Periodic Scheduling with Service Constraints

We consider the problem of servicing a number of objects in a discrete time environment. In each period, we may select an object that will receive a service in the period. Each time an object is serviced, we incur a servicing cost dependent on the time since the object's last service. Problems of this type appear in many contexts, e.g., multiproduct lot-sizing, machine maintenance, and several problems in telecommunications. We assume that at most one object can be serviced in a given period. For the general problem with m objects, which is known to be 𝒩𝒫-Hard, we describe properties of an optimal policy, and for the specific case of m = 2 objects, we determine an optimal policy.