Trains scheduling problem with multiple lines.

This study explores the problem of train scheduling (or) train timetabling and its impact on the administration of railway management. This is a highly dependable and effective public transportation system. The problem considers both single and multiple tracks along with multiple platforms with varying train capacities (like speed, passengers, and so on). Let the tracks link two major stations source and destination with intermediate stations. A subset of intermediate stations will function as junctions. A finite number of trains be available at the source for the passengers/cargo transit movements. The profit which is generated from the shipment of passengers between a pair of stations is known in advance. The train's travel time between the stations and halting time at the stations are predefined. The arrival and departure times at the stations are calculated through the travel time and halting time. All the trains should starts from the source station and continue the journey through the intermediate stations to reach the destination station. Overtaking of the trains is permitted only at the intermediate stations. The trains must make a halt at one or more intermediate stations before reaching the destination. Now the objective is to find the best train operating schedule that maximizes the profit within the admissible travel time threshold. A zero-one integer linear programming is used to model this problem mathematically. For a better understanding of this problem, a case study is considered from the Indian railway network with two major stations Chennai and Hyderabad. A branch and bound [Formula: see text] algorithm is proposed to determine an optimal operating schedule. In addition, the experiments are carried out on a wide range of randomly generated instances of small and medium sizes, to test the efficiency of the algorithm. The computational results indicate that the algorithm is capable of finding the optimal schedules within a reasonable amount of time.