Earliness-tardiness scheduling problems, I: Weighted deviation of completion times about a common due date

This paper and its companion (Part II) concern the scheduling of jobs with cost penalities for both early and late completion. In Part I, the authors 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. They 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. The authors prove that the recognition version of this problem is NP-complete in the ordinary sense. They 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. The authors 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.