I have to set up a linear programming problem corresponding to the following scenario:
If my understanding of the problem is correct, I use $mod$:
Let $i$ be $A$ or $B$.
Let $x$ be amount of raw material.
Let $x_i \ge 0$ be lbs of raw materials of $i$ purchased so $\sum_{i=A,B} x_i = x$
Define the cost of $x_i$ to be:
$$c_{i,1} = \frac{500,000}{2,000} x_i$$
MI yield:
$$m_A := (5,000) (x_A - x_A \ mod \ 5,000) (80\%)$$
Define the cost of $m_A$ to be:
$$c_{A,2} = \frac{100,000}{1} \frac{7}{1} m_A$$
MII yield:
$$m_B := (4,000) (x_B - x_B \ mod \ 5,000) (75\%)$$
Define the cost of $m_B$ to be:
$$c_{B,2} = \frac{175,000}{1} \frac{8}{1} m_B$$
Packing for A:
$$p_A := (15,000)(m_A - m_A \ mod \ 15,000)$$
$$c_{A,3} := (200,000)(10)p_A$$
Packing for B:
$$p_B := (8,000)(m_B - m_B \ mod \ 8,000)$$
$$c_{B,3} := (200,000)(10)p_B$$
Revenue for A:
$$r_A = 400p_A(\frac{1}{1/4})$$
Revenue for B:
$$r_B = 600p_B(\frac{1}{1/3})$$
It seems then that our LP has obj function
$$(max) z = \sum_{i = A, B} r_i - \sum_{j = 1,2,3} \sum_{i = A, B} c_{i,j}$$
s.t. the ff constraints:
$$x_A + x_B = x \ge 0$$
$$x_A , x_B \ge 0$$
So is that it? The role of input/hr is in the $mod$ thing I did above? So if we were to solve it, we might have to use this?
Okay looks too complicated so likely wrong. How do I go about this?
From Chapter 2 here.
Let $x_A$, $x_B$ the amount of products $A$ and $B$ (in units) that are produced and $X_A$, $X_B$ the corresponding raw material that you need to buy (in pounds).
You want to maximize profits: $$ 400x_A +600x_B -\frac{100000}{5000} X_A - \frac{175 000}{4000}X_B - 200 000 (x_A+x_B)-\frac{500000}{2000}(X_A+X_B) $$ (check that all terms are homogenous to dollars) subject to availability constraints: $$ \frac{X_A}{5000}\le 7\quad \mbox{(no more than 7 hours on machine A)}\\ \frac{X_B}{4000}\le 8\quad \mbox{(no more than 8 hours on machine B)}\\ \frac{x_A}{15000}+\frac{x_B}{8000}\le 10\quad \mbox{(no more than 10 hours of packing)} $$ and don't forget to ensure consistency between variables in units and pounds, with the machine output: $$ x_A=80\% \frac{X_A}{1/4}\quad \mbox{(check that you have units on both sides)}\\ x_B=75\% \frac{X_B}{1/3}\quad \mbox{(check that you have units on both sides, again)}\\ x_A,x_B,X_A,X_B\ge 0 $$