Summary and conclusion
Motivated by recent studies on real-time holding control for rail transit system, this paper presented a modified mathematical model for holding strategy applicable to urban bus transit operations. The developed model is flexible in adapting to two different bus-loading conditions (full and less than full capacity) when a given bus departs a stop along a given route. The cost function of the developed model measures the passengers’ total waiting time. This cost function is the sum of waiting times for new arrivals at stops and for those passengers that are left behind by a full-load bus. In the developed model, we proposed the concept of dwell time functions that calculate the delay times at stops based on the number of passengers boarding and alighting from the front and the rear door of the bus. A heuristic algorithm based on simulated annealing is presented for solving the model. The efficiency of the algorithm was tested using a numerical example. The first case assumes a regular service, and considered as a benchmark point, while the other case assumes an irregular service. When solving the numerical example for the irregular case, two decision situations where considered. First, a “do nothing” decision (the extreme case), that is, do not correct the irregularity of the bus service, with the second is to correct the irregularity of the service by using the holding strategy concept incorporated in the developed model. The results indicate that applying the holding strategy significantly improved the cost function by bringing its value closer to that of the benchmark case. The computational time to reach an optimal solution was found to be insignificant, which is what a manager desires when operating in a real-time environment.