## 5 Numerical Analysis
### Platform Order Dispatch Strategies Under Different Order Densities  
To investigate the optimal order allocation ratio, \(\alpha\), in various scenarios, this paper designs the following algorithmic procedure tailored to different travel market conditions. (1) Initialization: Input the order scale to generate the initial order set, \(D\). Input the ride-hailing vehicle set, \(V_{C}\), and the taxi set, \(V_{T}\), and initialize vehicle locations and statuses. (2) Order allocation: Calculate the offline order acceptance decision value for taxis to determine their order acceptance mode. Allocate orders to vehicles in the system based on the order allocation mechanism. (3) Parameter space traversal: Traverse the parameter combinations, \((T_{X},T,\alpha)\), and compute the objective function value for each combination. (4) Optimal \(\alpha\) selection: For each parameter pair \((T_{X},T)\), retain the \(\alpha\) value that maximizes the objective function and output it. Based on this algorithmic procedure, the optimal \(\alpha\) values for platform revenue, passenger utility, driver earnings, and their combined total under different regional order densities were plotted, as shown in Figure 6.  
From the perspective of overall social welfare, an increase in order density significantly affects situations where taxi arrival times exceed the penalty threshold for passenger order  
Figure 4: Passenger order cancellation probability under different proportions of taxis accepting offline orders.  
Figure 5: Passenger order cancellation probability under different order densities.  
cancellation, while ride-hailing vehicles from the platform take longer to arrive. When order density is low, ride-hailing platforms can adjust the number of cruising taxis in the road network by increasing the proportion of orders allocated to taxis, thereby regulating the probability of passenger order cancellations. However, as regional order density increases, both taxi and platform ride-hailing vehicle pick-up times increase significantly. In this case, increasing the proportion of orders allocated to platform ride-hailing vehicles instead helps maintain social welfare at a higher level.  
For ride-hailing platforms, allocating orders to taxis generates additional commissions. Therefore, in scenarios where passengers are unlikely to cancel orders, allocating more orders to taxis is always beneficial to the platform. However, as order density increases and the probability of taxi offline order acceptance rises, platforms should also consider allocating more orders to their own ride-hailing vehicles for situations where online vehicles  
Figure 6: Optimal order allocation ratio under different order densities in various regions. (**a**) The optimal order allocation ratio for maximizing platform revenue. (**b**) The optimal order allocation ratio for maximizing passenger utility. (**c**) The optimal order allocation ratio for maximizing driver earnings. (**d**) The optimal order allocation ratio for maximizing the sum of these three benefits.  
require longer pick-up times to reduce revenue losses due to passenger order cancellations. Nevertheless, passengers prefer that platforms allocate orders to their own ride-hailing vehicles, as this increases their chances of encountering empty cruising taxis. This preference becomes more apparent as order density increases, which is also related to the decrease in the number of taxis participating in online order dispatch during high order overflow. When order overflow is relatively mild, passengers are willing to wait longer for orders that are responded to online.