Approximation schemes for parallel machine scheduling with non-renewable resources

In this paper the approximability of parallel machine scheduling problems with resource consuming jobs is studied. In these problems, in addition to a parallel machine environment, there are non-renewable resources, like raw materials, energy, or money, consumed by the jobs. Each resource has an initial stock, and some additional supplies at a-priori known moments in time and in known quantities. The schedules must respect the resource constraints as well. The optimization objective is either the makespan, or the maximum lateness. Polynomial time approximation schemes are provided under various assumptions, and it is shown that the makespan minimization problem is APX-complete if the number of machines is part of the input even if there are only two resources.