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. Set-up times are required to switch from one family to another, but are not required to switch 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 fixed. The optimization is with respect to the weighted flowtime, and we treat problems both with and without a makespan-constraint. Practical examples based on this model are described. 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 set-ups. Without a makespan constraint and without preemption, we prove that our analysis of a deterministic model extends to stochastic set-up and processing times without loss of optimality. Managerial insights based on our technical results are provided.
