Single machine scheduling with step-deteriorating processing times

We study in this paper a scheduling model in which each task has a normal processing time which deteriorates as a step function if its starting time is beyond a given deteriorating date. We focus on problems with identical task deteriorating dates. We show that the flow time problem is NP-complete and suggest a pseudo-polynomial algorithm for the makespan problem. We also introduce a general method of solution, which facilitates the identification of solvable cases for some related problems. Finally, we provide a counter-example that invalidates the conjecture of optimality of a switching algorithm reported in the literature. Thus, we solve several unexplored or open problems and obtain a sharp boundary delineating the complexity of the considered problems.