Real-time cruising speed design approach for multiline bus systems

In this paper, we focus on controlling multiline buses operated in networks with curbside bus stops. In such networks, both bus bunching and bus queueing, which often result in passenger inconvenience as well as bus waiting delays, are frequently observed during bus operations. To address the adverse influences of these two phenomena, we propose a mixed integer programming (MIP) model to provide guidance on real-time bus cruising speeds based on the real-time state of the whole system. The proposed model can avoid bus bunching by coordinating bus cruising speeds and alleviate bus queueing congestion by restricting the number of bus arrivals at each stop. Specifically, to address bus bunching, the model has a quadratic objective function that minimizes the total expected passenger waiting time; while for bus queueing, the model has big-M and time-indexed constraints that restrict the number of bus arrivals in each time interval of length g based on the waiting capacity Qs of each stop s. Simulation experiments are conducted with different sizes of virtual networks, and the corresponding results show that the proposed model can lead to both a shorter average passenger waiting time and less bus congestion at stops. The improvement with respect to bus waiting delay is more significant: the percentage decrease can reach 46.9%. Moreover, the results of the average computing time show that the control model can be solved before a bus finishes service in most cases, which means decisions can be fed back to drivers in a timely manner; therefore, the proposed model can meet the requirements of real-time control. A sensitivity analysis with respect to parameters Qs and g is also performed. These two parameters are shown to only affect control performance with respect to bus waiting delays at stops, and shorter bus waiting delays can be achieved with relatively small Qs and large g. Further experiments are conducted within a real network, and the experimental results show that our method is suitable for implementation in practice.