Optimal timetables for public transportation

This paper analyzes the optimal timetable for a given number of public transport vehicles on a single transit line when riders differ with respect to the times at which they prefer to travel and the schedule delay costs they incur from traveling earlier or later than desired. The problem of minimizing riders' total schedule delay costs is formulated in continuous time and first-order optimality conditions are identified. An explicit solution is derived for the ‘line’ model in which preferred travel times are uniformly distributed in the population over part of the day and trips cannot be rescheduled between days. This solution is compared with the optimal schedule for the ‘circle’ model in which preferred travel times are uniformly distributed over the full 24 h day and trips can be rescheduled between days.