A step function for improving transit operations planning using fixed and variable scheduling

This work describes a highly informative graphical technique for the problem of finding the least number of vehicles required to service a given timetable of trips. The technique used is a step function, called a deficit function, which was introduced in the last 20 years as an optimization toot for minimizing the number of vehicles in a fixed trip schedule. However not much attention was given to the possibility of variable trip schedule, within given tolerances, and to the deficit function use for additional elements in the transit operations planning process. The objectives of this work are four fold: (a) to develop an improved lower bound to the fixed schedule fleet size problem, (b) to use the deficit function for minimizing the fleet size with variable schedules (possible shifts in departure times), (c) to allow for the combination of deadheading trip insertions and shifts in departure times in the fleet size minimization problem, and (d) to outline example applications of the deficit function use in designing better transit services. In addition this work covers the procedures to create the chains of trips (daily vehicle duty or block) where the number of these chains complies with the minimum fleet size derived. The algorithms developed are accompanied with examples. The approach used in this work provides immediate feedback on the value of shifting departure times, within given tolerances, as well as combining these shifts with the insertion of deadheading trips for reducing the fleet size, The value of embarking on such a technique is to achieve the greatest vehicle saving while complying with passenger demand. This saving isattained through a procedure incorporating a man/computer interface which would allow the inclusion of practical considerations that experienced transit schedulers may wish to introduce in the schedule.