Solving cyclic train timetabling problem through model reformulation: Extended time-space network construct and Alternating Direction Method of Multipliers methods

The cyclic train timetabling problem aims to synchronize limited operational resources toward a master periodic schedule of transport services. By introducing an extended time-space network construct, this paper proposes a new type of integer programming model reformulation for the cyclic train timetabling problem on a double-track railway corridor at the macroscopic level. This reformulation method also holds the promises to be applied in a broader set of routing and scheduling problems with periodic activity requirements. We also hope that this space-time network extension technique, as a special version of variable splitting methods in the dual decomposition literature, could potentially bridge the modeling gaps between cyclic and non-cyclic timetables. Specifically, the existing mathematical programming model for the periodic event scheduling problem (PESP) is transformed into a multi-commodity network flow model with two coupled schedule networks and side track capacity constraints. In addition, two dual decomposition methods including Lagrangian relaxation and Alternating Direction Method of Multipliers (ADMM), are adopted to dualize the side track capacity constraints. For each train-specific sub-problem in an iterative primal and dual optimization framework, we develop an enhanced version of forward dynamic programming to find the time-dependent least cost master schedule across the time-space network over multiple periods. ADMM-motivated heuristic methods with adjusted penalty parameters are also developed to obtain good upper bound solutions. Based on real-world instances from the Beijing-Shanghai high-speed railway corridor, we compare the numerical performance between the proposed reformulation and the PESP model that involves the standard optimization solver.