Simultaneous resource scheduling to minimize weighted flow times

Many scheduling problems in manufacturing and service firms involve tasks that require more than one resource to be used simultaneously to execute the task. Two formulations of this scheduling problem are given. The authors develop a Lagrangian relaxation for the problem that has an intuitive interpretation. The relaxation suggests heuristics and provides a lower bound on the optimal solution. Next a surrogate relaxation is developed that provides a second bound and a heuristic solution. An enumeration procedure to determine the optimal solution is described, and computational results on the effectiveness of the bounds and the heuristics are presented. The Lagrangian relaxation performs better on problems with a low degree of simultaneity, and the surrogate relaxation does better with a high degree of simultaneity.