@chartl Thank you very much for your detailed response - I am going to test it out and see how I go.
Just a couple things to clarify:
I actually know more than just the total cartons made - I know the quantity of each carton made as well. For example: x = (0, 0, 1000, 0, 488, 0), not just 1488.
The other thing that I am struggling to wrap my mind around is:
ϕ is the proportion of the 8 hour period that the product run for which I am estimating as well as r_n, but ϕ is not independent of r_n… Does that matter?