Solve the following linear programming problem using Two Phase Method:

Question

Minimize Z = x1 - 2x2 -3x3
Subject to: -2x1 + x2 +3x3 = 2
2x1 + 3x2 + 4x3 =1
where x1, x2, x3 ≥ 0

Answer 1

To solve this linear programming problem using the Two-Phase Method, we first need to convert it into standard form by introducing slack variables and a surplus variable.

The given problem is:
Minimize Z = x1 - 2x2 - 3x3

Subject to:
-2x1 + x2 + 3x3 = 2
2x1 + 3x2 + 4x3 = 1

To convert it into standard form, let's introduce slack variables s1 and s2:
-2x1 + x2 + 3x3 + s1 = 2
2x1 + 3x2 + 4x3 + s2 = 1

Now, let's create the initial tableau for the Two-Phase Method by including the auxiliary objective function:

Z = a1 + a2

Subject to:
-2x1 + x2 + 3x3 + s1 = 2
2x1 + 3x2 + 4x3 + s2 = 1
a1 - s1 = 0
a2 - s2 = 0

Now, we can proceed with the Two-Phase Method to solve this linear programming problem.