Minimum Weighted Completion Time ( 1999

The minimum weighted completion time problem involves (i) a set J of n jobs, a positive weight wj for each job j ∈ J , and a release date rj before which it cannot be scheduled; (ii) a set of m machines, each of which can process at most one job at any time, and (iii) an arbitrary set of positive values {pi,j}, where pi,j denotes the time to process job j on machine i. A schedule involves assigning jobs to machines and choosing an order in which they are processed. Let Cj denote the completion time of job j for a given schedule. The weighted completion time of a schedule is defined as ∑ j∈J wjCj , and the goal is to compute a schedule that has the minimum weighted completion time. In the scheduling notation introduced by Graham et al. [7], a scheduling problem is denoted by a 3-tuple α | β | γ, where α denotes the machine environment, β denotes the additional constraints on jobs, and γ denotes the objective function. In this article, we will be considered with the αvalues 1, P , R, and Rm which respectively denote one machine, identical parallel machines (i.e., for a fixed job j and for each machine i, pi,j equals a value pj that is independent of i), unrelated machines (pi,j ’s are dependent on both job i and machine j), and a fixed number m (not part of the input) of unrelated machines. The field β takes on the values rj which indicates that the jobs have release dates, and the value pmtn which indicates that preemption of jobs is permitted. Further, the value prec in the field β indicates that the problem may involve precedence constraints between jobs which poses further restrictions on the schedule. The field γ is either ∑ wjCj or ∑ Cj which denote total weighted and total (unweighted) completion times respectively. Some of the simpler classes of the weighted completion time scheduling problems admit optimal polynomial time solutions. They include the problem P | | ∑ Cj for which the shortest-job-first strategy is optimal, the problem 1 | | ∑ wjCj for which Smith’s rule [13] (scheduling jobs in their non-decreasing order of pj wj values) is optimal, and the problem R | | ∑ Cj which can be solved via matching techniques [2, 9]. With the introduction of release dates, even the simplest classes of the weighted completion time minimization problem becomes strongly NP-hard. In this article, we focus on the work of Afrati et al. [1] whose main contribution is the design of polynomial-time approximation schemes (PTASs) for several classes of scheduling problems to minimize weighted completion time with release dates. Prior to this work, the best solutions for minimizing weighted completion time with release dates were all O(1)-approximation algorithms (for example, [4, 5, 11]); the only known PTAS for a strongly NP-Hard problem involving weighted completion time was