We study here a flow-line which covers a wide range of real-world situations, from plastic molding to steel milling and the service industries, where the duration of a task performed on a job is a monotone non-decreasing function of the time the job spent in the system.

Our model is set in a deterministic environment, with the initial conditions (i.e. release times of the jobs) as control parameters. Restricting to the 2-machine case for simplicity, we prove a closed form formula for the optimal release times of the jobs for the minimum weighted sum of completion times of a finite sequence of jobs.