The Northwest Flower Company owns a greenhouse that supplies roses and carnations to florists in Oregon, Washington, and Idaho. The greenhouse can grow any combination of these two flowers. They sell the flowers in "bunches" with 25 blooms to a bunch. They have 10,000 square feet available for planting this year. Each bunch of roses requires about 4 square feet, and each bunch of carnations requires about 5 square feet. Special fertilizer is required for the flowers: roses need 5 pounds, and carnations need 2 pounds. The availability of fertilizer is limited to 5000 pounds. Sales commitments require the company to grow at least 500 bunches of roses. The profit contributions are $6 per bunch of roses and $8 per bunch of carnations.

a. Formulate this problem as a linear programming (LP) problem.
b. Graph this problem.
c. What are the corner points of the feasible region?
d. What is the optimal solution?

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To formulate this problem as a linear programming (LP) problem, define the variables, objective function, and constraints. Graph the constraints to find the corner points of the feasible region. Evaluate the objective function at the corner points to identify the optimal solution.

Explanation:

To formulate this problem as a linear programming (LP) problem, we need to define the decision variables, objective function, and constraints. Let's use R to solve this problem.

Decision Variables:

Let x be the number of bunches of roses and y be the number of bunches of carnations.

Objective Function:

Maximize the profit contribution: Z = 6x + 8y.

Constraints:

The greenhouse can't exceed the available square footage: 4x + 5y <= 10000.
There is a limited amount of fertilizer: 5x + 2y <= 5000.
There is a sales commitment for roses: x >= 500.
The number of bunches of roses and carnations can't be negative: x, y >= 0.

By graphing these constraints, we can identify the corner points of the feasible region. Using the corner points, we can find the optimal solution by evaluating the objective function.

Corner points of the feasible region:

(500, 0), (500, 1000), (900, 800), (0, 2000), (0, 0).

The optimal solution can be found by evaluating the objective function at each corner point and selecting the values that maximize it. The resulting optimal solution is (900, 800) with a profit contribution of $8,800.

Learn more about Formulating a Linear Programming Problem here:

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