It is your manager’s job to set you up for SUCCESS not failure. It sounds as though your function is integral to the operation and profitability of the company so unless something nefarious is afoot upper management should take your concerns more seriously.
Assuming they don’t WANT you to fail so they can use you as a fall guy here are a few suggestions:
1) Quantify the problem and appeal to their bottom line
Simply saying that you are overworked sounds like complaining and smells like ungratefulness.
SHOW them how with assistance YOU can MANAGE the inspections more efficiently, increase their profitability, and reduce loss due to failed inspections, downtime, etc.
For example, if machine X and Y pass inspection faster the company will save this dollar amount. If you can quantify possible gains in dollar figures and include this in a report it will get their attention.
2) Ask for feedback
Again, for upper management to be unaware of your situation doesn’t reflect very well on your company. However, they might be green or experiencing personal issues, etc. Whatever the case, it is clear that you must take ownership of your success. Ask your manager for feedback concerning your productivity and the quality of your work. If they didn’t make it clear how you would be evaluated when you were employed then NOW is the time for them to provide that criteria.
3) Make an honest self assessment
If the company doesn’t have any real system for evaluating employee performance or for addressing employee concerns it may be time to start looking for another job.