A short note on “A note on single-machine scheduling with job-dependent learning effects”

Abstract

Mosheiov & Sidney [5] showed that the single-machine makespan minimization problem with job-dependent learning effects can be formulated as an assignment problem and solved in O(n3) time. Later on, Koulamas [4] showed that the scheduling problem can be solved in O(nlogn) time by sequencing the jobs according to the shortest processing time (SPT) rule if the job-dependent learning rate are correlated with the level of sophistication of the jobs and bounded from below. The optimality of the SPT job sequence still holds when job-dependent learning rate are inversely correlated with the level of sophistication of the jobs and bounded from above. The main results in the paper are correct, however, there is an error in Corollary 1 and the proofs of Proposition 1 and Corollary 1 are not complete. In this note, a counter example for the latter case is given first and then a modified corollary is provided. Next, a lemma is provided to complete the proofs of Proposition 1 and Corollary 1. Finally, a simple algorithm is developed to solve the latter case in O(n2) time.