Problem:
The Delicious Apple Company has orchards in Red City, Golden, and Macville Estimated amounts of apples to be available from the three orchards are the following: Red City, 15,000 bushels; Golden, 12,000 bushels; and Macville, 9,000 bushels. Apple Products Company wants to buy up to 10,000 bushels, for which it will pay $2.90 a bushel. Fruit Processors, Inc., is interested in buying up to 12,000 bushels at $3.20 a bushel. Extragood Fruit Company will buy up to 18,000 bushels at $3.10 per bushel. Delicious Apple Company would like to distribute its apples to the buyers so as to maximize its profits. The costs of shipping form the orchards to the buyers' processing plants are given in the following table.
From Orchard Apple Products Fruit Processors Extragood Fruit
Red City $0.55 $0.65 $0.45
Golden $0.40 $0.45 $0.55
Macville $0.45 $0.45 $0.85
a) Formulate the linear programming model for this problem.
b) Solve the problem for the optimal answer and provide the value of the objective function and the variables at optimality.