Approximating Weighted Completion Time for Order Scheduling with Setup Times

Consider a scheduling problem in which jobs need to be processed on a single machine. Each job has a weight and is composed of several operations belonging to different families. The machine needs to perform a setup between the processing of operations of different families. A job is completed when its latest operation completes and the goal is to minimize the total weighted completion time of all jobs. We study this problem from the perspective of approximability and provide constant factor approximations as well as an inapproximability result. Prior to this work, only the NP-hardness of the unweighted case and the polynomial solvability of a certain special case were known.