Linear Programming
Linear programming questions ask you to formulate constraints, draw the feasible region, and optimise an objective function by the corner point method. A full 6-mark graphical problem is standard, sometimes with a short formulation part.
Method
- Identify decision variables x and y and write the objective Z = ax + by, stating whether it is to be maximised or minimised.
- Convert each condition into a linear inequality, including x ≥ 0, y ≥ 0.
- Draw each line, test a point such as the origin to decide which side is feasible, and shade the common region.
- Find the corner points by solving pairs of boundary lines simultaneously.
- Evaluate Z at every corner point. The optimum occurs at a corner; if two corners tie, every point on the joining edge gives the same value.
Worked example
Maximise Z = 3x + 2y subject to x + y ≤ 4, x ≤ 2, x ≥ 0, y ≥ 0. The feasible region has corners (0, 0), (2, 0), (2, 2) and (0, 4). Evaluating: Z = 0, 6, 10 and 8 respectively. The maximum value is 10, attained at (2, 2).
Traps
- Shading the wrong side of a constraint line; always test a point.
- Missing a corner formed by the intersection of two slanted lines.
- Optimising at a point that is not a corner of the feasible region.
- Neglecting non-negativity constraints, which often create corners.