Due-window assignment with unit processing-time jobs

In due-window assignment problems, jobs completed within a designated time interval are regarded as being on time, whereas early and tardy jobs are penalized. The objective is to determine the location and size of the due-window, as well as the job schedule. We address a common due-window assignment problem on parallel identical machines with unit processing time jobs. We show that the number of candidate values for the optimal due-window starting time and for the optimal due-window completion time are bounded by 2. We also prove that the starting time of the first job on each of the machines is either 0 or 1, thus introducing a fairly simple, constant-time solution for the problem.