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notsponge [240]
3 years ago
9

The Springer Dog Food Company makes dry dog food from two ingredients. The two ingredients (A and B) provide different amounts o

f protein and vitamins. Ingredient A provides 16 units of protein and 4 units of vitamins per pound. Ingredient B provides 8 units of protein and 8 units of vitamins per pound. Ingredients A and B cost $0.50 and $0.20 per pound, respectively. The company wants its dog food to contain at least 12 units of protein and 6 units of vitamins per pound and be as inexpensive as possible.
a. Reformulate an LP model for this probelm

b. Sketch the feasible region for this problem

c. Determine the optimal solution to the problem by enumerating the corner points.

d. How will the optimal solution change if the company requires only 5 units of vitamins per pound rather than 6?

Mathematics
1 answer:
Nataly [62]3 years ago
3 0

Answer:

a) Objective function (minimize cost):

C=0.50A+0.20B

Restrictions

Proteins per pound: 16A+8B\leq 12

Vitamins per pound: 4A+8B\leq 6

Non-negative values: A,B\geq0

b) Attached

c) The optimum solution (minimum cost) is 0 pounds of ingredient A and 0.75 pounds of ingredient B. The cost is $0.15 per ration.

d) The optimum solution changes. The cost is now 0 pounds of ingredient A and 0.625 pounds of ingredient B. The cost is $0.125 per ration.

Step-by-step explanation:

a) The LP formulation for this problem is:

Objective function (minimize cost):

C=0.50A+0.20B

Restrictions

Proteins per pound: 16A+8B\leq 12

Vitamins per pound: 4A+8B\leq 6

Non-negative values: A,B\geq0

b) The feasible region is attached.

c) We have 3 corner points. In one of them lies the optimal solution.

Corner A=0 B=0.75

C=0.50*0+0.20*0.75=0.15

Corner A=0.5 B=0.5

C=0.50*0.5+0.20*0.5=0.35

Corner A=0.75 B=0

C=0.50*0.75+0.20*0=0.375

The optimum solution (minimum cost) is 0 pounds of ingredient A and 0.75 pounds of ingredient B. The cost is $0.15 per ration.

d) If the company requires only 5 units of vitamins per pound rather than 6, one of the restrictions change.

The feasible region changes two of its three corners:

Corner A=0 B=0.625

C=0.50*0+0.20*0.625=0.125

Corner A=0.583 B=0.333

C=0.50*0.583+0.20*0.333=0.358

Corner A=0.75 B=0

C=0.50*0.75+0.20*0=0.375

The optimum solution changes. The cost is now 0 pounds of ingredient A and 0.625 pounds of ingredient B. The cost is $0.125 per ration.

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