Hint$($s$)$ Check My Work

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 $12$ft$.15$ft$.30$ft$.$ Trim Loss

$150110$ ft$.$

$200310$ ft$.$

$33024$ ft$.$

$43214$ ft$.$

$57101$ ft$.$

Trim loss is the leftover paper from a pattern $($e$.$g$.,$ for pattern $4,3(12)+2(15)+1(30)=96$ feet used resulting in foot of trim loss$).$ Orders in hand for the coming week are $5,67012-$foot rolls, $1,68015-$foot rolls, and $3,35030-$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.

$($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.

Let x j $=$ number of $100-$foot$-$wide rolls using cutting pattern j$,$ j $=1,2,3,4,5.$

Optimal Solution: