Corner Point Calculator
Our free linear algebra calculator solves corner point problems. Get worked examples, visual aids, and downloadable results.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Corner Point Calculator
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Formula: Solve pairs of constraint equations: a1x + b1y = c1 and a2x + b2y = c2 using Cramers rule
Worked example — Maximum Z = 42 at corner point (4, 6)
Formula
Solve pairs of constraint equations: a1x + b1y = c1 and a2x + b2y = c2 using Cramers rule
Each corner point is found by solving two constraint equations simultaneously. The intersection point (x, y) is computed as x = (c1*b2 - c2*b1)/(a1*b2 - a2*b1) and y = (a1*c2 - a2*c1)/(a1*b2 - a2*b1). Only feasible points satisfying all constraints and non-negativity are kept.
Worked Examples
Example 1: Maximize Profit with Resource Constraints
Problem:Maximize Z = 3x + 5y subject to: x + y <= 10, 2x + y <= 14, x + 3y <= 18, x >= 0, y >= 0
Solution:Find all constraint intersections: (1) and (2): x + y = 10, 2x + y = 14 gives x = 4, y = 6 -> Z = 42 (1) and (3): x + y = 10, x + 3y = 18 gives x = 6, y = 4 -> Z = 38 (2) and (3): 2x + y = 14, x + 3y = 18 gives x = 4.8, y = 4.4 -> Z = 36.4 Axes intersections: (0,0)->Z=0, (7,0)->Z=21, (0,6)->Z=30, (10,0)->infeasible Check all for feasibility
Result:Maximum Z = 42 at corner point (4, 6)
Example 2: Minimize Cost Problem
Problem:Minimize Z = 2x + 3y subject to: x + y <= 10, 2x + y <= 14, x + 3y <= 18, x >= 0, y >= 0
Solution:Use same corner points as above: (0, 0): Z = 0 (4, 6): Z = 8 + 18 = 26 (6, 4): Z = 12 + 12 = 24 (7, 0): Z = 14 (0, 6): Z = 18 Minimum is at the origin.
Result:Minimum Z = 0 at corner point (0, 0)
Frequently Asked Questions
What is a corner point in linear programming?
A corner point (also called a vertex or extreme point) is a point where two or more constraint boundaries intersect within the feasible region of a linear programming problem. The feasible region is formed by the intersection of all constraint inequalities, and it forms a convex polygon in two dimensions. The Fundamental Theorem of Linear Programming states that if an optimal solution exists, it must occur at a corner point. This dramatically simplifies optimization because instead of checking infinitely many feasible points, you only need to evaluate the objective function at the finite set of corner points.
How do you find corner points of a feasible region?
To find corner points, you solve every pair of constraint equations simultaneously as a system of two linear equations. For each pair, compute the intersection point using methods like substitution, elimination, or Cramers rule. Then check whether each intersection point satisfies all other constraints (feasibility check) and non-negativity requirements. Points that fail any constraint are discarded. The remaining feasible intersection points are the corner points of the feasible region. For n constraints in two variables (including non-negativity), there are at most n-choose-2 possible intersection points to check.
Why does the optimal solution occur at a corner point?
The optimal solution occurs at a corner point because the objective function is linear and the feasible region is a convex polygon. A linear function over a convex set achieves its maximum and minimum values at extreme points (vertices). Geometrically, imagine sliding a straight line (level curve of the objective function) across the feasible region. The last point the line touches as it exits the region must be a vertex. If the objective function line is parallel to a constraint boundary, the optimum may occur along an entire edge, but it still includes the corner points at both ends of that edge.
What happens when constraint lines are parallel?
When two constraint lines are parallel, they have no intersection point and therefore do not produce a corner point together. Parallel constraints have the same ratio of coefficients (a1/b1 = a2/b2) but different right-hand sides. In this case, one constraint may be redundant (the other is always more restrictive). If the constraints are contradictory (like x + y <= 5 and x + y >= 10), the feasible region may be empty, meaning no solution exists. The calculator handles parallel lines by checking if the determinant of the coefficient matrix is zero and skipping those pairs.
How does the objective function affect which corner point is optimal?
The objective function determines which corner point is optimal by assigning a value to each corner point. Different objective functions typically select different corner points as optimal. For maximization, the corner point with the highest objective value is chosen. For minimization, the lowest value is selected. Changing the objective function coefficients changes the slope of the objective function line, which may cause a different corner point to become optimal. Sensitivity analysis examines how much the coefficients can change before the optimal corner point switches, which is important for understanding the robustness of the solution.
What is the feasible region in linear programming?
The feasible region is the set of all points that satisfy every constraint simultaneously, including non-negativity constraints. In two-dimensional problems, it forms a convex polygon (or is unbounded in some cases). Each linear inequality constraint defines a half-plane, and the feasible region is the intersection of all these half-planes. If the intersection is empty, the problem is infeasible. If the region is unbounded in the direction of optimization, the problem may be unbounded (no finite optimal solution). The corners of this polygon are exactly the corner points where constraint boundaries meet.
Can a linear programming problem have multiple optimal solutions?
Yes, a linear programming problem can have multiple optimal solutions. This occurs when the objective function is parallel to one of the binding constraints at the optimal corner point. In this case, every point along the constraint edge between two adjacent optimal corner points is also optimal, giving infinitely many optimal solutions. All these points share the same optimal objective function value. The simplex method will find one optimal corner point, but it is important to recognize when alternative optima exist. In practice, having multiple optimal solutions provides flexibility in implementation, as any point along the optimal edge can be chosen.
How does Corner Point Calculator handle the non-negativity constraints?
Corner Point Calculator automatically includes the non-negativity constraints x >= 0 and y >= 0 by treating the x-axis and y-axis as additional constraint boundaries. When finding intersection points, the calculator pairs each user-defined constraint with the axis boundaries as well as with other constraints. Any intersection point with a negative x or y coordinate is automatically discarded as infeasible. This ensures that all reported corner points lie in the first quadrant. The origin (0, 0) is always checked as a potential corner point since it is the intersection of the two axis boundaries.
What is the difference between a corner point and a basic feasible solution?
In theory, corner points and basic feasible solutions are equivalent concepts viewed from different perspectives. A corner point is a geometric concept describing a vertex of the feasible region polygon. A basic feasible solution is an algebraic concept from the simplex method, where exactly as many constraints are active (satisfied as equalities) as there are variables. Every corner point corresponds to a basic feasible solution and vice versa. The simplex method moves from one basic feasible solution to an adjacent one, which geometrically corresponds to moving along an edge from one corner point to the next. This equivalence is fundamental to why the simplex method works.
How many corner points can a feasible region have?
For a two-dimensional linear programming problem with n constraints (including non-negativity), the maximum number of corner points is n-choose-2, which equals n(n-1)/2. However, many intersection points will be infeasible, so the actual number of corner points is usually much smaller. For three user-defined constraints plus two non-negativity constraints (5 total), there are at most 10 possible intersection points but typically 4-6 feasible corner points. In higher dimensions, the number of potential vertices grows combinatorially, which is one reason the simplex method is preferred over exhaustive enumeration for large problems.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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