Estimation of Fleet Size for Variable Bus Schedules

Abstract

A highly informative graphical technique addresses the problem of finding the lower bound of the number of buses required to service a given timetable of trips. The technique is based on a step function that has been applied over the past 20 years as an optimization tool for minimizing the number of vehicles in a fixed trip schedule. The step function is called deficit function (DF), for it represents the deficit number of vehicles required at a particular terminal in a multiterminal transit system. The initial lower bound on the fleet size with deadheading (DH) (empty) trip insertions was found to be the maximum of the sum of all DFs. An improved lower bound was established later; it was 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 calculate and improve the lower bound by considering variable departure times along with possible insertions of DH trips. Certainly, the use of shifting departure times opens up more opportunities to reduce bus fleet size than does allowing only for DH trip insertions. This work shows how to apply possible shifts in departure times, within their tolerances, to improving the lower bound. The approach presented will enable bus schedulers and transit decision makers to decide more promptly how far the fleet size can be reduced by DH trip insertions and allowing shifts in departure times.