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

TL;DR

This paper revisits Hurdle's 1973 scheduling problem, introducing a novel calculus of variations approach to derive closed-form solutions for optimal dispatch rates and fleet size, extending the original work to more complex bus line scenarios.

Contribution

It formalizes a new mathematical solution method for Hurdle's scheduling problem, generalizes the results, and extends the model to complex bus line configurations with closed-form solutions.

Findings

Closed-form solutions for optimal dispatch rates and fleet size.

The approach outperforms graphic analysis in efficiency for large-scale problems.

Solutions for shuttle/feeder lines are special cases of the general bus line results.

Abstract

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.