Transit ridership is growing and the green movement is gaining momentum. In order to encourage current users to continue using urban bus systems and to attract new users to urban bus systems, the most comprehensive, affordable, and reliable system possible must be created. While a number of components are imperative to the success of a bus system, this research focuses on passengers' experiences and briefly considers an agency's operating costs. In terms of operating costs, the discussion centers around the determination of headways that optimize passenger loads. These optimal headways will also satisfy a minimum passenger comfort requirement. The majority of this research looks at passengers' experiences by answering the question, how can scheduled running times on low frequency bus routes be optimized in order to minimize the costs passengers incur due to in-vehicle travel time and waiting time? Passenger loads are optimized on an existing route by taking into account each vehicle's maximum desirable load and the number of passengers who demand bus service at each timepoint. This information is used to determine how frequently buses must depart from the route's initial timepoint to ensure that the maximum passenger load on each bus is as close as possible to its desired level. By bringing passenger loads as close as possible to their desired levels, transit agencies minimize their operating costs while guaranteeing a predetermined level of passenger comfort. This optimization process is explained and applied to a San Francisco Municipal Transportation Agency (SFMTA) bus route. The optimal headways from this optimization are compared with the headways used by the SFMTA. Three models, a linear programming model, a quadratic programming model, and a numerical model, are presented for determining the scheduled segment running times that minimize the net cost on low frequency bus routes. The net cost includes late boarding and alighting costs, an early dragging cost, and a reward for delivering passengers to their destination earlier than scheduled. These costs depend on the number of boarding and alighting passengers at each timepoint, the number of through passengers on each segment, the minimum and maximum segment and end-to-end running times, and the per-passenger costs associated with in-vehicle travel time and waiting time. The theory behind each model is discussed and a case study is presented in which data from the Chicago Transit Authority's (CTA) Route #152 are used in each model to determine the route's optimal scheduled running times. The linear and quadratic programming models are solved with Maple while a brute-force computer algorithm is developed and used with the numerical model. The optimal scheduled running times determined by the numerical model, which is the best of the three models presented here, produce a timetable that has a net cost that is nearly 10% less than the net cost associated with the previous version of the #152's timetable and more than 70% less than the net cost associated with the #152's current timetable. While the models developed in this thesis do not capture all factors relevant to the determination of optimal scheduled running times, they provide an excellent starting point for setting scheduled running times. Further, these models are easily adaptable to any route's unique characteristics and any transit agency's valuation of late and early costs.


Ramsay, John R.




Applied Mathematics

Publication Date


Degree Granted

Bachelor of Arts

Document Type

Senior Independent Study Thesis



© Copyright 2009 Samuel W. Hickey