Scheduling to minimize the total resource consumption with a constraint on the sum of completion times

We consider the problem of scheduling a set of jobs on a single machine where the release time of a job is related to the amount of resource consumed. The objective is to minimize the total amount of resource consumed subject to a constraint on the sum of completion times of the jobs. We show that the problem is NP-hard in general and can be solved efficiently when the resource consumption function is linear. The parallel machine case is also discussed.