On scheduling pseudo just-in-time jobs on single machine

We are dealing with the problem of maximizing the weighted number of pseudo just-in-time jobs on single machine model. The concept of pseudo just-in-time is a relaxation of just-in-time. In our formulation, each job is associated with a time-window instead of due date, and the lengths of all such time-windows are bounded above by a certain value given in advanced. A job is called pseudo just-in-time if it is completed within its time-window. We show that the problem is polynomial time solvable, and moreover, we show some results on computational complexity for more general settings.