Answer to Question #301471 in Operations Research for JASNA KV

Question

Solve the following LPP using two phase method

Max Z= 2x1+3x2+10x3

subject to x1+2x3=0

x2+x3=1

x1,x2,x3≥0

Answer 1

Solution:

Max Z= 2x1+3x2+10x3

subject to x1+2x3$\le$0

x2+x3$\ge$ 1

x1,x2,x3≥0

-->Phase-1<--

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 slack variable S1

2. As the constraint-2 is of type '≥' we should subtract surplus variable S2 and add artificial variable A1

After introducing slack,surplus,artificial variables

Max Z=A1

subject to

x1+2x3+S1=0

x2+x3-S2+A1=1 and x1,x2,x3,S1,S2,A1≥0

Since all Zj-Cj≥0

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

x1=0,x2=1,x3=0

Max Z=0

-->Phase-2<--

we eliminate the artificial variables and change the objective function for the original,

Max Z=2x1+3x2+10x3+0S1+0S2

Variable S2 should enter into the basis, but all the coefficients in the S2 column are negative or zero. So S2 can not be entered into the basis.

Hence, the solution to the given problem is unbounded.