Properties of Optimal-Weighted Flowtime Policies with a Makespan Constraint and Set-up Times

We characterize optimal policies for the problem of allocating a single server to a set of jobs from N families. Each job is an instance of demand for an item and is associated with a family, a holding cost rate, and a mean processing time. Setup times are required to switch from one family to another, but not within a family. We consider the case in which the order of jobs within the family is unconstrained and a variation in which the order is xed. The optimization is with respect to the weighted owtime, and we treat problems both with and without a makespan-constraint. We partially characterize an optimal policy by means of a Gittins reward rate index and a similar switching index derived from multi-armed bandit theory. For deterministic problems with a makespan constraint, we present an optimization algorithm for the special case of two families and at most three setups. Without a makespan constraint and without preemption, we prove that our analysis of a deterministic model extends to stochastic setup and processing times without loss of optimality.