Chapter 12, Problem 1P

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:

Trim loss is the leftover paper from a pattern (e.g., for pattern 4, 2(12) + 1(15) + 2(30) = 99 feet used results in 100 – 99 = 1 foot of trim loss). Demands this week are 5,670 12-foot rolls, 1,680 15-foot rolls, and 3,350 30-foot rolls. Develop an all-integer model that will determine how many 100-foot rolls to cut into each of the live patterns in order to meet demand and minimize trim loss (leftover paper from a pattern).

To determine

To create an all-integer model.

Explanation of Solution

In order to create an all-integer model let’s assume that:

x_j be the number of 100 foot wide rolls using cutting pattern j,  j =1, 2, 3, 4, 5

The all-integer model will be:

$Min10x_1+10x_2+4x_3+x_4+x_5$

s.t.

$\begin{array}{l}0x_1+0x_2+8x_3+2x_4+7x_5 \ge 5670\hfill \\ 6x_1+0x_2+0x_3+ x_4 + x_5\ge 1680\hfill \\ 0x_1+3x_2+0x_3+2x_4 +0x_5\ge 3350 \hfill \\ x_1,x_2,x_3,x_4,x_5\ge 0\hfill \end{array}$

The solution for the model will be:

Formula:

Solver:

Output:

Trim Loss will be 2007 feet

Pattern	Number used
1	0
2	0
3	0
4	1675
5	332