A model for real-time bus holding subject to vehicle capacity limits

Two distinct directions of research have emerged for the vehicle holding problem: (i) single variable optimization approaches that determine the holding time of a single vehicle when it is about to depart from a bus stop; and, (ii) multivariable, periodic optimization approaches that use rather complex mathematical programs to determine the holding times of all running vehicles. Compre-hensive mathematical programs that consider multiple decision variables cannot be easily solved in real time, and are typically reserved for periodic control in longer time horizons. For this reason, this study focuses on single variable optimization approaches which determine the holding time of a vehicle when it arrives at a control point stop. Up to now, single variable optimization methods resort to rather simple, rule-based control logics. One of them is the one-headway-based logic which determines the holding time of a bus based on its headway with its preceding bus without addressing other implications, such as overcrowding. To rectify this, we introduce a new nonlinear 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 we prove that it can be solved to global optimality. A simulation-based investigation of the performance of our model against the performance of classic control logics that do not consider vehicle capacity limits is finally performed in bus line 302 in Singapore.