Question

A company blends two materials: A and B to produce two types of fertilizers. Fertilizer 1 must be at least 50% of A and sells for $65 per kilo gram. Fertilizer 2 must be at least 70% of B and sells for $48 per kilogram. The price of martial A is $10 per 100 kilo grams and the price of martial B is $14 per 100 kilo grams if they purchased over 10,000 kilo gram the price will be reduced by 10%. Total budget of the company to spend on raw martial is $2000.

a) Write the linear optimization model for the company to make the best decision.

b) Solve the model and present the results and interpret them.

c)Rewrite the model if 10% discount only apply to the amount purchased over 10000 kilo grams (For example if the company purchases 10001 kg of A, the total price is 10000*10+1*9).

Answer 1

Ans:-

Given:-

Fertilizer 1 must be at least 50% of A and sells for $65 per kilo gram.

Fertilizer 2 must be at least 70% of B and sells for $48 per kilogram.

The price of martial A is $10 per 100 kilo grams and the price of martial B is $14 per 100 kilo grams

                                                   Explanation:-

a) Write the linear optimization model for the company to make the best decision.

Max Z = 65 * F1 + 48 * F2

Subject to the constraints

Total Budget: 10 * F1 + 14 * F2 < = 2000

Raw material A: F1 > = 50

Raw material B: F2 > = 70

F1, F2, > = 0

b) Solve the model and present the results and interpret them.

The model is solved in excel solver and the output is given below

	Fertilizer 1	Fertilizer 2
Profits	65	48
Units	62	70

Max

7390

Budget                              10                 14                   2000<=          2000

Raw material A                   1                  0                     62>=             50

Raw material B                   0                 1                      70>=             70

From the above solver output, we see that, on producing 62 units of Fertilizer 1 and 75 units of Fertilizer 2, the maximum profit earned is $ 7390

c) Rewrite the model if 10% discount only apply to the amount purchased over 10000 kilo grams (For example if the company purchases 10001 kg of A, the total price is 10000*10+1*9).

As the total budget is less, it is not possible to purchase 1000 kg of raw materials

	Fertilizer 1	Fertilizer 2
Profits	65	48
Units	0	177.77778

Max

8533.333

Budget                              10                 9                   2000<=                  2000

Raw material A                   1                  0                     0>=                      500

Raw material B                   0                 1                177.77778>=             700