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The problem involves solving a linear programming problem (LPP) using the Big M method. Here’s a breakdown of the problem as shown in the image:

### Problem Statement
**Objective:**  
Maximize \( Z = 6x_1 + 4x_2 \)

**Subject to the constraints:**
\[
2x_1 + 3x_2 \leq 30
\]
\[
3x_1 + 2x_2 \leq 24
\]
\[
x_1 + x_2 \geq 3
\]
\[
x_1, x_2 \geq 0
\]

### Solution Approach: Big M Method
The Big M method is used to solve linear programming problems involving inequalities with both “≤” and “≥” constraints. Here are the steps:

1. **Convert the inequalities into equalities**:
   - For \(\leq\) constraints, add a slack variable.
   - For \(\geq\) constraints, subtract a surplus variable and add an artificial variable with a large penalty \( M \) in the objective function.
   
   Let's rewrite the constraints with these additional variables:

   1. \(2x_1 + 3x_2 + s_1 = 30\) (Add slack variable \( s_1 \geq 0 \))
   2. \(3x_1 + 2x_2 + s_2 = 24\) (Add slack variable \( s_2 \geq 0 \))
   3. \(x_1 + x_2 - s_3 + a_1 = 3\) (Subtract surplus variable \( s_3 \geq 0 \) and add artificial variable \( a_1 \))

2. **Rewrite the Objective Function:**
   Modify the objective function \( Z \) to account for the artificial variable:
   \[
   Z = 6x_1 + 4x_2 - M \times a_1
   \]

3. **Set up the Initial Simplex Table with the Big M penalty term:**
   - The artificial variable \( a_1 \) is included in the basis with a large penalty of \( -M \) in the objective row.
   - You will need to use the simplex method iteratively to eliminate the artificial variable from the basis while maximizing \( Z \).

4. **Iterate Using the Simplex Method:**
   - Perform the simplex iterations, choosing pivot elements to maximize the objective function.
   - Continue until all artificial variables are removed from the basis and optimality conditions are met (all coefficients in the objective function row are non-negative).

5. **Obtain the Optimal Solution:**
   - Once the tableau indicates optimality, read the values of \( x_1 \) and \( x_2 \) from the final tableau.
   - If any artificial variable remains in the basis with a non-zero value, the original problem is infeasible.

Would you like to see a step-by-step setup of the initial simplex tableau or the detailed iterative steps for solving this using the Big M method?

### Follow-up Questions:
1. How does the Big M method differ from the Two-Phase method in dealing with artificial variables?
2. Can the Big M method be used to solve minimization problems as well?
3. What are some potential drawbacks of using very large values for \( M \)?
4. How do we know if a solution is infeasible when using the Big M method?
5. How can we interpret the shadow prices from the final simplex tableau?

### Tip:
When choosing the value of \( M \), ensure it is sufficiently large compared to other coefficients in the objective function to prevent artificial variables from remaining in the optimal solution.

Solve the following LPP using the Big M method: Maximize Z = 6x1 + 4x2, subject to: 2x1 + 3x2 <= 30, 3x1 + 2x2 <= 24, x1 + x2 >= 3, and x1, x2 >= 0.

Learn how to solve the LPP with the objective function Z = 6x1 + 4x2 and constraints 2x1 + 3x2 <= 30, 3x1 + 2x2 <= 24, x1 + x2 >= 3 using the Big M method. This solution involves linear programming, artificial variables, and the simplex method. Suitable for students studying operations research and optimization techniques.

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