STAR Co. provides paper to smaller companies with volumes that 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.

Pattern Number	12ft	15ft	30ft	Trim Loss (ft)
1	0	6	0	10
2	0	0	3	10
3	8	0	0	4
4	3	0	2	4
5	7	1	0	1

Trim loss is the leftover paper from a pattern (e.g., for pattern 4, 3(12) + 0(15) + 2(30) = 96 feet used results in 100 96 = 4 foot of trim loss). Orders in hand for the coming week are 5,660 12-foot rolls, 1,690 15-foot rolls, and 3,360 30-foot rolls. Any of the three types of rolls produced in excess of the orders in hand will be sold on the open market at the selling price. No inventory is held.

(a) Formulate an integer programming model that will determine how many 100-foot rolls to cut into each of the five patterns in order to minimize trim loss. (Let P_i = number of rolls cut into pattern i, for i = 1, 2, 3, 4, 5.)

Min_________

s.t

12-ft _________

15-ft ___________

30-ft _________

P₁, P₂, P₃, P₄, P₅ 0

(b)Solve the model formulated in part (a). What is the minimal amount of trim loss (in ft)?

________ ft

How many of each pattern should be used?

Pattern 1 _______rolls

Pattern 2________ rolls

Pattern 3 _______rolls

Pattern 4 ________rolls

Pattern 5________ rolls

How many of each type of roll will be sold on the open market?

12-ft ________rolls

15-ft __________rolls

30-ft __________rolls