# Math Help - linear programming

1. ## linear programming

A logging firm gets logs from its supplier of $10'$ and $20'$ (and can get as many of each as he needs). The firm needs to produce logs of lengths $5'$, $7'$, and $9'$ and needs $1000$, $3000$, and $2000$ of these lengths respectively. The firm needs to know how to cut the logs up so the total excess is minimized. How should the firm cut up the logs? The firm can stockpile any excess logs of $5'$, $7'$, or $9'$ lengths at no cost.

So let $x_{1} =$ number of $5'$ logs cut, $x_2 =$ number of $7'$ logs cut and $x_3 =$ number of $9'$ logs cut.

Then we want to minimize: $\text{max} \ z = x_{1}-x_{2}-x_{3}$ subject to constraints $x_1 \geq 0, \ x_2 \geq 0, \ x_3 \geq 0$.

Can you rule out the $10'$ case altogether because $5' + 5' = 10'$ (excess = 0)? Then formulate as $\text{min} \ z = 20-x_{1}-x_{2}-x_{3}$ s.t. the same constraints as above?

Is this correct?