EARLINESS-TARDINESS SCHEDULING PROBLEMS .1. WEIGHTED DEVIATION OF COMPLETION TIMES ABOUT A COMMON DUE DATE

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.