Minmax scheduling and due-window assignment with position-dependent processing times and job rejection

We study scheduling and due-window assignment problems with the objective function of a minmax type, i.e., the goal is to minimize the largest cost among all scheduled jobs. We assume that the processing times of jobs are position-dependent in the most general way. For a single machine and for a proportionate flow shop environment we present polynomial time solution procedures that are based on solving a linear assignment problem as a subroutine. We further extend the single machine model by allowing job-rejection, provided that the processing times deteriorate, i.e., the position-dependent processing times are non-decreasing functions of the job position. For this setting, the scheduler may decide not to process certain jobs, and each rejected job is penalized accordingly. The problem with the objective that additionally involves a maximum rejection cost component is also shown to be solvable in polynomial time.