Minimizing non-decreasing separable objective functions for the unit-time open shop scheduling problem

In this paper we study both the non-preemptive and preemptive versions of the popular unit-time open shop scheduling problem. For the set of feasible schedules which satisfy a predetermined order of job completion times, we construct the linear description of the convex hull of the vectors of the job completion times. Based on the properties of the resulting scheduling polydehdron we show that the problem of constructing an optimal schedule minimizing an arbitary non-decreasing separable cost function of job completion times is polynomially solvable.