Minimizing the number of late jobs on a single machine under due date uncertainty

Abstract

We study the problem of minimizing the number of late jobs on a single machine where job processing times are known precisely and due dates are uncertain. The uncertainty is captured through a set of scenarios. In this environment, an appropriate criterion to select a schedule is to find one with the best worst-case performance, which minimizes the maximum number of late jobs over all scenarios. For a variable number of scenarios and two distinct due dates over all scenarios, the problem is proved NP-hard in the strong sense and non-approximable in pseudo-polynomial time with approximation ratio less than 2. It is polynomially solvable if the number s of scenarios and the number v of distinct due dates over all scenarios are given constants. An O(nlog n) time s-approximation algorithm is suggested for the general case, where n is the number of jobs, and a polynomial 3-approximation algorithm is suggested for the case of unit-time jobs and a constant number of scenarios. Furthermore, an O(n ^s+v−2/(v−1)^v−2) time dynamic programming algorithm is presented for the case of unit-time jobs. The problem with unit-time jobs and the number of late jobs not exceeding a given constant value is solvable in polynomial time by an enumeration algorithm. The obtained results are related to a min-max assignment problem, an exact assignment problem and a multi-agent scheduling problem.