4. The Longhorn Feeding Company (LFC) operates a feedlot to which cattle are bro
ID: 465917 • Letter: 4
Question
4. The Longhorn Feeding Company (LFC) operates a feedlot to which cattle are brought for the final fattening process. Since LFC’s cattle production averages 100,000, it is important to feed the cattle in a way that meets their nutritional requirements at a minimum cost.
The mixture LFC feeds the cattle is a blend of four feedstuffs: corn, wheat, barley, and hay. The table below provides the relevant dietary and cost information per pound of each feedstuff, as well as a cow’s daily nutritional requirement. For example, for each pound of corn a cow consumes, it receives 2 grams of fat, 20 grams of protein, 4 milligrams of iron and 200 calories at a cost of $0.60.
Units of Nutrient per Pound of Feedstuff
Nutrient
Corn
Wheat
Barley
Hay
Minimum Daily Requirement
Fat (gm)
2
1
3
4
25 gm
Protein (gm)
20
15
15
10
400gm
Iron (mg)
4
7
6
5
125 mg
Calories
200
400
300
500
6,000
Cost per pound
$0.60
$0.40
$0.35
$0.05
Assuming a cow must consume exactly 24 pounds of feedstuff daily, write the linear programming formulation to determine the best feed mix for each cow daily.
Units of Nutrient per Pound of Feedstuff
Nutrient
Corn
Wheat
Barley
Hay
Minimum Daily Requirement
Fat (gm)
2
1
3
4
25 gm
Protein (gm)
20
15
15
10
400gm
Iron (mg)
4
7
6
5
125 mg
Calories
200
400
300
500
6,000
Cost per pound
$0.60
$0.40
$0.35
$0.05
Explanation / Answer
Decision variable:
Let,
C = amount of corn per pound of feedstuff
W = amount of wheat per pound of feedstuff
B = amount of Barley per pound of feedstuff
H = amount of Hay per pound of feedstuff
Objective is to minimize the cost of Feedstuff;
Min Z = $0.60C + $0.40W + $0.35B + $0.05H
Subjected to:
Fats requirement:
2C + W + 3B + 4H 25
Protein Requirement
20C + 15W + 15B + 10H 400
Iron Requirement
4C + 7W + 6B + 5H 125
Calories Requirement:
200C + 400W + 300B + 500H 6000
Consumption Requirement of Feedstuff:
C + W + B +H = 24
Non-negativity Constraint:
C, W, B, H 0
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