  The scheduling problem in Hurdle (1973) was formulated in a general form that
simultaneously concerned the vehicle dispatching, circulating, fleet sizing,
and patron queueing. As a constrained variational problem, it remains not fully
solved for decades. With technical prowess in graphic analysis, the author
unveiled the closed-form solution for the optimal dispatch rates (with key
variables undetermined though), but only suggested the lower and upper bounds
of the optimal fleet size. Additionally, such a graphic analysis method lacks
high efficiency in computing specific scheduling problems, which are often of a
large scale (e.g., hundreds of bus lines). In light of this, the paper proposes
a novel mathematical solution approach that first relaxes the original problem
to an unconstrained one, and then attacks it using calculus of variations. The
corresponding Euler-Lagrange equation yields the closed-form solution of the
optimal dispatching rates, to which Hurdle's results are a special case. Thanks
to the proposed approach, the optimal fleet size can also be solved. This paper
completes the work of Hurdle (1973) by formalizing a solution method and
generalizing the results. Based on that, we further make two extensions to the
scheduling problem of general bus lines with multiple origins and destinations
and that of mixed-size or modular buses. Closed-form results are also obtained
with new insights. Among others, we find that the solutions for shuttle/feeder
lines are a special case to our results of general bus lines. Numerical
examples are also provided to demonstrate the effectiveness and efficiency of
the proposed approach.
