Bus scheduling is a well-known NP-hard problem, and it is addressed with the use of heuristic solution methods or graphical approaches. In this study, the author proposes an improved formulation of the bus scheduling problem that considers the vehicle availability, the vehicle capacity and the allowed headway variability among successive trip dispatches. His formulation expands the classic bus scheduling model formulation by including the aforementioned features. In his study, the bus scheduling problem is understood as the problem of setting the optimal dispatching times for a set of pre-determined daily trips of a particular bus line. His model facilitates the search of solutions that can improve the waiting times of passengers while meeting the operational requirements and avoiding overcrowding. His proposed mathematical program is proved to be non-convex, and it is solved with heuristic solution methods because numerical optimisation approaches cannot guarantee a globally optimal solution. The performance of his approach is tested in a case study using real operational data from bus line 302 in Singapore. A simulation-based evaluation demonstrates potential gains of up to 20% on average passenger waiting times and a major reduction in refused passenger boardings because of overcrowding.

中文翻译：

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考虑出行时间，车辆可用性和容量的公交时刻表

总线调度是一个众所周知的NP难题，可以通过启发式解决方案方法或图形方法来解决。在这项研究中，作者提出了一种公交车调度问题的改进公式，该问题考虑了车辆的可用性，车辆的容量以及相继出行调度之间允许的车距变化。他的表述通过包括上述功能扩展了经典的公交车调度模型表述。在他的研究中，公交车调度问题被理解为为特定公交线路的一组预定每日行程设置最佳调度时间的问题。他的模型促进了解决方案的搜索，这些解决方案可以提高乘客的等候时间，同时满足运营要求并避免拥挤。他提出的数学程序被证明是非凸的，并且通过启发式求解方法得以解决，因为数值优化方法无法保证全局最优解。在案例研究中，使用来自新加坡公交302线的实际运行数据对他的方法的性能进行了测试。基于仿真的评估表明，平均乘客等待时间可提高20％，并且由于拥挤而导致的拒绝登机的情况大大减少。