I suggest that your administrators create a new assignment status of ‘Closed’. You can then set this when the assignment has reached a conclusion. This keeps the information readily available whilst making it clear that it is no longer in progress.
After a period of time (perhaps 6 months?) you can archive the assignment. This removes the assignment from reports and hides it from view, but it can still be retrieved if necessary.
After a further period of time, you can delete it entirely.