Collaborative optimization of last-train timetables with accessibility: A space-time network design based approach

Document Type

Journal Article

Publication Date


Subject Area

mode - rail, mode - subway/metro, place - asia, place - urban, operations - scheduling, operations - performance, planning - methods, planning - network design, ridership - behaviour


Last train timetable, Space-time network design, Accessibility, Lagrangian relaxation


To improve the accessibility of the metro network during night operations, this study aims to investigate a collaborative optimization for the last train timetable in an urban rail transit network. By using a space-time network framework, all the involved transportation activities are well characterized in an extended space-time network, in which the train space-time travel arcs, passenger travel arcs, transfer arcs, etc., are all taken into account. Two performance measures are proposed to evaluate the network-based timetable of the last trains. Through considering the route choice behaviors, the problem of interest is formulated as 0–1 linear programming models from the perspective of a space-time network design. To effectively solve the proposed models, we dualize the hard constraints into the objective function to produce the relaxed models by introducing a set of Lagrangian multipliers. Then, the sub-gradient algorithm is proposed to iteratively minimize the gap of the lower and upper bounds of the primal models. Finally, two sets of numerical experiments are implemented in an illustrative network and the Beijing metro network, respectively, and experimental results demonstrate the efficiency and performance of the proposed methods.


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


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