An analytic solution for real-time bus holding subject to vehicle capacity limits

This study focuses on single variable optimization approaches which determine the holding time of a vehicle when it is ready to depart from a bus stop. Up to now, single variable optimization methods resort to rule-based control logics to equalize the inter-departure headways or adhere to the target headway values. One of them is the two-headway-based control logic which determines the holding time of a bus based on its headway with its preceding and following bus without addressing other implications, such as overcrowding. To rectify this, we introduce a new model for the single variable bus holding problem that considers the passenger demand and vehicle capacity limits. Then, we reformulate this problem to an easier-to-solve program with the use of slack variables and introduce an analytic solution that can determine the holding time of a vehicle at the respective bus stop. Our analytic solution does not add a computational burden to the two-headway-based control logic and can be applied in real time. The operational benefit of our bus holding approach compared to other analytic solutions that do not consider the vehicle capacity is investigated using actual data from bus line 302 in Singapore.