algebra question

Question

Genko Olive Oil Company has decided to diversify, and begin making and selling flour. It has decided to specialise in two types; bread flour and self-raising flour. Genko would like to determine how much of each type of flour it should pro- duce each day in order to maximise its profits. The profit made on the sale of 1 kg of bread flour is $2, while the profit made on the sale of 1 kg of self-raising flour is $1.80. The company is capable of making up to a total of 200 kg of flour per day. However, due to an ongoing shortage of baking powder (a key input in the production of self-raising flour) Genko can only produce up to 150 kg of the self-raising flour per day. In addition, company management are anticipating a high demand for self-raising flour (a key ingredient in cakes and puddings) as the holiday season approaches. For this reason, they have decided that the company should produce at least twice as much self-raising flour as bread flour.

(a) Formulate this problem as an optimization problem.

(b) Draw the feasible region.

(c) Solve this linear optimization problem using the graphical method

Answer 1

This question is most easily set up if we do one sentence at a time.

Genko Olive Oil Company has decided to diversify, and begin making and selling flour. It has decided to specialise in two types; bread flour and self-raising flour.

This sentence identifies the variables. Let B = kg of Bread Flour and R = kg of selfRising flour.

Second sentence:

Genko would like to determine how much of each type of flour it should pro- duce each day in order to maximise its profits.

This sentence says that we are looking for the best profit.

Third sentence:

The profit made on the sale of 1 kg of bread flour is $2, while the profit made on the sale of 1 kg of self-raising flour is $1.80.

This sentence gives the profit formula: P = 2 B + 1.8 R

Fourth sentence:

The company is capable of making up to a total of 200 kg of flour per day.

This is one of the conditions. B + R ≤ 200

Fifth sentence:

due to an ongoing shortage of baking powder (a key input in the production of self-raising flour) Genko can only produce up to 150 kg of the self-raising flour per day.

This is another condition. R ≤ 150

Sentences 6 and 7:

company management are anticipating a high demand for self-raising flour (a key ingredient in cakes and puddings) as the holiday season approaches. For this reason, they have decided that the company should produce at least twice as much self-raising flour as bread flour.

This is the last condition. R ≥ 2B

Put the conditions into a graph letting the horizontal axis be B, and the vertical axis be R.

If you wish you may use DESMOS Graphing Calculator. It will shade. Your feasible region is the area where all shadings overlap.

Identify the coordinates of the corners:

(0, 0). (0, 150). (50, 150) (66, 133)

Now put each pair of coordinates into the Profit equation for B and R to calculate the profit.

(0, 0). P = 2(0) + 1.8(0) = 0

(0, 150). P = 2(0) + 1.8(150) = 270

(50, 150). P = 2(50) + 1.8(150) = 370

(66, 133). P = 66(2) + 1.8 (133) = 371.40

The most profit is the 66 kg of Bread Flour and 133 kg of Self Rising Flour.