To solve the given linear programming (LP) problem using the two-phase method, we will follow these steps:

Problem Recap:

Maximize $z = 4x_1 + \frac{1}{2}x_2 + x_3$

Subject to:

1. $x_1 + 2x_2 + 3x_3 \geq 2$
2. $2x_1 + x_2 + x_3 \leq 5$
3. $x_1, x_2, x_3 \geq 0$

Step 1: Convert the inequalities into equalities

1. Convert the first inequality $x_1 + 2x_2 + 3x_3 \geq 2$ by subtracting a surplus variable $s_1$: $x_1 + 2x_2 + 3x_3 - s_1 = 2, \quad s_1 \geq 0$
2. Convert the second inequality $2x_1 + x_2 + x_3 \leq 5$ by adding a slack variable $s_2$: $2x_1 + x_2 + x_3 + s_2 = 5, \quad s_2 \geq 0$

Now, we have the following system:

1. $x_1 + 2x_2 + 3x_3 - s_1 = 2$
2. $2x_1 + x_2 + x_3 + s_2 = 5$
3. $x_1, x_2, x_3, s_1, s_2 \geq 0$

Step 2: Introduce artificial variables

Since the first constraint has a surplus variable, we introduce an artificial variable $A_1$ to ensure feasibility in Phase I:

1. $x_1 + 2x_2 + 3x_3 - s_1 + A_1 = 2$

Step 3: Set up Phase I Objective Function

The objective of Phase I is to minimize the sum of artificial variables $A_1$. The Phase I objective function will be: $\text{Minimize } W = A_1$

The constraints for Phase I become:

1. $x_1 + 2x_2 + 3x_3 - s_1 + A_1 = 2$
2. $2x_1 + x_2 + x_3 + s_2 = 5$

Step 4: Solve Phase I

• We perform the simplex method to minimize $W$. If $W = 0$ at the optimal solution, proceed to Phase II. Otherwise, the problem is infeasible.

Step 5: Set up Phase II Objective Function

Once Phase I is complete, in Phase II, the original objective function is maximized: $\text{Maximize } z = 4x_1 + \frac{1}{2}x_2 + x_3$

Subject to:

1. $x_1 + 2x_2 + 3x_3 - s_1 = 2$
2. $2x_1 + x_2 + x_3 + s_2 = 5$

Step 6: Solve Phase II

Perform the simplex method again to find the optimal solution for the original objective function.

Would you like me to continue by solving each step in detail, or do you have specific questions about the steps?

Here are 5 related questions to explore further:

1. What is the difference between surplus and slack variables in linear programming?
2. How do artificial variables help in the two-phase method?
3. Can every linear programming problem be solved using the two-phase method?
4. What happens if the Phase I objective function is not minimized to zero?
5. How does the simplex method work in optimizing a linear programming problem?

Tip: In the two-phase method, always ensure that all artificial variables are removed before moving to Phase II. If any artificial variables remain in the basis at zero value, it may indicate degeneracy or a special case.