Answer to Question #130261 in Operations Research for mj

Question

A health food manufacturer makes two types of protein shakes, Whey and Vegan. Each mixture
contains vitamins, protein, and carbohydrates, in the following proportions:
Carbohydrates (g) Protein (g) Vitamin (g)
Whey 100 400 300
Vegan 200 300 100
Minimum dietary requirements are that a protein shake contain at least 0.9kg of vitamins, 2.2kg of
protein, and 0.8kg of carbohydrate. Whey costs N$20 per kilogram to produce, and Vegan costs
$12.50 per kilogram to produce. Find the number of kilograms of each protein shake that should be
produced to minimize cost.

Answer 1

Let amount of Whey in kilograms be w and amount of Vegan be v. Then we can make a system of inequalities based on task conditions:

$0.1w+0.2v \geq 0.8$

$0.4w+0.3v \geq 2.2$

$0.3w+0.1v \geq 0.9$

If we plot these, we will get

We can see that the the minimum amount of shakes that satisfy nutritional conditions corresponds to the bold line. If we produce more, the cost will be higher.

Now we need to find where the cost is the lowest along the line on the picture above. We know that total price will be $C = 20w+12.5v$. To explore the behaviour of price, let's find $\frac{dC}{dv} = 20 \frac{dw}{dv} + 12.5$

Differentiating the equations of lines (or from the slope, that is basically the same) we have $w'_1 = -2, w_2' =-0.75, w_3' = -0.33$.

$C_1' = -27.5, C_2' = -2.5, C_3' = 5.83$

We see that from v = 0 to v = 2 the price is decreasing, from v=2 to v=6 it continues to decrease, and starting from v = 6 it increases. $C'<0$ means that higher v corresponds to the lower price. So for the segments with $C'<0$ the minimum price will be at v=6. At the same time for v>6 the price starts to increase. Thus, the minimum price is in $(v,w)=(6,1)$.

Answer: 6 kg of Vegan and 1 kg of Whey.