Integrated optimization on train scheduling and preventive maintenance time slots planning

Xiaojie Luan, Jianrui Miao*, Lingyun Meng, Francesco Corman, Gabri Lodewijks

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

63 Citations (Scopus)


We address the problem of simultaneously scheduling trains and planning preventive maintenance time slots (PMTSs) on a general railway network. Based on network cumulative flow variables, a novel integrated mixed-integer linear programming (MILP) model is proposed to simultaneously optimize train routes, orders and passing times at each station, as well as work-time of preventive maintenance tasks (PMTSs). In order to provide an easy decomposition mechanism, the limited capacity of complex tracks is modelled as side constraints and a PMTS is modelled as a virtual train. A Lagrangian relaxation solution framework is proposed, in which the difficult track capacity constraints are relaxed, to decompose the original complex integrated train scheduling and PMTSs planning problem into a sequence of single train-based sub-problems. For each sub-problem, a standard label correcting algorithm is employed for finding the time-dependent least cost path on a time-space network. The resulting dual solutions can be transformed to feasible solutions through priority rules. Numerical experiments are conducted on a small artificial network and a real-world network adapted from a Chinese railway network, to evaluate the effectiveness and computational efficiency of the integrated optimization model and the proposed Lagrangian relaxation solution framework. The benefits of simultaneously scheduling trains and planning PMTSs are demonstrated, compared with a commonly-used sequential scheduling method.

Original languageEnglish
Pages (from-to)329-359
JournalTransportation Research. Part C: Emerging Technologies
Publication statusPublished - 2017


  • Cumulative flow variables
  • Integrated optimization
  • Lagrangian relaxation
  • Preventive maintenance time slots (PMTSs) planning
  • Train scheduling


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