Answer:
see below
Step-by-step explanation:
In order for the problem to make sense, we have to assume typos in Patterns 2, 3, and 4. We suppose they should be (0, 0, 3, 10), (8, 0, 0, 4) and (2, 1, 2, 1).
If we let p, q, r, s, t represent the number of rolls cut to patterns 1 to 5, respectively, then we want ...
minimize 10p +10q +4r +s +t . . . . . . . total trim loss
subject to ...
0p +0q +8r +2s +7t = 5672 . . . . . . number of 12-ft rolls
6p +0q +0r +1s +1t = 1670 . . . . . . . . number of 15-ft rolls
0p +3q +0r +2s +0t = 3200 . . . . . . number of 30-ft rolls
p≥0, q≥0, r≥0, s≥0, t≥0
_____
The solution works out to be (p, q, r, s, t) = (1, 0, 253, 1600, 64) with a trim loss of 2686 feet (equivalent to 27 100-ft rolls).
Note that we have endeavored to fill the order exactly. We don't know if there are cut choices that would minimize loss further, but result in a few rolls extra of some width or another.
