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Abstract. Train scheduling has been one of the signi cant issues in the railway industry in recent years since it has an important role in e cacy of railway infrastructure. In this paper, the timetabling problem of a multiple-tracked railway network is discussed. More speci cally, 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 nd a lower bound. Finally, comprehensive numerical experiments on the Tehran Metro case are reported. Results show the e ciency of the heuristic rules and also the Lagrangian Relaxation method in a way that the optimum values are obtained for all analyzed problems.

In this paper, a metaheuristic approach for the two-machine flow-shop problem with a common due date and the weighted late work performance measure (F2|d j =d|Y w ) are presented. The late work criterion estimates the quality of a solution with regard to the duration of the late parts of jobs, not taking into account the quantity of the delay for the fully late activities. Since the problem mentioned is known to be NP-hard, a trajectory methods, namely GRASP is proposed based on the special features of the case under consideration. Then, the results of computational experiments are reported, in which the metaheuristic solution is compared with exact approach and three other heuristic methods' results.

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