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	6	0	10 ft.
2	0	0	3	10 ft.
3	8	0	0	4 ft.
4	2	1	2	1 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

Answer 1

Please please please rate the answer.It really helps me.Thanks!

Answer 2

To formulate the integer programming model for this problem, we need to define the decision variables and constraints.

Let: x1 = Number of 100-foot rolls cut into Pattern 1 x2 = Number of 100-foot rolls cut into Pattern 2 x3 = Number of 100-foot rolls cut into Pattern 3 x4 = Number of 100-foot rolls cut into Pattern 4 x5 = Number of 100-foot rolls cut into Pattern 5

Objective: Minimize trim loss (trim loss is the leftover paper from a pattern)

Objective Function: Minimize 10x1 + 10x2 + 4x3 + x4 + x5

Constraints:

1. Demand constraint for 12-foot rolls: x1 + x2 + 8x3 + 2x4 + 7x5 ≥ 5,670

2. Demand constraint for 15-foot rolls: 6x1 + 3x3 + x4 + x5 ≥ 1,680

3. Demand constraint for 30-foot rolls: 9x2 + 2x4 ≥ 3,350

4. Non-negativity constraint: x1, x2, x3, x4, x5 ≥ 0 (since the number of rolls cannot be negative)

5. Integer constraint: x1, x2, x3, x4, x5 are integers (since the number of rolls must be whole numbers)

The objective is to minimize the trim loss (the amount of leftover paper) while meeting the demand for 12-foot, 15-foot, and 30-foot rolls. The constraints ensure that enough rolls of each pattern are cut to satisfy the orders in hand, and the non-negativity and integer constraints specify that the number of rolls cannot be negative and must be whole numbers.