Markov Process/Objective Function

Question

Consider the problem:

“United World Animal League (UWAL) is trying to save the endangered Madagascar bob-tailed bear. They have two pairs in their mating program and want to provide them with an appropriate diet. Daily, they must provide them with at least 225 units of vitamin I, at least 375 units of vitamin II, and at most 150 units of mineral beta. They have three standard bear chows available to them. Each pound of Purina Bear Chow provides 80 units of vitamin I, 2 units of vitamin II, 60 units of mineral beta, and 30 calories. Each pound of Science Diet Bear Food provides 5 units of vitamin I, 120 units of vitamin II, 40 units of mineral beta, and 50 calories. Each pound of Super Bear provides 40 units of vitamin I, 40 units of vitamin II, 50 units of mineral beta, and 60 calories. If UWAL wants to minimize the calories provided, how much of each bear chow must they give each bear?”

(a) Find the objective function.

(b) Find the constraints.

Answer 1

This is a minimization problem, so the objective function will be based on minimizing calories:

Let's have P represent the number of pounds of Purina mix, D represent the number of pounds of Science Diet mix, and B represent the number of pounds of Super Bear mix. We want to minimize the calories consumed, and there are 30 calories in a pound of P, 50 in a pound of D, and 60 in a pound of B, so the equation we're minimizing is...

30P + 50D + 60B

There are many constraints though. First, we have to give the bears AT LEAST 225 units of Vitamin I (V1), 375 units of Vitamin II (V2) and AT MOST 150 units of mineral beta (M). We can write that as:

V1 >= 225

V2 >= 375

M <= 150

But V1, V2, and M will be determined by the mix of P, D, and B. So let's rewrite these constraints as a function of the pounds of each mix we ultimately give the bears:

80P + 5D + 40B >= 225

2P + 120D + 40B >= 375

60P + 40D + 50B <= 150

So the final write-up of the objective function and its constraints is:

Minimize 30P + 50D + 60B s.t

80P + 5D + 40B >= 225

2P + 120D + 40B >= 375

60P + 40D + 50B <= 150

Done!