Weighted throughput in a single machine preemptive scheduling with continuous controllable processing times

We consider the problem of weighted throughput in the single machine preemptive scheduling with continuous controllable processing times. A set of tasks can be scheduled on a single machine. Each task j is associated with a nonnegative weight \(w_{j}\), a release date, a due date, and an interval of possible processing times. A task j can either be scheduled with a total processing time \(p_j\) which is in the given interval, or rejected (not participating in the schedule). The reward for processing j for \(p_{j}\) time units is \(w_{j}p_{j}\), and we are interested in constructing a feasible preemptive schedule such that the sum of rewards is maximized. We present a dynamic programming algorithm that solves the problem in pseudo-polynomial time and use it to obtain an FPTAS. Afterward, as our main contribution we propose an interesting efficient frontier approach for improved complexity bounds.