How this calculator works
In two variables, a linear objective reaches its best value at a corner of the feasible region, provided the region is bounded in the improving direction. The feasible region is the set of points satisfying every inequality, together with the usual non-negativity conditions x >= 0 and y >= 0. This calculator intersects every pair of boundary lines, keeps the feasible intersections, evaluates the objective at each one and returns the best. It also checks whether an improving direction escapes to infinity, which would make the problem unbounded.
Formula
Maximize or minimize p = px x + py y subject to ai x + bi y <= ci or >= ci, with x >= 0 and y >= 0.
Worked example
Maximizing 3x + 2y subject to x + y <= 4 and x <= 2, with x and y non-negative, gives corner points (0, 0), (0, 4), (2, 0) and (2, 2). Their objective values are 0, 8, 6 and 10, so the maximum is 10 at (2, 2).
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Frequently asked questions
Why are x and y assumed non-negative?
The corner point method in introductory linear programming uses the first quadrant. Any non-negativity constraint you enter is still applied, so the assumption only adds the standard x >= 0 and y >= 0 boundaries.
What does unbounded mean?
The feasible region allows the objective to improve without limit. There is no finite best value, even though corner points may exist.
Can a constraint be redundant?
Yes. A redundant constraint does not change the feasible region. Its boundary intersections still produce feasible points, but they cannot beat the true optimum.