There's info. available on where and how notches can be cut in joists, such as
http://www.eastdevon.gov.uk/txt/bc-guidancenote7-notchingofjoists.pdf
It's unlikely a 1 1/2" waste can be notched into a joist without seriously weakening it. You
might get away with boring a hole on the centre line of the joist, but again only in certain areas, and depending on the size of the joist (see link above), but you may lose the fall on the waste, and water doesn't run up hill.
Generally, the practical solution is to build a raised floor under the tray (eg. 25mm ply on 4X2's) so the waste can run above floor level.
It can make access to the trap and waste easier if it does get blocked.