1. Stat 261
Two phase method

2. Usage Two-phase algorithm:
If there is any constraint with = or ≥ sign , then we can use
two-phase method.
Two-phase algorithm:
1- Convert system to standard form
2- Add artificial variables to constraint with = or ≥ sign.
3- Start phase one where we minimize or maximize objective of artificial variables.
4- Solve as a simplex method until to obtain optimal table in phase two.

3. Example :
Consider the following example? Minimize Z=4X1+X2 Subject to 3X1+X2=3 4X1+3X2≥6 X1+2X2≤4 X1,X2 ≥ 0 We need to two artificial variables.

4. Solution by Tora:
Step 1: open Tora select linear programming from menu input number of variables and number of constraints click enter input coefficients for objective and constraints as the we shown in the next table.

6. Step 2: Entering variable Leaving variable
click on solve menu select solve problem from menu algebraic iterations Two-phase method go to input screen Click on all iterations and we get 5 iterations as shown below.
Entering variable
Leaving variable

8. Thanks for all