Big-M Method (Edexcel A-Level Further Mathematics): Revision Notes

Problem

Maximise:

Subject to:

1. $x_1 + x_2 \leq 5$
2. $2x_1 + 3x_2 \geq 12$
3. $x_1, x_2 \geq 0$

Step 1: Add Variables

• Add a slack variable for $\leq$

• Add a surplus variable and an artificial variable for $\geq$

Step 2: Modify the Objective Function

Add -M for the artificial variable $t_1$:

Step 3: Set Up Initial Tableau

Step 4: Solve Using Simplex

Apply the Simplex algorithm to:

4. Remove artificial variables ($t_1$)
5. Maximise $Z$ by optimising the remaining variables.

Common Mistakes

1. Incorrect $M$-term signs: Using +M for maximisation or -M for minimisation.
2. Forgetting surplus variables: Omitting surplus variables in $\geq$ constraints.
3. Artificial variable errors: Not penalising artificial variables in the objective function.
4. Arithmetic errors: Mistakes during pivoting steps.
5. Ignoring non-negativity: Forgetting $x, t, s \geq 0$

Key Formulas

1. Objective function with $M$:

2. Slack and surplus variables:

3. Feasibility check: Artificial variables must exit the basis to achieve an optimal solution.