@article{zhou_zhong_2007, title={Single-track train timetabling with guaranteed optimality: Branch-and-bound algorithms with enhanced lower bounds}, volume={41}, ISSN={["1879-2367"]}, DOI={10.1016/j.trb.2006.05.003}, abstractNote={A single-track train timetabling problem is studied in order to minimize the total train travel time, subject to a set of operational and safety requirements. This research proposes a generalized resource-constrained project scheduling formulation which considers segment and station headway capacities as limited resources, and presents a branch-and-bound solution procedure to obtain feasible schedules with guaranteed optimality. The search algorithm chronologically adds precedence relation constraints between conflicting trains to eliminate conflicts, and the resulting sub-problems are solved by the longest path algorithm to determine the earliest start times for each train in different segments. This study adapts three approaches to effectively reduce the solution space. First, a Lagrangian relaxation based lower bound rule is used to dualize segment and station entering headway capacity constraints. Second, an exact lower bound rule is used to estimate the least train delay for resolving the remaining crossing conflicts in a partial schedule. Third, a tight upper bound is constructed by a beam search heuristic method. Comprehensive numerical experiments are conducted to illustrate the computational performance of the proposed lower bound rules and heuristic upper bound construction methods.}, number={3}, journal={TRANSPORTATION RESEARCH PART B-METHODOLOGICAL}, author={Zhou, Xuesong and Zhong, Ming}, year={2007}, month={Mar}, pages={320–341} }