Earliness-Tardiness Scheduling Problems, I: Weighted Deviation of Completion Times About a Common Due Date

TLDR

The paper studies scheduling of jobs with penalties for both early and late completion, where job weights vary but are independent of timing, and Part II extends the results to the unweighted case with arbitrary due dates. The study aims to minimize the weighted sum of earliness and tardiness for jobs on a single processor around a common due date, assuming the due date is not early enough to constrain scheduling. The authors provide optimality conditions, a computationally efficient dynamic programming algorithm, and identify four special cases that are polynomially solvable. The recognition problem is NP‑complete, but a fully polynomial approximation scheme exists when job weights are polynomially bounded.

Abstract

This paper and its companion (Part II) concern the scheduling of jobs with cost penalties for both early and late completion. In Part I, we consider the problem of minimizing the weighted sum of earliness and tardiness of jobs scheduled on a single processor around a common due date, d. We assume that d is not early enough to constrain the scheduling decision. The weight of a job does not depend on whether the job is early or late, but weights may vary between jobs. We prove that the recognition version of this problem is NP-complete in the ordinary sense. We describe optimality conditions, and present a computationally efficient dynamic programming algorithm. When the weights are bounded by a polynomial function of the number of jobs, a fully polynomial approximation scheme is given. We also describe four special cases for which the problem is polynomially solvable. Part II provides similar results for the unweighted version of this problem, where d is arbitrary.