Improved Lower-Bound Fleet Size for Transit Schedules

This work describes a highly informative graphical technique for the problem of finding the lower bound of the number of vehicles required to service a given timetable of trips. The technique is based on a step function that has been applied over the last 20 years as an optimization tool for minimizing the number of vehicles in a fixed-trip schedule. The step function is called a Deficit Function (DF), as it represents the deficit number of vehicles required at a particular terminal in a multi-terminal transit system. The initial lower bound on the fleet size with deadheading (empty) trip insertions was found to be the maximum of the sum of all DFs. An improved lower bound was established later, based on extending each trip’s arrival time to the time of the first feasible departure time of a trip to which it may be linked or to the end of the finite time horizon. The present work continues the effort to improve the lower bound by introducing a simple procedure to achieve this improvement that uses additional extension possibilities for a certain trip’s arrival times.