Scheduling with step-improving processing times

We study a single machine scheduling problem of minimizing total completion time with step-improving processing times. For step-improving processing times, job processing times reduce by a job-dependent amount after a common critical date.
We show that the problem is NP-hard in general. We formulate a Mixed Integer Programming model and develop an LP-based heuristic for the problem. Finally, we perform computational experiments showing that the proposed heuristic provides effective solutions.