Train Timetabling on double track and multiple station capacity railway   with useful upper and lower bounds
  Train scheduling is one of the significant issues in the railway industry in recent years since it has an important role in efficacy of railway infrastructure. In this paper, the timetabling problem of a multiple tracked railway network is discussed. More specifically, a general model is presented here in which a set of operational and safety requirements is considered. The model handles the trains overtaking in stations and considers the stations capacity. The objective function is to minimize the total travel time. Unfortunately, the problem is NP-hard and real size problems cannot be solved in an acceptable amount of time. In order to reduce the processing time, we presented some heuristic rules, which reduce the number of binary variables. These rules are based on problem's parameters such as travel time, dwell time and safety time of stations and try to remove the impracticable areas of the solution space. Furthermore, a Lagrangian Relaxation algorithm model is presented in order to find a lower-bound. Finally, comprehensive numerical experiments on the Tehran Metro case are reported. Results show the efficiency of the heuristic rules and also the Lagrangian Relaxation method in a way that for all analyzed problems the optimum value are obtained. 
Afshin Oroojlooy Jadid, Kourosh Eshghi
math.OC