Parallel-machine scheduling with controllable processing times

We consider a problem of scheduling n independent and simultaneously available jobs on m unrelated parallel machines. The job processing times can be compressed through incurring an additional cost, which is a convex function of the amount of compression. Two problems are formulated as assignment problems, which can be solved in O (n(3)m + n(2)m log(nm)) time. One is to minimize the total compression cost plus the total flow time. The other is to minimize the total compression cost plus the sum of earliness and tardiness costs.