Answer to Question #203223 in Operations Research for Raj Kumar

Question

Solve the following LPP by the two-phase simplex method.

 Max Z = x₁ + x₂ − x₃

 Subject to

4x₁ + x₂ + x₃ = 4

3x₁ + 2x₂ - x₄ = 6

x₁,x₂,x₃ ≥ 0

Answer 1

Solution:

The problem is converted to canonical form by adding slack, surplus and artificial variables as appropiate

1. As the constraint-1 is of type '=' we should add artificial variable A₁

2. As the constraint-2 is of type '=' we should add artificial variable A₂

3. As the constraint-3 is of type '≤' we should add slack variable S₁

4. As the constraint-4 is of type '≤' we should add slack variable S₂

5. As the constraint-5 is of type '≥' we should subtract surplus variable S₃ and add artificial variable A₃

After introducing slack,surplus,artificial variables:

Max Z=-A₁-A₂-A₃ subject to

$4x_1+x_2+x_3+A_1=4 \\ 3x_1+2x_2+A_2=6 \\ x_1+S_1=0 \\ x_2+S_2=0 \\x_3-S_3+A_3=0$

and 

x₁,x₂,x₃,S₁,S₂,S₃,A₁,A₂,A₃≥0

Since all Z_j-C_j≥0

Hence, optimal solution has arrived with the value of variables as :

x₁=0,x₂=0,x₃=4

Max Z=0

But this solution is not feasible

because the final solution violates the 2nd constraint  3 x₁ + 2 x₂    = 6.

and the artificial variable A2 appears in the basis with a positive value of 6.

So phase-2 is not possible.