A dairy produces cheese, milk, sour cream, and yogurt.
Suppose:
Every 100 lbs of cheese requires 2 units of plant capacity, 3 workers, and 7 gallons of culturing additive, and gives $1,500 in profit.
Every 100 gallons of milk requires 1 unit of plant capacity, 1 worker, and no culturing additive, and gives $600 in profit.
Every 100 gallons of sour cream requires 5 units of plant capacity, 3 workers, and no culturing additive, and gives $900 in profit.
- Every 100 gallons of yogurt requires 0.6 units of plant capacity, 0.25 workers, and 1 gallon of culturing additive.
During the next week, one has 20 units of plant capacity, 24 workers, and 70 gallons of culturing additive available.
The goal is to maximize profit.
Let
x= amount of cheese produced (in 100's of pounds)
y=amount of milk (in 100's of gallons)
z= amount of sour cream (in 100's of gallons)
w= amount of yogurt (in 100's of gallons)
p=profit
I have to model this as a linear programming problem. This is what I have:
$p=1500x+600y+900z+200w$
These are the constraints:
$2x+y+5z+0.6w \leq20$
$3x+y+3z+0.25w \leq 24$
$7x+w \leq 70$
Are my equations correct?
Also, I put in slack variables ($a,b,c$) into the inequalities above and tried the simplex method. However, once I set the non-basic variables equal to zero, I got an optimal solution, because there weren't any negative coefficients. That seems too simple, so I'm wondering if I messed up with my equations along the way.
Any help is appreciated. Thank You.