Corner Point Calculator

Linear programming optimizes a linear objective while respecting linear constraints. In a problem with two decision variables, each inequality represents a half-plane on a coordinate graph. Their shared area is the feasible region. When that region is bounded and nonempty, a maximum or minimum objective value occurs at one or more of its corner points, also called vertices or extreme points. The corner point method uses this fact to replace a continuous search with a finite set of evaluations. This calculator accepts an objective of the form cx times x plus cy times y and two constraints of the form ax times x plus by times y less than or equal to a limit. It also enforces standard nonnegativity conditions, so x and y cannot be negative. It constructs candidate vertices from the origin, intercepts on both coordinate axes, and the intersection of the two constraint boundary lines. Candidates that violate either inequality are removed. After finding feasible corners, the calculator substitutes every point into the objective function. For maximization it keeps the largest value; for minimization it keeps the smallest. The displayed corner list lets you verify the comparison. If two adjacent vertices produce the same best value, every point on the segment joining them is also optimal, even though this compact result names one of those vertices. The graphical method is especially useful for production planning, resource allocation, diet models, and introductory operations research. Decision variables can represent quantities of products, hours of work, or amounts of material. Constraint coefficients describe how each decision consumes a limited resource, while objective coefficients represent profit, cost, time, or another performance measure. A correct model matters as much as arithmetic. Confirm that every constraint uses compatible units and that the inequality direction matches the practical limit. This calculator specifically handles two less-than-or-equal constraints plus nonnegativity. Problems containing greater-than-or-equal or equality constraints may need algebraic conversion and can create feasible regions not represented by this input form. The corner point method is not a substitute for the simplex method on large models, but it makes the underlying geometry clear. It also provides a quick check for hand-drawn feasible regions. Decimal results are rounded to six places for readability. If coefficients produce exact fractions, retain those fractions in formal work to avoid accumulating rounding error.