LINEAR PROGRAMMING
Maximising Profit & Minimising Cost
Complete Study Notes with Worked Examples and Graphs
1. What is Linear Programming?
Linear Programming (LP) is a mathematical method used to find the best possible outcome (maximum
profit or minimum cost) given a set of linear constraints. It is widely used in business, economics,
engineering, and logistics.
Key Terms to Know:
Decision variables: The unknowns you solve for (e.g., x and y representing units of two
products).
Objective function: The expression you want to maximise or minimise (e.g., Profit P = 3x +
2y).
Constraints: Limitations expressed as inequalities (e.g., 2x + y <= 18 means labour hours are
limited).
Feasible region: The shaded area on the graph where ALL constraints are satisfied
simultaneously.
Corner points (vertices): The corners of the feasible region — the optimal solution always
occurs at one of these.
2. General Method — Steps to Solve Any LP Problem
Step-by-Step Procedure:
1. Define your variables. State clearly what x and y represent.
2. Write the objective function. Identify what you are maximising or minimising.
3. Write all constraints as inequalities. Include non-negativity: x >= 0, y >= 0.
4. Graph each constraint by treating it as an equation first, then plotting the line.
5. Identify the feasible region — shade the area satisfying ALL constraints.
6. Find all corner points (use simultaneous equations for intersections).
7. Substitute each corner point into the objective function.
8. Select the optimal value: LARGEST for maximisation, SMALLEST for minimisation.
3. Maximising Profit
When you are told to maximise profit, you want the corner point giving the highest value. Constraints
are typically <= (at most), so the feasible region lies below and to the left of the boundary lines.
Linear Programming Notes | Page 1
, Worked Example — Maximise Profit
Problem:
A factory makes two products, X and Y. Each unit of X requires 2 hours of labour and 1 hour
of machine time. Each unit of Y requires 1 hour of labour and 3 hours of machine time.
There are 18 labour hours and 21 machine hours available. Profit on X is R3 per unit and R2
per unit on Y. How many of each should be made to maximise profit?
Step 1 — Define Variables
Let x = number of units of Product X
Let y = number of units of Product Y
Step 2 — Objective Function
Maximise: P = 3x + 2y
Step 3 — Constraints
Labour: 2x + y <= 18
Machine time: x + 3y <= 21
Non-negativity: x >= 0, y >= 0
Step 4 & 5 — Graph and Feasible Region
Plot each line using intercepts. Test the origin (0,0) to confirm which side to shade (for <= constraints,
the origin-side is usually feasible). The blue shaded region below satisfies all constraints
simultaneously.
Linear Programming Notes | Page 2
Maximising Profit & Minimising Cost
Complete Study Notes with Worked Examples and Graphs
1. What is Linear Programming?
Linear Programming (LP) is a mathematical method used to find the best possible outcome (maximum
profit or minimum cost) given a set of linear constraints. It is widely used in business, economics,
engineering, and logistics.
Key Terms to Know:
Decision variables: The unknowns you solve for (e.g., x and y representing units of two
products).
Objective function: The expression you want to maximise or minimise (e.g., Profit P = 3x +
2y).
Constraints: Limitations expressed as inequalities (e.g., 2x + y <= 18 means labour hours are
limited).
Feasible region: The shaded area on the graph where ALL constraints are satisfied
simultaneously.
Corner points (vertices): The corners of the feasible region — the optimal solution always
occurs at one of these.
2. General Method — Steps to Solve Any LP Problem
Step-by-Step Procedure:
1. Define your variables. State clearly what x and y represent.
2. Write the objective function. Identify what you are maximising or minimising.
3. Write all constraints as inequalities. Include non-negativity: x >= 0, y >= 0.
4. Graph each constraint by treating it as an equation first, then plotting the line.
5. Identify the feasible region — shade the area satisfying ALL constraints.
6. Find all corner points (use simultaneous equations for intersections).
7. Substitute each corner point into the objective function.
8. Select the optimal value: LARGEST for maximisation, SMALLEST for minimisation.
3. Maximising Profit
When you are told to maximise profit, you want the corner point giving the highest value. Constraints
are typically <= (at most), so the feasible region lies below and to the left of the boundary lines.
Linear Programming Notes | Page 1
, Worked Example — Maximise Profit
Problem:
A factory makes two products, X and Y. Each unit of X requires 2 hours of labour and 1 hour
of machine time. Each unit of Y requires 1 hour of labour and 3 hours of machine time.
There are 18 labour hours and 21 machine hours available. Profit on X is R3 per unit and R2
per unit on Y. How many of each should be made to maximise profit?
Step 1 — Define Variables
Let x = number of units of Product X
Let y = number of units of Product Y
Step 2 — Objective Function
Maximise: P = 3x + 2y
Step 3 — Constraints
Labour: 2x + y <= 18
Machine time: x + 3y <= 21
Non-negativity: x >= 0, y >= 0
Step 4 & 5 — Graph and Feasible Region
Plot each line using intercepts. Test the origin (0,0) to confirm which side to shade (for <= constraints,
the origin-side is usually feasible). The blue shaded region below satisfies all constraints
simultaneously.
Linear Programming Notes | Page 2
