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.