Corner Point Calculator

About the Corner Point Calculator

The corner-point method (also called the vertex method) is the foundational technique for solving two-variable linear programming problems. The key insight behind it is the Fundamental Theorem of Linear Programming: if a linear objective function has an optimal value on a bounded feasible region, that optimum occurs at one of the region's corner points — the vertices where constraint boundary lines intersect.

This page automates the full graphical workflow. You enter a system of linear inequalities (one per line, using familiar notation like 2x+y<=20 or x>=0), and it parses each constraint into a boundary line. It then computes every pairwise intersection of those lines, checks which intersections satisfy all constraints (i.e., lie inside the feasible region), and lists the feasible corner points in a detailed evaluation table.

Next, you specify a linear objective function such as 5x+4y and choose whether to maximize or minimize. The calculator evaluates the objective at every corner point and highlights the optimal vertex. A polygon visualization draws the feasible region with labeled corners, and a bar chart compares the objective values side by side.

Whether you are a student working through homework on LP, a teacher preparing examples, or an analyst prototyping a resource-allocation model, the output keeps the feasible region, candidate vertices, and objective values in one structured view. Five built-in presets — a simple triangle, a classic four-constraint LP, a minimization problem, a square, and a diamond — let you load standard cases quickly.

When This Page Helps

Use this page when you want the corner-point method laid out cleanly: feasible intersections, rejected intersections, objective values, and the final optimum. It is useful for classroom LP problems, teaching the graphical method, and checking two-variable models before moving to larger optimization methods.

How to Use the Inputs

1. Enter your system of linear inequalities in the text area, one constraint per line.
2. Use format like x>=0, y>=0, 2x+y<=20, x+3y<=24.
3. Enter the objective function (e.g. 5x+4y) in the input field.
4. Choose whether to maximize or minimize the objective.
5. Read the number of corner points found and the optimal solution.
6. Review the evaluation table showing each corner point and its objective value.
7. See the feasible region polygon and the highlighted optimal vertex.
8. Use preset buttons to load common textbook examples.

Example Calculation

Tips & Best Practices

• Always include non-negativity constraints (x >= 0, y >= 0) unless the problem explicitly allows negative values.
• If no corner points are found, your constraints may be inconsistent (empty feasible region).
• The corner-point method only works for linear objectives; for non-linear objectives, the optimum may be in the interior.
• For minimization, you can also negate the objective and maximize, then negate the result.
• If two corner points share the same optimal value, every point on the line segment between them is also optimal.

The Fundamental Theorem of Linear Programming

The theorem states that if a linear objective function has an optimal value on a bounded convex feasible region, that optimum is attained at one (or more) vertices. The proof relies on the convexity of the feasible region: any interior point can be written as a convex combination of vertices, and because the objective is linear, its value at that interior point is a weighted average of its values at the vertices — which can never exceed the maximum vertex value. This is why checking only the corners is sufficient: you are guaranteed not to miss a better solution hiding in the interior.

Solving an LP Graphically: Step by Step

For two-variable problems, the graphical method is both intuitive and rigorous. First, graph each constraint as a boundary line and shade the halfplane that satisfies the inequality. The feasible region is the intersection of all shaded halfplanes. Next, identify every corner point by solving pairs of boundary-line equations simultaneously. Discard any intersection that violates another constraint. Finally, evaluate the objective function at each remaining vertex and pick the best value. This calculator automates all three steps, but understanding the geometry helps you catch errors and build intuition for larger problems.

From Two Variables to the Simplex Method

Real-world LPs often involve hundreds or thousands of variables and constraints, far beyond what graphical methods can handle. The Simplex method (developed by George Dantzig in 1947) generalises the corner-point idea to higher dimensions: it moves along edges of the feasible polytope from vertex to adjacent vertex, always improving the objective, until it reaches the optimum. Interior-point methods offer an alternative by traversing the interior of the polytope. Both approaches rely on the same theorem — the optimum is at a vertex — making the two-variable graphical method a perfect conceptual entry point for understanding industrial-strength optimisation.

Frequently Asked Questions

• What is a corner point?

  A corner point (vertex) is a point where two or more constraint boundary lines intersect and the point satisfies all constraints — it lies inside or on the boundary of the feasible region.

• Why does the optimum always occur at a corner point?

  Because a linear function on a convex polygon achieves its extreme values at the vertices. This is the Fundamental Theorem of Linear Programming.

• What if the feasible region is unbounded?

  For maximization, an unbounded region in the direction of increasing objective means the problem has no finite maximum. The calculator will still list all finite corner points found.

• Can I enter more than 4 constraints?

  Yes — enter as many constraints as you like, one per line. The calculator checks all pairwise intersections.

• What format should constraints use?

  Use the form ax+by<=c or ax+by>=c. Examples: x>=0, y>=0, 2x+y<=20. Coefficients of 1 can be omitted (x instead of 1x).

• Does this solve integer programming?

  No — corner points may have fractional coordinates. For integer LP, you would need branch-and-bound or cutting planes on top of this.