Problem Set 1 Thomas Ratzler is the manager of the Metropolitan Zoo which has a

Question

Problem Set 1 Thomas Ratzler is the manager of the Metropolitan Zoo which has a wide variety of animals. The animals' pattern of feeding is such that they must have at least 30 grams of protein and at least 20 grams of fat per feeding period. These nutrients are derived from two types of food. Type 1 food, which costs 18 cents per unit and supplies 2 grams of protein and 4 of fat; and Type 2 food, which costs 12 cents per unit and has 6 grams of protein and 2 of fat. Type 2 food is bought under long-term contract requiring that at least 2 units of 2 to be used per serving. How much of each food must be bought to produce the minimum cost per serving? Show all the corner points in your solution.

Explanation / Answer

Problem Set 1:

Here the 2 decision variables given would be the the number of units of Type I food consumed and number of units of the Type 2 food consumed. Let that be X and Y units respectively.

According to the given information, the constraints given here would be obtained as:

Total proteins > = 30 therefore 2X + 6Y > = 30 that is X + 3Y > = 15

Total fat > = 20 therefore 4X + 2Y > = 20 that is 2X + Y > = 10

Also, we are given that Y > = 2 and X > = 0

Now for all the above 4 constraints the corner points are computed by finding the intersection points of the equations as:

multiplying the first equation by 2 and subtracting the second one from the first, we get:

6Y - Y = 20

Y = 4 and therefore X = 3 which also satisfies X > = 0 and Y> = 2

Now total cost in this case is 18X + 12Y = 54 + 48 = 102 cents

For X = 0, we have Y = 5

but here 2X + Y > = 10 is not satisfied therefore this wont be a corner point that is considered

X = 0 and Y = 10.

X + 3Y >= 15 therefore all constraints are satisfied here.

Total cost = 10*12 = 120 cents.

Therefore the optimal solution to the given problem here is

X = 3 and Y = 4 that is 3 units of Type 1 food and 4 units of Type II food. Total min optimal cost as 102 cents.

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