Optimal bus services on express basis in the case of balking and reneging

This paper considers the problem of dispatching vehicles (buses, trains, aeroplanes, etc.) to serve passengers from some origin to a given destination. The authors assume that the passengers arrive at the origin continuously with deterministic arrival rate. They further assume that the customers leave the queue at a rate which depends upon the time as well as queue length. The vehicles are dispatched at discrete times in such a way as to minimize a cost function which is the sum of total operating cost of transportation, loss of revenue due to vacant seats and reneging of the customers, and waiting cost of passengers. Dynamic programming has been used to find out the optimal dispatching policy consisting of number of passengers in different trips, dispatch times of the vehicles and the number of vehicle trips.