Optimal scheduling in a periodic environment

Abstract

We consider the problem of scheduling a sequence of actions when the benefit obtained from an action depends on only the current time modulo a period and the time since the previous action. A simple example is a model of a shuttle bus service where the profit from running a bus depends on the time of day and the time since the previous bus. Mathematically, such problems can be formulated as maximizing V(tn-1,tn) over schedules (tn)n0 as tN goes to infinity, for functions V(t,t´) satisfying the periodicity condition V(t+T,t´+T) = V(t,t´), for some T>0. The problem is related to Aubry-Mather theory in the dynamics of area-preserving maps. We extend this theory in order to prove the existence of optimizing schedules for each initial condition t0, to characterize the properties of such schedules and to analyse their dependence on t0 and on the parameters of the model.