High School

The Okanagan Flower Company owns a greenhouse, which furnishes roses and carnations to florists in Vernon, Kelowna, and Westbank. The greenhouse can grow any combination of the two flowers. They sell the flowers in "bunches". They have 10,000 square feet available for planting this year. Each bunch of roses takes about 4 square feet and each bunch of carnations about 5 square feet. Special fertilizer is required for flowers: roses need 5 pounds and carnations 2 pounds. The availability of the fertilizer is limited to 5000 pounds. Sales commitments require the company to grow at least 500 bunches of roses. Profit contributions are $6 per bunch of roses and $8 per bunch of carnations. In the space provided below: a. Clearly define the decision variables (1 mark) b. Write a complete linear program for this problem (4 marks)

Answer :

Maximize linear programming problem $6R + $8C subject to 4R + 5C ≤ 10,000, 5R + 2C ≤ 5,000, R ≥ 500, R ≥ 0, and C ≥ 0.

Formulate a linear programming problem to maximize profit by determining the optimal quantities of roses (R) and carnations (C) to be grown, given the constraints 4R + 5C ≤ 10,000, 5R + 2C ≤ 5,000, R ≥ 500, R ≥ 0, and C ≥ 0?

Decision variables:

Let's define the decision variables for this problem:

Let R represent the number of bunches of roses to be grown.

Let C represent the number of bunches of carnations to be grown.

Linear program formulation:

Objective function:

Maximize profit: $6R + $8C

Constraints:

Area constraint: The total area used for planting cannot exceed the available 10,000 square feet.

4R + 5C ≤ 10,000

Fertilizer constraint: The total amount of fertilizer used cannot exceed the available 5,000 pounds.

5R + 2C ≤ 5,000

Sales commitment constraint: At least 500 bunches of roses need to be grown.

R ≥ 500

Non-negativity constraint: The number of bunches of roses and carnations cannot be negative.

R ≥ 0, C ≥ 0

Mathematically, the linear program can be written as:

Maximize 6R + 8C

Subject to:

4R + 5C ≤ 10,000

5R + 2C ≤ 5,000

R ≥ 500

R ≥ 0, C ≥ 0

To maximize profit, the objective function is defined as the sum of the profit contributions from roses and carnations, which is $6R + $8C. The goal is to determine the optimal values of R and C that maximize this objective.

The area constraint ensures that the total area used for planting (4 square feet per bunch of roses and 5 square feet per bunch of carnations) does not exceed the available 10,000 square feet.

The fertilizer constraint ensures that the total amount of fertilizer used (5 pounds per bunch of roses and 2 pounds per bunch of carnations) does not exceed the available 5,000 pounds.

The sales commitment constraint requires that at least 500 bunches of roses be grown, ensuring the fulfillment of sales commitments.

The non-negativity constraints state that the number of bunches of roses and carnations cannot be negative since we cannot have negative quantities.

By solving this linear program, the optimal values of R and C can be obtained, representing the number of bunches of roses and carnations to be grown, respectively, in order to maximize profit.

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