Joint optimization of seat allocation and pricing with multi-ride season tickets in high-speed railway services

The rapid expansion of high-speed railway (HSR) networks and advancements in service capabilities have prompted operators to introduce innovative service offerings, such as multi-ride season tickets, to accommodate frequent inter-city travelers. These tickets allow passengers to take a predefined number of trips within a specified period, often at discounted rates and without requiring advance seat reservations. While such services enhance flexibility and cost savings for users, their effectiveness relies on sophisticated seat allocation and pricing strategies. Otherwise, the benefits of reserved-ticket holders may be compromised by potentially insufficient seat availability, thereby undermining overall system efficiency. This challenge is further complicated by the intricate nature of railway operations, where multiple origin-destination (O-D) pairs and diverse ticket types (e.g., reserved vs. season tickets) compete for finite shared train capacity. To address this, we propose a joint optimization framework for seat allocation and pricing in HSR systems that integrates both regular and multi-ride season ticket services. The problem is formulated as a bi-level optimization model: the upper level maximizes the operator's profit through strategic seat allocation and pricing, and the lower-level models passenger behavior using a joint user equilibrium framework that captures the demand patterns of both reserved and season ticket options. We analyze the existence and uniqueness properties of the equilibrium solutions. To solve the bi-level problem, we develop a gradient projection-based algorithm that combines sensitivity analysis-driven iterations with backtracking line search techniques. Numerical experiments across HSR networks of varying scales demonstrate the effectiveness of the proposed approach. Results indicate that integrating multi-ride season tickets into seat allocation and pricing strategies achieves profits equal to or exceeding those in the reserved-ticket-only case. The findings also highlight the potential benefits of jointly optimizing seat allocation and pricing strategies.