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L06_Artificial Variable techniques - Two Phase Method

# L06_Artificial Variable techniques - Two Phase Method -...

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Minimize 2 1 7 5 x x z + = Subject to the constraints 0 , 18 60 4 3 42 3 2 2 1 2 1 2 1 2 1 + + + x x x x x x x x Ans: x 1 =12, x 2 =6 Soln(= Min z ) = 102

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Introducing the surplus and artificial variables, R 1 , R 2 , the LPP is modified as follows: Minimize Subject to the constraints 1 2 1 1 1 2 2 2 1 2 3 3 1 2 1 2 3 1 2 3 2 3 42 3 4 60 18 , , , , , , , 0 x x s R x x s R x x s R x x s s s R R R + - + = + - + = + - + = 3 2 1 2 1 7 5 MR MR MR x x z + + + + =
z 1 -5 -7 0 0 0 -M -M -M 0 Basic z x1 x2 s1 s2 s3 R1 R2 R3 Sol -5+6M -7+8M -M -M -M 0 0 0 120M R1 0 2 3 -1 0 0 1 0 0 42 R2 0 3 4 0 -1 0 0 1 0 60 R3 0 1 1 0 0 -1 0 0 1 18 x2 0 2/3 1 -1/3 0 0 1/3 0 0 14 z 1 0 -M -M 0 0 98+8M 1 2 3 3 M - + 7 5 3 3 M - + R2 0 1/3 0 4/3 -1 0 -4/3 1 0 4 R3 0 1/3 0 1/3 0 -1 -1/3 0 1 4 7 8 3 3 M -

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z 1 0 -M -M 0 0 98+8M Basic z x1 x2 s1 s2 s3 R1 R2 R3 Sol x2 0 2/3 1 -1/3 0 0 1/3 0 0 14 R2 0 1/3 0 4/3 -1 0 -4/3 1 0 4 R3 0 1/3 0 1/3 0 -1 -1/3 0 1 4 x2 0 3/4 1 0 -1/4 0 0 1/4 0 15 z 1 0 0 -M - M 0 1 05+3M 1 2 3 3 M - + 7 5 3 3 M - + s1 0 1/4 0 1 -3/4 0 -1 3/4 0 3 R3 0 1/4 0 0 1/4 -1 0 -1/4 1 3 7 8 3 3 M - 1 4 4 M + 7 4 4 M - + 7 5 4 4 M -
z 1 0 0 -M -M 0

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L06_Artificial Variable techniques - Two Phase Method -...

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