STAR Co. provides paper to smaller companies whose volumes are not large enough to warrant dealing directly with the paper mill. STAR receives 100-feet-wide paper rolls from the mill and cuts the rolls into smaller rolls of widths 12, 15, and 30 feet. The demands for these widths vary from week to week. The following cutting patterns have been established: Number of:Pattern 12ft. 15ft. 30ft. Trim Loss1 0 6 0 10 ft.2 0 0 2 10 ft.3 7 0 0 4 ft.4 1 2 2 1 ft.5 7 1 0 1 ft.Trim loss is the leftover paper from a pattern (for example, for pattern 4, 1(12) + 2(15) + 2(30) = 99 feet used resulting in 100-99 = 1 foot of trim loss). Demands this week are 5,672 12-foot rolls, 1,67015-foot rolls, and 3,200 30-foot rolls. Develop an all-integer model that will determine how many 100-foot rolls to cut into each of the five patterns in order to meet demand and minimize trim loss

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

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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.