Pricing Analysis for Railway Multi-Ride Tickets: An Optimization Approach for Uncertain Demand within an Agreed Time Limit

A multi-ride ticket with a certain period of validity and maximum number of uses has been introduced into railway transport. The key to pricing the railway multi-ride ticket is determining the uncertain demand within an agreed time limit. Unfortunately, limited studies have focused on this pricing issue. Therefore, we focused on railway multi-ride ticket pricing optimization in two different scenarios: a single train with multiple stops and multiple trains with multiple stops. First, the expected coefficient and incentive coefficient were introduced to describe the decision-making process for multi-ride tickets and simulate the change in passengers' travel behavior after purchasing multi-ride tickets. Then, passenger demand functions based on a normal distribution were developed to establish the pricing models with maximized revenue. Finally, we adopted improved particle swarm optimization (PSO) to solve the models. Two numerical cases were used to verify the models separately for two application scenarios. The results revealed that the multi-ride ticket pricing problem is not a simple summation of pricing for one-time travel of passengers. In the situation of a single train with multiple stops, the expected coefficient is positively related to the total income, whereas the incentive coefficient has limited influence on the optimal price and total revenue. Furthermore, a multi-ride ticket should allow the passenger to take trains eight times at most in 8 days at the price of CNY 4922 (abbreviated as 4922 (8, 8)) rather than 3785 (8, 6). Railway enterprises should cautiously limit the scope of trains available for multi-ride tickets in the case of multiple trains with multiple stops.