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	5	0	1	10 ft.
2	0	0	3	10 ft.
3	3	0	2	4 ft.
4	3	2	1	4 ft.
5	7	1	0	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.
	Min x1 + x2 + x3 + x4 + x5 s.t. x1 + x2 + x3 + x4 + x5 - Select your answer -≤≥=Item 11 12-foot rolls x1 + x2 + x3 + x4 + x5 - Select your answer -≤≥=Item 18 15-foot rolls x1 + x2 + x3 + x4 + x5 - Select your answer -≤≥=Item 25 30-foot rolls x1, x2, x3, x4, x5 are integers
(b)	Solve the model formulated in part a. What is the minimal amount of trim loss?
	Trim Loss:  feet
	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”.
	Pattern Number Rolls Used 1 2 3 4 5

Answer 1

(a) The integer linear programming model based on given data is as follows:

Objective Function:

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

Subject to Constraints:

5x1 + 0x2 + 3x3 + 3x4 + 7x5 >= 5670

0x1 + 0x2 + 0x3 + 2x4 + 1x5 >= 1680

1x1 + 3x2 + 2x3 + 1x4 + 0x5 >= 3350

where x1, x2, x3 , x4 , x5 >= 0

Greater than sign is used in above constraints because order limits are given and company has to alteast make these much rolls. Less roles will lead to loss in sales.

(b) Now Solving the model in excel solver

Therefore, the minimum amount of trim loss is 8380 ft

The variables are:

x1 = 0

x2 = 0

x3 = 1255

x4 = 840

x5 = 0