A logging firm gets logs from its supplier of $\displaystyle 10' $ and $\displaystyle 20' $ (and can get as many of each as he needs). The firm needs to produce logs of lengths $\displaystyle 5' $, $\displaystyle 7' $, and $\displaystyle 9' $ and needs $\displaystyle 1000 $, $\displaystyle 3000 $, and $\displaystyle 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 $\displaystyle 5' $, $\displaystyle 7' $, or $\displaystyle 9' $ lengths at no cost.

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

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

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

Is this correct?