A Train Scheduling Model by Considering the Train Stops for Prayer

Considering the large amount of investment in railway transportation systems, effective use of resources is highly important. Typically, in order to utilize the line capacity, train scheduling plays a substantial role and thus attracts lots of attention all over the world.
Various train-scheduling models are introduced, according to different conditions of different countries. However, for Iran, no comprehensive scheduling model has been proposed yet, due to obligatory stops for the passenger trains to say their prayers.
In this paper, for first time, a comprehensive mathematical model by considering the constraint of obligatory stops for prayer is presented.
The constraint of obligatory stops for prayer is a domestic and religious attribute. In fact, Muslims believe it as essential to pray to God, in the certain times, with special rules or conditions. One of these rules is the restricted period of daily prayer. Another rule of praying is to keep still and calm, meanwhile. Moreover, the person must keep his/her body toward a certain direction, toward the City of Mecca, and should not turn away all during their prayers.
All the rules and conditions, above, necessitate stopping of the train in a certain period and in a suitable place. In this paper, the constraint of praying is modeled in three steps:
1. Recognition of the occurrence of praying conditions for each train at each station.
2. Recognition of stopping necessity for praying for each train.
3. Selection of the most appropriate station for stop.
At step 1, in order to recognize whether a train will reach a certain station in the permissible time of praying or not, it must confirm that the train arrives at the station after the beginning of this period and before the end of it. Only in the existence of both these conditions, the train can reach to the certain station in the permissible period and therefore can make a stop there.
At step 2, it verifies that the train has to stop for praying or not. In order to confirm the necessity or lack of necessity for stop at the intermediate station for praying, three groups of constraints are designed and integrated in the model. The first group of constraints verifies that the passengers have enough time for praying at the origin station. The second group of constraints verifies that the passengers have enough time at the destination station. Finally, the third group of constraints combines the results from the other two groups of constraints and determines in which trains, passengers do not have enough time for praying in either stations of origin and destination and therefore they must make a stop for praying on their way.
At step 3, based on consequences from steps 1 and 2, the best station is determined for making a stop on their ways. Finally, a small example and 11 sample problems are solved and the results are presented.