LP model constraint formulation

Question

We have a production plant that for each ton of $a$ requires $p_x$ tons of $x$ and $p_y$ tons of $y$ and we must decide how much material to ship to this plant. Is it just

$a = y/p_y = x/p_x$?

Do I need more constraints?

Answer 1

You could indeed formulate this as an LP, e.g.

$$ maximize~: ~~a$$ subject to $$\frac{1}{p_y} * y - a = 0\\ \frac{1}{p_x} * x -a = 0$$ and $$ a,x,y \geq 0$$

This constraints are equivalent to those you mentioned in your question, yet, you might have troubles implementing them in your form (if you planned to do this). I also added non-negativity constraints (while they are not really required in this setting).

Note that I assumed that you want to maximize your total production and that both $p_x$ and $p_y$ are constants (and not variables). Obviously, $y$ and $x$ are not bounded, so $a$ can be increased up to infinity. Are you sure you didn't miss any additional constraints (e.g. production cost, transportation capacity, ...)?

Best, miweiss