Answer to Question #201595 in Operations Research for Pratik

Question

Solve using dual simplex method

Minimize z 2x₁ + 2x₂ + 4x3

2x+3x2 + 5x3 2 2

Subject to 3x1 + x₂ + 7x3 <3

x1 + 4x₂ + 6x3 ≤ 5

Answer 1

In order to apply the dual simplex method, convert minZ to maxZ:

$maxZ=-2x_1-2x_2-4x_3$

subject to

$2x_1+3x_2+5x_3\le2$

$3x_1+x_2+7x_3\le3$

$x_1+4x_2+6x_3\le5$

$x_1,x_2,x_3\ge0$

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

•  as the constraint-1 is of type '$\le$ ' we should add slack variable S₁
•  as the constraint-2 is of type '$\le$ ' we should add slack variable S₂
•  as the constraint-1 is of type '$\le$ ' we should add slack variable S₃

After introducing slack variables:

$maxZ=-2x_1-2x_2-4x_3+0S_1+0S_2+0S_3$

$2x_1+3x_2+5x_3+S_1=2$

$3x_1+x_2+7x_3+S_2=3$

$x_1+4x_2+6x_3+S_3=5$

$x_1,x_2,x_3,S_1,S_2,S_3\ge0$

Since all $Z_j-C_j\ge0$ and all $X_{Bi}\ge0$ thus the current solution is the optimal solution:

$x_1=x_2=x_2=0$