Corner Point Calculator

How Corner Point Calculator Works?

The Corner Point Calculator solves a two-variable linear programming problem by finding feasible vertices of the region defined by up to three linear constraints. The objective function is formed from the two entered coefficients:

Here, $c_x$ and $c_y$ are the objective coefficients entered for $x$ and $y$. Each constraint has the general form:

where ⧣ can be ≤, =, or ≥. The calculator examines the boundary lines associated with the three constraints, calculates their intersections, and retains points that satisfy all selected inequalities. These retained intersections are the feasible corner points.

For a pair of boundary equations, the intersection can be calculated using a two-by-two linear system:

If the determinant $D=a_1{b}_2-a_2{b}_1$ is nonzero, the lines have a unique intersection:

The objective function is then evaluated at each feasible point. For a point $(x_j,y_j)$, its objective value is:

The calculator compares these values and reports the coordinates and objective value of the best point according to the selected optimization type.

This method follows the corner-point principle: when a linear objective has a finite optimum over a nonempty polygonal feasible region, at least one optimum occurs at a corner point.

Practical Use Cases

Students can use the calculator to check graphical linear programming exercises, including the intersection of constraint boundaries, feasibility testing, and evaluation of maximum or minimum objective values.

Operations researchers can model small two-variable allocation problems, such as selecting quantities of two activities while satisfying three resource, demand, or capacity constraints.

Engineers can compare feasible design combinations when two decision variables are restricted by linear performance or material limits and the objective is to maximize output or minimize a linear cost.

Business analysts can evaluate simple product-mix, staffing, or budget scenarios by entering the contribution coefficients, constraint relations, and right-side limits.

Manufacturing planners can identify the best combination of two products under several production limits, then review every reported corner point and its corresponding objective value.

Researchers and instructors can use the separate corner-point outputs to demonstrate how changing a coefficient, inequality direction, or right-hand-side value changes the feasible region and selected optimum.

Accuracy and Limitations

The calculator uses numerical intersection and comparison of linear equations, so results are generally reliable for ordinary finite decimal inputs within the supported coefficient and right-side range of -1,000,000 to 1,000,000. Very close or parallel constraint boundaries can produce rounding-sensitive coordinates, and coincident or parallel lines may not create a unique corner point. It is designed for two-variable linear programs with three entered constraints and reports up to five identified feasible corner points; it does not model nonlinear expressions, integer-only decisions, more than two decision variables, or additional constraints. If the feasible region is empty, unbounded, or has an optimum along an entire edge, the listed corner-point results should be interpreted carefully rather than as a complete general-purpose linear-programming proof.

Quick Examples