Scheduling with target start times

We address the single-machine problem of scheduling n independent jobs subject to target start times. Target start times are essentially release times that may be violated at a certain cost. The objective is to minimize a bicriteria objective function that is composed of total completion time and maximum promptness, which measures the observance of these target start times. We show that in case of a linear objective function the problem is solvable in O(n4) time if preemption is allowed or if total completion time outweighs maximum promptness.