Single machine scheduling subject to deadlines and resource dependent processing times

The problem of scheduling n jobs on a single machine is studied. Each job has a deadline and a processing time which is a linear decreasing function of the amount of a common resource allocated to the job. The objective is to find simultaneously a sequence of the jobs and a resource allocation so as the deadlines are satisfied and the total weighted resource consumption is minimized. The problem is shown to be solvable in O(n log n) time if the resource is continuously divisible. If the resource is discrete, then the problem is proved to be binary NP-hard. Some special cases are solvable in O(n log n) time. A fully polynomial approximation scheme is presented for the general problem with discrete resource.