Question #272112

Use big 𝑀 method solve. 𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑍 = 2𝑦1 + 4𝑦2 𝑆𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 2𝑦1 – 3𝑦2 ≥ 2,



−𝑦1 + 𝑦2 ≥ 3; 𝑦1



, 𝑦2 ≥ 0

1
Expert's answer
2021-11-29T17:00:17-0500

𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑍=2𝑦1+4𝑦2𝑍 = 2𝑦_1 + 4𝑦_2

𝑆𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜

2𝑦13𝑦222𝑦_1 – 3𝑦_2 ≥ 2

𝑦1+𝑦23−𝑦_1 + 𝑦_2 ≥ 3

𝑦1,𝑦20𝑦_1, 𝑦_2 ≥ 0


After introducing surplus,artificial variables

Min Z=2x1+4x2+0S1+0S2+MA1+MA2

subject to

2x1-3x2-S1+A1=2-x1+x2-S2+A2=3

and x1,x2,S1,S2,A1,A2≥0



Positive maximum Zj-Cj is M-2 and its column index is 1. So, the entering variable is x1.

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

∴ The pivot element is 2.

Entering =x1, Departing =A1, Key Element =2



Since all Zj-Cj≤0

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

x1=1,x2=0

Min Z=2

But this solution is not feasible

because the final solution violates the 2nd constraint -  x1 +  x2 ≥ 3.

and the artificial variable A2 appears in the basis with positive value 4


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