The Stability Box in Interval Data for Minimizing the Sum of Weighted Completion Times

We consider a single-machine scheduling problem, in which the processing time of a job can take any value from a given segment. The criterion is to minimize the sum of weighted completion times of the n jobs, a positive weight being associated with a job. For a job permutation, we study the stability box, which is a subset of the stability region. We derive an O(n logn) algorithm for constructing a job permutation with the largest volume and dimension of a stability box. The efficiency of a permutation with the largest dimension and volume of a stability box is demonstrated via a simulation on a set of randomly generated instances with 1000≤ n≤ 2000. If several permutations have the largest volume of a stability box, the developed algorithm selects one of them due to one of three simple heuristics: a lower-point heuristic, an upper-point heuristic or a mid-point heuristic.