How many people touched me? Every time paperwork passes through a person's hands, it can be delayed. Figure out the minimum number of people who need to handle the invoice, and cut out any extras.
Did I get stuck anywhere waiting to be handled? For each stage of the process, set a time limit by which the paperwork needs to be processed. Otherwise, it risks sitting in a pile on someone's desk for days.
Was I prepared without error? Or did someone have to fix me? Create a checklist for employees so they can make sure the document is complete before handing it off.