Optimizing scheduled times, allowing for behavioural response

We consider transport activities for which time has to be allocated or scheduled in advance. When the schedule is implemented, the time actually taken by each activity is subject to random variation, hence can exceed the scheduled time. To reduce such over-runs or lateness, and improve reliability and costs, some extra time is usually allowed for some, or all, activities in the schedule. However, it is well known that if more time is allocated for an activity then the activity often tends to take longer. Because of this behavioural response, some or all of the benefits (in reliability, costs, etc.) of the extra time allowance are lost. To compensate for this, should even more time be allowed for each activity, or should less be allowed, and if so, how much? We consider this question here and in particular we discuss the effect of such behavioural response on expected costs and on the optimal time to allow for an activity. We find that the optimal time to allow depends, in very simple ways, on a behavioural response ratio, and on the ratio of scheduled time costs to lateness costs. The model is applicable to computing optimal times for public transport timetables, for buses, trains or airlines. It is also relevant to choosing how much time to allow for each of a set of operations in production scheduling or service scheduling.