Train scheduling for minimizing passenger waiting time with time-dependent demand and skip-stop patterns: Nonlinear integer programming models with linear constraints

Document Type

Journal Article

Publication Date


Subject Area

mode - rail, operations - scheduling, planning - service improvement, ridership - demand


Train timetable, Passenger waiting time, Time-varying demand, Skip-stop pattern, Nonlinear mixed integer programming


This paper focuses on how to minimize the total passenger waiting time at stations by computing and adjusting train timetables for a rail corridor with given time-varying origin-to-destination passenger demand matrices. Given predetermined train skip-stop patterns, a unified quadratic integer programming model with linear constraints is developed to jointly synchronize effective passenger loading time windows and train arrival and departure times at each station. A set of quadratic and quasi-quadratic objective functions are proposed to precisely formulate the total waiting time under both minute-dependent demand and hour-dependent demand volumes from different origin–destination pairs. We construct mathematically rigorous and algorithmically tractable nonlinear mixed integer programming models for both real-time scheduling and medium-term planning applications. The proposed models are implemented using general purpose high-level optimization solvers, and the model effectiveness is further examined through numerical experiments of real-world rail train timetabling test cases.


Permission to publish the abstract has been given by Elsevier, copyright remains with them.


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