Answer to Question #283388 in Operations Research for Andinet

Question #283388

6. 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)

1
Expert's answer
2022-01-10T14:24:28-0500

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 Zj-Cj 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 Zj-Cj 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 Zj-Cj≤0

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

x1=233.3333, x2=0

Min Z=1166.6667


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