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 of times and in known quantities. The schedules must respect the resource constraints as well. The objective is the minimum schedule length or makespan. Polynomial time approximation schemes are provided under various assumptions, and it is shown that the problem becomes APX-hard if the number of machines is part of the input even if there are only two resources