Maximizing the weighted number of just-in-time jobs on a single machine with position-dependent processing times

Abstract

We study the problem of maximizing the weighted number of just-in-time jobs on a single machine with position-dependent processing times. Unlike the vast majority of the literature, we do not restrict ourselves to a specific model of processing time function. Rather, we assume that the processing time function can be of any functional structure that is according to one of the following two cases. The first is the case where the job processing times are under a learning effect, i.e., each job processing time is a non-increasing function of its position in the sequence. In the second case, an aging effect is assumed, i.e., each job processing time is a non-decreasing function of its position in the sequence. We prove that the problem is strongly $\mathcal{N }\mathcal{P }$-hard under a learning effect, even if all the weights are identical. When there is an aging effect, we introduce a dynamic programming (DP) procedure that solves the problem with arbitrary weights in $O(n^{3})$ time (where $n$ is the number of jobs). For identical weights, a faster optimization algorithm that runs in $O(n\log n)$ time is presented. We also extend the analysis to the case of scheduling on a set of $m$ parallel unrelated machines and provide a DP procedure that solves the problem in polynomial time, given that $m$ is fixed and that the jobs are under an aging effect.