Linear programming decides how to make the most profit or the least cost under limits on time, materials or budget. With two variables, the corner point method finds the answer by checking the vertices of the feasible region.
How to use the Corner Point Calculator
- Objective coefficient pₓ
- Objective coefficient p_y
- Maximise or minimise?
- Number of constraints
- Include x ≥ 0 and y ≥ 0?
- Constraint 1: a
- b
- sign
- c
- Constraint 2: a
- b
- sign
- c
- Constraint 3: a
- b
- sign
- c
- Constraint 4: a
- b
- sign
- c
- Constraint 5: a
- b
- sign
- c
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How it works
Each pair of boundary lines is intersected; intersections satisfying every constraint are corner points. The objective is evaluated at each, and the best is reported.
Formula
Optimise P = pₓx + p_y·y subject to aᵢx + bᵢy (≤, ≥, =) cᵢ
The optimum of a bounded feasible region is at a corner point
Worked example
Inputs — Objective coefficient pₓ: 3; Objective coefficient p_y: 5; Maximise or minimise?: Maximise; Number of constraints: 3; Include x ≥ 0 and y ≥ 0?: Yes; Constraint 1: a: 1; b: 2; sign: ≤; c: 14; Constraint 2: a: 3; b: -1; sign: ≥; c: 0; Constraint 3: a: 1; b: -1; sign: ≤; c: 2.
Results — Optimal solution = Maximum P = 38 at (6, 4); Corner points = x y P = 3x + 5y
2 6 36
6 4 38
0 0 0
2 0 6; Number of corner points = 4; Note = If the feasible region is unbounded, the optimum may not exist even though corner values are listed..
Key terms
- Feasible region
- All points satisfying every constraint.
- Objective function
- The quantity to maximise or minimise.
Frequently asked questions
Why is the optimum at a corner?
A linear objective increases steadily in one direction, so over a convex polygon it peaks at a vertex.
What if the feasible region is unbounded?
A maximum may not exist; check the direction of the objective.
What if two corners give the same value?
Every point on the edge between them is optimal.
How many variables can this method handle?
Two; larger problems use the simplex method.