Linear Programming
What is tested
Linear Programming tests how you build an objective function and constraints from a word problem, solve a two-variable problem graphically, and read the optimal value correctly from the feasible region.
Core rules
- Name the decision variables first, then write Z = ax + by and every constraint as a linear inequality, including the implicit x ≥ 0, y ≥ 0.
- Draw each constraint as a line, test a point (usually the origin) to pick the half-plane, and shade the common region.
- Corner Point Theorem: for a bounded feasible region, the maximum and minimum of Z occur at corner points; evaluate Z at each corner and compare.
- For an unbounded region, evaluate Z at the corners, then check whether the open half-plane Z > (or <) that corner value meets the feasible region.
Worked example
Maximise Z = 3x + 2y subject to x + y ≤ 4, x + 2y ≤ 6, x ≥ 0, y ≥ 0. The two lines intersect at (2, 2). Corners and values: (0,0) → 0; (4,0) → 12; (2,2) → 10; (0,3) → 6. The maximum is 12 at (4, 0). Note that (0,4) is not a corner, since x + 2y ≤ 6 fails there.
Marks-losing traps
Omitting the non-negativity constraints; evaluating a point lying outside the feasible region; listing corners from one constraint only; assuming an unbounded region always has a maximum; quoting the optimal Z without the coordinates, or the coordinates without Z.