Answer to Question #283924 in Operations Research for Sam

Question

There are two types of fertilizers F1 and F2. F1 consists of 10% nitrogen and 6% phosphoric acid and F2 consists of 5% nitrogen and 10% phosphoric acid. After testing the soil conditions, a farmer finds that she needs at least 14 kg of nitrogen and 14 kg of phosphoric acid for her crop. If F1 costs Birr 6/kg and F2 costs Birr 5/kg, determine how much of each type of fertilizer should be used so that nutrient requirements are met at a minimum cost. What is the minimum cost? (solve through simplex method)

Answer 1

minimize cost:

$Z=6x_1+5x_2$

subject to:

$0.1x_1+0.05x_2\ge 14$ - amount of nitrogen

$0.06x_1+0.1x_2\ge 14$ - amount of phosphoric

x₁ is amount (kg) of fertilizers F1

x₂ is amount (kg) of fertilizers F2

After introducing surplus, artificial variables:

Min Z=5x₁+10x₂+0S₁+0S₂+MA₁+MA₂

subject to

0.1x₁+0.05x₂-S₁+A₁=14

0.06x₁+0.1x₂-S₂+A₂=14

and x₁,x₂,S₁,S₂,A₁,A₂≥0

Positive maximum Z_j-C_j is 0.16M-5 and its column index is 1. So, the entering variable is x₁.

Minimum ratio is 140 and its row index is 1. So, the leaving basis variable is A₁.

∴ The pivot element is 0.1.

Entering =x₁, Departing =A₁, Key Element =0.1

Positive maximum Z_j-C_j is 0.6M-50 and its column index is 3. So, the entering variable is S₁.

Minimum ratio is 9.3333 and its row index is 2. So, the leaving basis variable is A₂.

 ∴ The pivot element is 0.6.

Entering =S₁, Departing =A₂, Key Element =0.6

Since all Z_j-C_j≤0

Hence, optimal solution is arrived with value of variables as :

x₁=233.3333, x₂=0

Min Z=1166.6667