Minimizing the sum of weighted completion times with unrestricted weights

Given a set of tasks with associated processing times, deadlines and weights unrestricted in sign, we consider the problem of determining a task schedule on one machine by minimizing the sum of weighted completion times. The problem is NP-hard in the strong sense. We present a lower bound based on task splitting, an approximation algorithm, and two exact approaches, one based on branch-and-bound and one on dynamic programming. An overall exact algorithm is obtained by combining these two approaches. Extensive computational experiments show the effectiveness of the proposed algorithm. © 1995