Revisit the scheduling problem in Hurdle, V.F., 1973: A novel mathematical solution approach and two extensions

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.

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