orders is given by Equation (11).  
\[P_{C}=\int_{0}^{T_{p}}f(T_{X})dT_{X}=1-e^{-\frac{[N-(1-\theta_{T})D][\pi]}{2S} T_{p}} \tag{11}\]