Single machine scheduling with step-learning

In this paper, we study scheduling with step-learning, i.e., a setting where the processing times of the jobs started after their job-dependent learning-dates are reduced. The goal is to minimize makespan on a single machine. We focus first on the case that idle times between consecutive jobs are not allowed. We prove that the problem is NP-hard, implying that no polynomial-time solution exists and, consequently, propose a pseudo-polynomial time dynamic programming algorithm. An extensive numerical study is provided to examine the running time of the algorithm with different learning-dates and job processing time ranges. The special case of a common learning-date for all the jobs is also studied, and a (more efficient) pseudo-polynomial dynamic programming is introduced and tested numerically. In the last part of the paper, the more complicated setting in which idle times are allowed is studied. An appropriate dynamic programming is introduced and tested as well.