The image shows a linear programming problem with constraints and the solution process for Phase 1 using surplus and artificial variables.

Problem Setup

The initial constraints are:

1. $2X_1 + X_2 \geq 4$
2. $X_1 + 2X_2 \geq 3$
3. $X_1, X_2 \geq 0$

Solution - Phase 1

To convert these inequalities into equalities, surplus variables $S_1$ and $S_2$, and artificial variables $A_1$ and $A_2$ are added:

1. $2X_1 + X_2 - S_1 + A_1 = 4$
2. $X_1 + 2X_2 - S_2 + A_2 = 3$

The initial basic feasible solution is:

• $A_1 = 4$
• $A_2 = 3$

Objective Function

The objective function is defined as: $\text{Maximize } Z = -A_1 - A_2$ This form is typically used in the first phase of the two-phase method to drive artificial variables out of the basis.

Would you like a detailed step-by-step guide on solving this problem or any specific part of the solution explained?

Further Questions

1. How are surplus and artificial variables used in solving linear programming problems?
2. What is the two-phase method in linear programming, and why is it used?
3. How does the addition of artificial variables affect the initial solution?
4. What is the significance of maximizing $Z = -A_1 - A_2$ in Phase 1?
5. How is the final solution found once the artificial variables are removed?

Tip

When dealing with inequalities in linear programming, converting them to equalities with surplus or slack variables allows using matrix-based methods like the Simplex method effectively.