He makes $3$ products for his shop: large bowls, small bowls, and pots. Each large bowl uses $3$ pounds of clay and $6$ fluid ounces of glaze. Each small bowl uses $2$ pounds of clay and $6$ fluid ounces of glaze. Each pot requires $4$ pounds of clay and $2$ fluid ounces of glaze.
He must use up $72$ pounds of old clay and $108$ fluid ounces of old glaze and can order more if necessary.
If he can make a large bowl in $5$ hours, a small bowl in $6$ hours, and a pot in $4$ hours, determine how many of each he should make to minimize his time and what the minimum time will be.
So this is a linear optimization problem and I'm trying to set up the constraints first.
What answer did folks get on this?
There are three products: So lets have
Large bowl = $x_{1}$
Small bowl = $x_{2}$
Pot = $x_{3}$
The objective function is:
min $5x_{1}+6x_{2}+4x_{3}$
Subjected to the constraints:
$3x_{1}+2x_{2}+4x_{3} \geq 72$
$6x_{1}+6x_{2}+2x_{3} \geq 108$
$x_{1}, x_{2}, x_{3} \geq 0$