Subsidy Allocation Problem with Bus Frequency Setting Game: A Trilevel Formulation and Exact Algorithm.

This article focuses on optimizing government subsidy allocation for urban bus systems by explicitly modeling the competition among multiple bus operators and passengers' route choices within a trilevel programming framework. The study formulates the subsidy allocation problem as a biobjective trilevel model involving the government (upper level), profit-maximizing bus operators (middle level), and passengers minimizing travel costs (lower level), and transforms it into a single-objective bilevel programming problem with equilibrium constraints. An exact algorithm with acceleration techniques is developed to solve the resulting mixed-integer quadratic constrained problem efficiently. Numerical experiments, including a real-world case study of Suzhou, China, reveal that competition typically occurs only on lines jointly served by different operators, which generally do not require subsidies; moreover, competition reduces social costs more effectively in cities where passengers have a higher value of time. The findings suggest that subsidy policies and competition regulation should consider local economic factors, operator profitability sensitivity, fare structures, and transfer discounts to balance service quality, operator incentives, and social welfare.