Linear Programming
Linear Programming tests formulation of an objective and constraints, graphical solution of two-variable problems, feasible regions, and the corner-point theorem. The exam expects a clear feasible region and evaluation at vertices.
Core rules
Write the objective as Z=ax+by, with constraints as linear inequalities and non-negativity x≥0, y≥0. Graph each inequality; the feasible region is the common shaded side. A bounded feasible region has maximum and minimum values of Z at corner points. The corner-point method evaluates Z at every vertex, including intersections of boundary lines and axis intercepts. If the region is unbounded, check whether the objective can be increased indefinitely; a corner value is not automatically optimal. The objective line ax+by=k has slope −a/b, which helps confirm the optimum.
Worked example
Maximise Z=3x+2y subject to x+y≤4, x≤2, x≥0, y≥0. Vertices: (0,0), (2,0), (2,2), (0,4). Values: Z=0, 6, 10, 8. The maximum is 10 at (2,2).
Traps
Reverse the inequality when multiplying by a negative number. Shade the feasible side, not the side containing the origin by habit. Include intersection points of constraints; missing a vertex can change the answer. For 'at least' constraints, the feasible region may be unbounded. Do not ignore non-negativity.