The problem presented is a linear programming problem (LPP) that needs to be solved using the Big M method. Here is a breakdown of the given LPP:

Objective Function: $\text{Maximise } Z = 6x_1 + 7x_2$

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

Big M Method Setup:

1. Introduce Slack and Surplus Variables:
   Convert inequalities into equalities by introducing slack variables for the ≤ constraints and surplus variables for the ≥ constraint.

   • For $2x_1 + 3x_2 \leq 30$, add slack variable $s_1$:
     $2x_1 + 3x_2 + s_1 = 30$

   • For $3x_1 + 2x_2 \leq 24$, add slack variable $s_2$:
     $3x_1 + 2x_2 + s_2 = 24$

   • For $x_1 + x_2 \geq 3$, subtract surplus variable $s_3$ and introduce an artificial variable $A_1$:
     $x_1 + x_2 - s_3 + A_1 = 3$

2. Objective Function with Artificial Variable:
   Since we are introducing an artificial variable $A_1$ for the Big M method, we modify the objective function to penalize this variable: $Z = 6x_1 + 7x_2 - M \cdot A_1$ where $M$ is a very large positive number.

3. Reformulated Problem: Now the problem is: $\text{Maximise } Z = 6x_1 + 7x_2 - M \cdot A_1$ Subject to the following system of equalities: $2x_1 + 3x_2 + s_1 = 30$ $3x_1 + 2x_2 + s_2 = 24$ $x_1 + x_2 - s_3 + A_1 = 3$ and $x_1, x_2, s_1, s_2, s_3, A_1 \geq 0$.

4. Apply Simplex Algorithm with Big M:

   • Start with the initial simplex tableau by introducing the slack, surplus, and artificial variables.
   • Iteratively apply the Simplex algorithm, ensuring that artificial variables are driven out of the basis to achieve an optimal solution.

Would you like me to continue with the detailed steps of the Simplex tableau or explain the next steps in more detail?

Here are 5 follow-up questions for better understanding:

1. What are the key differences between slack, surplus, and artificial variables in linear programming?
2. How does the Big M method handle artificial variables in the objective function?
3. Why do we assign a large penalty $M$ to artificial variables in the objective function?
4. What are the stopping criteria for the Big M method during the simplex iterations?
5. How would the solution differ if we used the two-phase simplex method instead of the Big M method?

Tip: Always check if a linear programming problem involves ≥ constraints, as they often require the use of surplus and artificial variables in methods like the Big M method.