Swapping the order of scheduled services to minimize expected costs of delays

In operating scheduled public transport services, by bus, train, or airline, any delay in the start time of one activity may cause "knock-on" delays to the next activity: for example, a train departure delay may delay the next train. In view of this, dispatchers and operators often have to decide whether delays and costs may be reduced by swapping the order of the delayed activities. We consider this decision problem here, with trains as an example. We analyse a minimal information model in which the only information about departure delays is the probability distribution of the "ready to depart" times. We show that in this case the optimal (cost minimizing) swapping policy usually reduces to one of two "bang-bang" policies: swap immediately or never swap. We develop simple heuristics for deciding which of these two policies is best. We also consider the effect of new or updated information about "ready to start" times becoming available as that time approaches, and extend this to a full information model in which the exact delays are known in advance. We discuss the application of these methods developed for pairs of trains to multiple trains and stations.