Discrete Mathematics

Overview

The Simplex Algorithm is a powerful algebraic method used to solve linear programming problems. While you may have used the graphical method for problems with two variables, the Simplex Algorithm allows you to solve complex problems with three or more variables by moving systematically along the edges of a feasible region to find the optimal solution.

In GCSE Further Mathematics, examiners test your ability to translate a real-world problem into standard form, construct an initial simplex tableau, and accurately perform row operations. This topic connects deeply with algebra, inequalities, and optimisation. Exam questions typically range from 4 to 8 marks, rewarding methodical working and precise arithmetic.

Listen to the companion podcast for an in-depth walkthrough of the core concepts and common exam pitfalls:

Key Concepts

Concept 1: Standard Form and Slack Variables

Before applying the algorithm, you must convert inequalities into equations. This is done by adding slack variables (s_1, s_2, \dots). A slack variable represents the unused capacity of a constraint. For example, if a constraint is 2x + y \le 10, we rewrite it as 2x + y + s_1 = 10. The objective function must also be rearranged so all terms are on one side, typically Z - cx - dy = 0.

Example: Maximise Z = 3x + 2y subject to x + y \le 5.
Standard form: Z - 3x - 2y = 0 and x + y + s_1 = 5.

Concept 2: The Initial Tableau

The initial simplex tableau is a matrix representing the standard form equations. The columns represent the variables (x, y, s_1, Z) and the Right-Hand Side (RHS). The basic variables (initially the slack variables) form an identity matrix within the tableau.

Concept 3: Identifying Pivot Column and Row

To improve the objective function, we must choose an entering variable and a leaving variable.

• Pivot Column (Entering Variable): The column with the most negative value in the objective row (Z row). This variable will increase Z the most.
• Pivot Row (Leaving Variable): Perform the ratio test by dividing the RHS by the positive values in the pivot column. The row with the smallest non-negative ratio is the pivot row.

Concept 4: Row Operations

Once the pivot element (intersection of pivot row and column) is identified, use row operations to make the pivot element 1 and all other elements in the pivot column 0. This algebraically moves you to an adjacent, superior vertex on the feasible region.

Concept 5: Identifying the Optimal Solution

The algorithm terminates when there are no negative values remaining in the objective row. At this point, the optimal solution is found. The values of the basic variables are read from the RHS column, and all non-basic variables are 0.

Mathematical Relationships

• Objective Function: Z = ax_1 + bx_2 + \dots (The function to be maximised)
• Constraints: a_{11}x_1 + a_{12}x_2 \le b_1 (Resource limitations)
• Standard Form: a_{11}x_1 + a_{12}x_2 + s_1 = b_1 (Where s_1 \ge 0)
• Ratio Test: \text{Ratio} = \frac{\text{RHS}}{\text{Pivot Column Value}} (Only calculate for values > 0)

Practical Applications

The Simplex Algorithm is widely used in logistics, manufacturing, and finance to optimise resources. For example, an airline might use it to minimise fuel costs while meeting passenger demand and flight schedule constraints.