This paper presents a simultaneous approach to the school bus routing and scheduling problem with stochastic time-dependent travel times that guarantees arriving on-time at a single school for all buses with a required reliability level, where bus stop locations, transportation demand and school bell time are known in advance. Dealing with the scheduling problem, previous studies have focused on determining the dispatch times of buses. We extend the meaning of scheduling to a problem in which the pickup times for all bus stops are determined as well. The proposed simultaneous routing and scheduling problem is treated as a bi-level problem and solved by a heuristic algorithm that is a combination of Ant Colony Optimization and a proposed Route Decomposition heuristic algorithm. To discover the importance of the effect of the time-dependent character of travel times, a comparative small example has been solved under two cases: (i) stochastic and time-dependent travel times and (ii) stochastic -only travel times. Results show that the aggregation errors in the stochastic-only case, which are due to aggregating all travel time records into a single probability distribution, are remarkable and should not be neglected. Furthermore, a second example has been solved to show the applicability of the proposed method in large instances.
