12.3.3 Big-M Method

Introduction

The Big-M method is an alternative to the two-stage Simplex method for solving problems with $\geq$ or equality constraints. It introduces artificial variables and assigns them a very large penalty ( $M$ ) in the objective function to ensure they are removed from the solution.

The Role of $M$

Artificial variables are penalised in the objective function to discourage their inclusion in the final solution:

For maximisation: $M > 0$ is added with a negative coefficient.
For minimisation: $M > 0$ is added with a positive coefficient.

Process

Formulate the problem by adding slack, surplus, and artificial variables.
Modify the objective function to include $M$ terms for artificial variables.
Solve the problem using the Simplex method.

Worked Example

infoNote

Problem

Maximise:

Z = 4x_1 + 3x_2

Subject to:

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

Step 1: Add Variables

Add a slack variable for $\leq$

x_1 + x_2 + s_1 = 5

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

2x_1 + 3x_2 - s_2 + t_1 = 12

Step 2: Modify the Objective Function

Add -M for the artificial variable $t_1$ :

Z = 4x_1 + 3x_2 - Mt_1

Step 3: Set Up Initial Tableau

\begin{array}{c|cccccc|c} \text{Basic Variable} & x_1 & x_2 & s_1 & s_2 & t_1 & Z & \text{RHS} \\ \hline s_1 & 1 & 1 & 1 & 0 & 0 & 0 & 5 \\ t_1 & 2 & 3 & 0 & -1 & 1 & 0 & 12 \\ \hline Z & -4 & -3 & 0 & M & -M & 1 & -12M \\ \end{array}

Step 4: Solve Using Simplex

Apply the Simplex algorithm to:

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

Note Summary

infoNote

Common Mistakes

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

infoNote

Key Formulas

Objective function with $M$ :

Z = \text{original objective} - M(\text{artificial variables})

Slack and surplus variables:

a_1x_1 + a_2x_2 \geq b \quad \Rightarrow \quad a_1x_1 + a_2x_2 - s + t = b

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

Big-M Method Simplified Revision Notes for A-Level Edexcel Further Maths Decision Maths 1

12.3.3 Big-M Method

Introduction

The Role of $M$

Process

Worked Example

Problem

Note Summary

Common Mistakes

Key Formulas

500K+ Students Use These Powerful Tools to Master Big-M Method For their A-Level Exams.

Other Revision Notes related to Big-M Method you should explore

Big-M Method Simplified Revision Notes for A-Level Edexcel Further Maths Decision Maths 1

12.3.3 Big-M Method

Introduction

The Role of MMM

Process

Worked Example

Problem

Note Summary

Common Mistakes

Key Formulas

500K+ Students Use These Powerful Tools to Master Big-M Method For their A-Level Exams.

Other Revision Notes related to Big-M Method you should explore

The Role of $M$