Linear Programming Calculator
Solve a linear programming problem in two variables by the corner-point method: the feasible region's corners, Z at each one, and the optimum, including unbounded regions.
About the Linear Programming Calculator
If a linear programming problem has an optimal value, it occurs at a corner of the feasible region (the corner-point theorem). So find the corners, by solving pairs of boundary lines and keeping the points that satisfy every constraint, and evaluate Z at each.
When the feasible region is unbounded, the largest corner value M is only the maximum if the half-plane Z > M has no point in common with the region.
The formula
Optimal Z occurs at a corner point of the feasible region
Worked example
Maximise Z = 3x + 2y, x + 2y ≤ 8, 3x + 2y ≤ 12, x, y ≥ 0
- Corners: (0, 0), (4, 0), (2, 3), (0, 4)
- Z = 0, 12, 12, 8
- Maximum 12 at (4, 0) and (2, 3): every point between them too
Questions people ask
What does 'unbounded' mean?
The feasible region goes on forever in some direction, and Z keeps growing (or falling) along it, so there's no maximum (or minimum).
Why is the best value always at a corner?
The objective is linear, so it changes steadily across the feasible region. Its largest and smallest values must occur at a vertex, or along a whole edge when the objective line is parallel to it.