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The scheduling problem in Hurdle (1973) was formulated in a general form thatsimultaneously concerned the vehicle dispatching, circulating, fleet sizing,and patron queueing. As a constrained variational problem, it remains not fullysolved for decades. With technical prowess in graphic analysis, the authorunveiled the closed-form solution for the optimal dispatch rates (with keyvariables undetermined though), but only suggested the lower and upper boundsof the optimal fleet size. Additionally, such a graphic analysis method lackshigh efficiency in computing specific scheduling problems, which are often of alarge scale (e.g., hundreds of bus lines). In light of this, the paper proposesa novel mathematical solution approach that first relaxes the original problemto an unconstrained one, and then attacks it using calculus of variations. Thecorresponding Euler-Lagrange equation yields the closed-form solution of theoptimal dispatching rates, to which Hurdle's results are a special case. Thanksto the proposed approach, the optimal fleet size can also be solved. This papercompletes the work of Hurdle (1973) by formalizing a solution method andgeneralizing the results. Based on that, we further make two extensions to thescheduling problem of general bus lines with multiple origins and destinationsand that of mixed-size or modular buses. Closed-form results are also obtainedwith new insights. Among others, we find that the solutions for shuttle/feederlines are a special case to our results of general bus lines. Numericalexamples are also provided to demonstrate the effectiveness and efficiency ofthe proposed approach.

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