Train timetabling on double track and multiple station capacity railway with useful upper and lower bounds

Train scheduling has been 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-sized 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 the optimum values are obtained for all analyzed problems.