To enable invoice back ordering, click the gear icon, then setup, then options.
Scroll down and click the check box next to "Process Back Orders". Then save.
Now, when you associate an open PO with an invoice (screenshot 1), you will see an option to back order the item (screenshot 2). If you opt to back order the item, reciProfity will calculate the difference between what was ordered and what was received and create a PO with the back ordered items (using the same PO number).