Minimizing a Separable Convex Function on Parallel Machines with Preemptions

The basic scheduling problem we are dealing with in this paper is the following one. A set of jobs has to be scheduled on a set of parallel uniform machines. Each machine can handle at most one job at a time. All jobs have the same execution requirement. Each machine has a known speed. The processing of any job may be interrupted arbitrarily often and resumed later on any machine. The goal is to find a schedule that minimizes ∑ fj, where fj are convex non-decreasing functions such that fi−fj are all monotonic functions. Thus, we consider problem Q | pj = p, pmtn | ∑ fj. We show that problem Q | pj = p, pmtn | ∑ fj is equivalent to the problem of minimizing a convexseparable function under linear constraints. We use this representation to solve problems Q | pj = p, pmtn | ∑ Tj and Q | pj = p, pmtn | ∑ wjCj in polynomial time. Note that both problems Q | pj = p, pmtn | ∑ wjCj and Q | pj = p, pmtn | ∑ Tj had an open complexity status. Recently, Tian et al. [10] proposed a polynomial algorithm for problem 1 | rj, pj = p, pmtn | ∑ Tj. We show that both the problem P | pmtn | ∑ Tj of minimizing total tardiness on a set of parallel machines with allowed preemptions and the problem P | rj, pj = p, pmtn | ∑ Tj of minimizing total tardiness on a set of parallel ∗Corresponding author. Tel.:+0391 6712025; fax:+0391 6711171. E-mail address: [email protected] (F.Werner).