OPTIMIZATION GUIDE

Graphical method for two-variable linear programming

For a continuous two-variable linear program, an optimum—when bounded—occurs at a feasible corner point. The graphical method makes those points visible.

FORMULA

Maximize or minimize Z = c₁x + c₂y subject to linear constraints

WORKED EXAMPLE

Maximize Z = 3x + 5y subject to 2x + y ≤ 18, 2x + 3y ≤ 42, 3x + y ≤ 24, and x,y ≥ 0.

STEP BY STEP

Write the objective

Choose whether the model maximizes or minimizes a linear expression in x and y.

Add constraints

Each constraint defines a half-plane. Include non-negative restrictions when the quantities cannot be negative.

Inspect corner points

Evaluate the objective at feasible intersections. Also check whether the feasible region could be unbounded before treating a corner-point result as complete.