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	4	1	10 ft.
2	4	3	0	7 ft.
3	8	0	0	4 ft.
4	2	1	2	1 ft.
5	2	3	1	1 ft.

Trim loss is the leftover paper from a pattern (e.g., for pattern 4, 2(12) + 1(15) + 2(30) = 99 feet used resulting in 100-99 = 1 foot of trim loss). Orders in hand for the coming week are 5,670 12-foot rolls, 1,680 15-foot rolls, and 3,350 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.

Optimal Solution:

(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. If your answer is zero enter “0” and if the constant is "1" it must be entered in the box.

(b)	Solve the model formulated in part a. What is the minimal amount of trim loss?
	Trim Loss: in feet (c) How many of each pattern should be used and how many of each type of roll will be sold on the open market? If your answer is zero enter “0”.

Answer 1

(a) Integer programming model is following:

Let x1, x2, x3, x4, x5 be the number of 100-ft rolls to be cut into pattern 1,2,3,4,5 resp

Minimize 10x1+7x2+4x3+1x4+1x5

s.t.

0x1+4x2+8x3+2x4+2x5 >= 5670

4x1+3x2+0x3+1x4+3x5 >= 1680

1x1+0x2+0x3+2x4+1x5 >= 3350

x1, x2, x3, x4, x5 integers

(b) Solution of the model using LINDO is following:

Optimal solution:

x4 = 2835

All other variables = 0

Trim loss = 2835 feet

(c) 2835 rolls of 100 ft should be cut in pattern 4.

Excess rolls produced of 12-ft size and sold in open market = 2835*2 - 5670 = 0

Excess rolls produced of 15-ft size and sold in open market = 2835*1 - 1680 = 1155

Excess rolls produced of 30-ft size and sold in open market = 2835*2 - 3350 = 2320