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. It is acceptable to have fractional bushels.
|
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 formulation for this problem.
b) Using any linear programming software, solve the problem for the optimal answer and provide the value of the objective function and the variables at optimality. Provide a copy of your linear programming output. In addition to providing this output, you are to specifically identify the value of the objective function and the values of the variables at optimality.
c) Using the sensitivity analysis (ranging) output provided as part of the linear programming output, provide two sensitivity analysis interpretations. Provide one based on the objective function and one based on a constraint. Provide a copy of the sensitivity analysis output supporting your interpretations.
d) Now suppose that Delicious Apple has signed a contract with Extragood Fruit that ensures that Extragood Fruit will receive in total at least as many apples as Fruit Processors. Provide the modification to the linear programming formulation provided in a) for this additional requirement. It is not necessary to solve the linear programming problem for the solution.
e) Write a constraint that will ensure that Apple Products obtains no more than 30% of its total number of bushels from the Golden orchard. You only need formulate this constraint. You do not need to solve the linear programming problem for the solution.