Cattle are sent to a feedlot to be grain-fed before being processed into beef. The owners of a feedlot seek to determine the amounts of cattle feed to buy so that the minimum nutritional standards are satisfied to ensure proper weigh gain, while total feed costs are minimized. The feed mix used is made up of three grains that contain the following nutrients per pound of feed. Nutrient (ounces per pound of feed)
FEED A B C D Cost of Mix (per lb.)
Feed Mix X 4 2 3 6 $5.00
Feed Mix Y 2 3 0 8 $4.50
Feed Mix Z 5 2 2 5 $3.50
The minimum requirement per cattle per day is 6 lbs. of Nutrient A, 5 lbs. of Nutrient B, 2 lbs. of Nutrient C, and 10 lbs. of Nutrient D. Another restriction is that only 500 lbs. of Feed Mix Z is available per day. Because there are usually 100 cattle to feed at any given time, this means that no more than 5 lbs. of Feed Mix Z can be counted on for use in the feed mix each day. Formulate and solve this problem in Excel to determine the number of lbs. of each type of Feed Mix to include in the total mix that meet the requirements and minimize the cost (Hint: Make sure your LHS and RHS of your constraints are the same units: 1 lb. = 16 ounces)
a) How many decision variables does this problem have?
b) Not counting the non-negativity constraint - how many constraints does this problem have?
c) What is the minimum cost based on your optimal solution (the value of the objective function)?
d) How many lbs. of Feed Mix X should be included in the mix based on your optimal solution?
e) How many lbs. of Feed Mix Y should be included in the mix based on your optimal solution?
f) How many lbs. of Feed Mix Z should be included in the mix based on your optimal solution?
g) What is the total ounces of Nutrient A included in the mix based on your optimal solution (the value of the LHS of the constraint)?
h) What is the total ounces of Nutrient B included in the mix based on your optimal solution (the value of the LHS of the constraint)?
i) What is the total ounces of Nutrient C included in the mix based on your optimal solution (the value of the LHS of the constraint)?
j) What is the total ounces of Nutrient D included in the mix based on your optimal solution (the value of the LHS of the constraint)?