A new perspective on single-machine scheduling problems with late work related criteria

Abstract

This paper provides two new perspectives on single-machine scheduling problems in which the objective involves penalties regarding late work. Both of this perspectives have been neglected in the previous literature. We begin by presenting a parameterized complexity analysis of the $\mathcal{N}\mathcal{P}$-hard problem of minimizing the total late work on a single machine. We do so with respect to the following four parameters: (i) the number of different processing times ($\upsilon _{p}$); (ii) the number of different due dates ($\upsilon _{d}$); (iii) the maximal processing time $({p}_{\max });$ and (iv) the maximal due date ($d_{\max }$). We use results from the literature to conclude that the problem is hard with respect to (wrt.) parameter $\upsilon _{d}$ and is tractable (i.e., solvable in FPT time) wrt. $p_{\max }$. We then provide two FPT algorithms showing that the problem is also tractable wrt. to $\upsilon _{p}$ and $d_{\max }$. We continue by analyzing a single-machine scheduling problem with assignable due dates where the cost function to be minimized includes penalties due to weighted early and tardy work. We assume that each job can be assigned a different due-date, the value of which is subject to a job-dependent upper bound. We provide an efficient method to optimally assign due dates for a given job schedule. We then use this result to reduce the problem to a purely combinatorial problem, which we show is $\mathcal{N}\mathcal{P}$-hard in general, but solvable in either FPT time or polynomial time for some special cases.