Ling for trains). (6) A passenger can only be delayed as soon as and may only transfer once. (7) Every passenger waiting at a station boards the train when it arrives at the station. (eight) Express trains overtake nearby trains at overtaking stations with out stopping. three. Modeling According to the above modeling assumptions, a 0 integer programming model (IPM) for E/L mode is established, using the stop program for express trains and also the variety of express/local trains that depart through peak hours (i.e., operating frequency) as choice variables. By traversing all of the operating schedules for E/L mode, parameters (e.g., the train operating timetable, the amount of passengers waiting at each and every station, the number of passengers on board in each and every inter-station segment, plus the quantity of delayed passengers) are calculated for every schedule. Then, ttotal and also the total number of operating trains Ctotal in every single schedule are obtained. The optimal schedule is located through comparison. This study examines the morning peak period along with the cyclic mode of operation. The study period TR can be divided into various minimum cycles Tc . Let I and 1: n be the number of trains as well as the ratio with the quantity of departing express trains towards the quantity of departing neighborhood trains in each and every cycle, respectively. Table 1 summarizes the main model parameters and their definition.Table 1. Definition of model parameters. Parameter i j N xi,j Definition Train quantity Station quantity Number of stations Whether train i stops at station j (xi,j = 1 indicates that train i stops at station j; xi,j = 0 implies that train i doesn’t cease at station j)Appl. Sci. 2021, 11,4 ofTable 1. Cont. Parameter yi,j Iaa Idd Ida Ita Iat Idt TR TC odr,s Definition Whether or not train i overtakes the preceding neighborhood train at station j (yi,j = 1 indicates that train i overtakes the preceding regional train; yi,j = 0 means that train i will not overtake the preceding regional train) Minimum interval in between two successive trains arriving at the exact same station (seconds, s) Minimum interval involving two successive trains departing in the very same station (s) Minimum interval amongst one particular train departing from a station as well as the next train arriving in the same station (s) Minimum interval amongst a train passing via a station devoid of stopping along with the subsequent train arriving in the same station (s) Minimum interval amongst 1 train arriving at a station plus the subsequent train passing by means of the same station without the need of stopping (s) Minimum interval amongst one particular train departing from a station and the subsequent train passing by way of the identical station with out stopping (s) Duration with the study period (s) Duration of 1 cycle (s) Passenger OD volume from station r to station s in the course of the study period (persons)three.1. Train Timetable Calculations Establishing a train timetable demands the calculation with the interval at which trains depart from the origin station and their travel occasions along each inter-station segment with the route, PF-05381941 Description determined by which the arrival and departure occasions of each train at every single station could be obtained. Calculated timetables should satisfy the tmin constraint. It is also necessary to figure out no matter whether overtaking happens and, in that case, to adjust the arrival and departure times to account for the overtaking. 3.1.1. Calculation on the Arrival and Departure Instances of Trains at Each and every Station 1. Departure intervals A train schedule ought to be formulated to satisfy the passenger flow demand on the route. The minimum operating frequency will depend on the maximum sectional.