Extend timetable to complete day

The preprocessing is done using python and bash to come up with a AMPL file which is then solved using Gurobi. The constraints elaborated by Agrawal et. al. [8] such as headway, frequency, symmetry, turnaround, rake-linking are modelled in AMPL. The OD file is designed (manually) such that the number of rakes to be used and their starting locations are specified in the same file. This is done on the basis of last services of morning peak timetable. (Services terminating at Vashi/Belapur are stabled and not used in offpeak timetable) Along with that few more constraints are added in order to fill the noon and night gap in the timetable. Given below is the explanation of the whole scenario :

• Departure Time Fixation Constraint

This constraint is used when linking the morning peak end services to off peak start services. For eg - If 14 services are terminating at CST during morning peak, than the same needs to start from CST and is divided into CST-PNVL, CST-AND, and CST- BA services for off peak hours. In addition to headway, dwell, traversal, turnaround, rake-linking and symmetric constraints another ’departure time fixation’ constraint is added. Since we have the terminating times of the 14 services at CST, this time fixation constraint maps the 14 services with another 14 services in off peak hour such that the starting time of these services in off peak period is greater than 3 minutes but less than 6 minutes from the mapped service arrival in morning peak. In order to model the constraints in AMPL we consider two sets, Offstart , which contain all the departure events from a particular station in off peak and another set Morningend which contains the timings of events that are terminating at the same station at the end of morning peak. We define a binary variable link dep [i, j] where i varies for all the elements in set Offstart and j varies for all elements in Morningend

• Arrival Time Fixation Constraint

This constraint is for fixing the arrival times of the services ending in the offpeak so that they get linked to evening peak start. The evening peak timetable is input for this constraint. The origin stations and departure times of all the rakes are taken into consideration. The constraint can be explained as follows by taking CST into account - We define a set called as Offend that contains the event numbers of all the services terminating at CST during the course of offpeak period. Now let’s say that 14 ser- vices are starting from CST in evening peak (Evestart) and we have the information on their starting times and stations using evening peak timetable. Now these 14 ser- vices(Evestart) are to be mapped with services from the larger set of arrival events at CST(Offend ). Again the constraint is defined such that the event timings of the linked services (arr[Offend]) must be no more than 6 minutes and no less than 3 minutes as it has to get linked with the evening service. The best possible 14 services are chosen from set (Offend ) and are declared as sinks so that they get linked with evening peak services. We define a binary variable link arr [i, j] where i varies for all the elements in set Offend and j varies for all elements in Evestart

The output of the algorithm above is times associated with the departure and arrival events and the rake linkages. Post processing scripts have been written in order to convert the output to the desired format. Also scripts have been written so that we get a vertical tabular display of all the services with time-breakup at the major stations. This is the output required by the Central Railways.