Question

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 Loss
1	0	0	3	10 ft.
2	0	6	0	10 ft.
3	3	4	0	4 ft.
4	3	0	2	4 ft.
5	2	5	0	1 ft.

Trim loss is the leftover paper from a pattern (e.g., for pattern 4, 3(12) + 0(15) + 2(30) = 96 feet used resulting in 100-96 = 4 foot of trim loss). Demands this week are 5,686 12-foot rolls, 1,660 15-foot rolls, and 3,490 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 (leftover paper from a pattern).

Optimal Solution:

Pattern	Number Rolls Used
1 =
2 =
3 =
4 =
5 =

Trim Loss: __ feet

Answer 1

Xi: Number of rolls cut from pattern i

i belongs to [1,2,3,4,5]

Objective function (Minimize T)

T= 10*X1+ 10*X2+ 4*X3 +4*X4 + x5

12*(3* X3+3* X4+2* X5)>=5686*12

15*(6* X2+4* X3+ 5* X5)>=1660*15

30*(3* X1+2* X4)>=3490*30

Xi>=0, Xi is an integer

Pattern	No of rolls used
1	0
2	0
3	0
4	1745
5	332

Trim Lost= 7312 feet