🎯 Learning Objectives
- Form inequalities from practical constraints.
- Identify a feasible region.
- Evaluate an objective function at vertices.
Introduction
This lesson develops Linear Programming from meaning and patterns before moving to calculations. Look for what the quantities, shapes or symbols represent, then connect that idea to the method.
🤔 Start with an Idea
A small business has limited time, money and materials. It wants the greatest profit possible without breaking any limit. Linear programming finds the best permitted choice.
📖 Key Notes
Let variables represent quantities. Translate every limit into an inequality, include non-negativity conditions, graph the constraints, then test the objective function at each vertex.
🏆 Optimisation Journey
The gold polygon represents choices satisfying every constraint.
🧠 Let's Think Together
A workshop makes tables x and chairs y. Constraints are x + y ≤ 8, 2x + y ≤ 12, x ≥ 0, y ≥ 0. Profit is P = 30x + 20y.
- Draw the feasible region.
- Find its vertices.
- Evaluate P at every vertex.
- Select the vertex giving the largest profit.
✍️ Worked Examples
At (4,4), P = 200.
At (6,0), P = 180.
Maximum: 200 at (4,4).
🎯 Practice Questions
Easy
Write the non-negativity constraints for quantities x and y.
Show answer
x ≥ 0 and y ≥ 0.
Challenge
Why are vertices tested?
Show answer
A linear objective reaches its maximum or minimum at a vertex of the feasible polygon.
⚠️ Common Mistakes
- Omitting non-negativity constraints.
- Optimising over a region that does not satisfy all constraints.
- Testing random interior points instead of all vertices.
- Giving a mathematical maximum without interpreting it in context.
📝 Exam Focus
Show the formulation, graph, feasible region, vertex table and final contextual statement. These are separate opportunities to earn marks.
Summary
- Define variables.
- Form constraints.
- Graph the common feasible region.
- Test the objective at every vertex.
- Interpret the optimum.