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

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 NP-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 (pmax); and (iv) the maximal due date (dmax). 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. pmax. We then provide two FPT algorithms showing that the problem is also tractable wrt. to upsilon p and dmax. 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 NP-hard in general, but solvable in either FPT time or polynomial time for some special cases.

Automated summary

This paper introduces two new perspectives on single-machine scheduling problems with penalties for late work, which have not been considered in previous literature. Firstly, a parameterized complexity analysis is conducted for the NP-hard problem of minimizing total late work on a single machine, considering four parameters. Secondly, two FPT algorithms are proposed to solve the problem, showing its tractability in relation to certain parameters. Furthermore, the paper explores a single-machine scheduling problem with assignable due dates and penalties for early and tardy work, providing an efficient method to assign due dates and identifying the problem's NP-hardness as well as solvability for special cases.