Hong Life CalculatorsUnderstanding the corner point calculator — concept and principle
This calculator solves two-variable linear programming problems by the corner-point method. Enter an objective z = c₁x + c₂y to maximize or minimize and a list of linear constraints; it finds every corner of the feasible region, evaluates z at each and picks the best. Use it for finite-math and business-math problems about production, scheduling or diets.
How it is calculated
- Each constraint a·x + b·y ≤ c (or ≥, =) has a boundary line a·x + b·y = c; x ≥ 0 and y ≥ 0 can be added.
- Intersect every pair of boundary lines.
- Keep only the intersection points that satisfy all constraints: these are the corners.
- Evaluate z = c₁x + c₂y at each corner and choose the largest (maximize) or smallest (minimize).
- If two corners tie, every point on the edge between them is optimal. If the region is unbounded in an improving direction, there is no optimum.
The principle
Linear constraints cut out a convex region, a polygon with straight edges. The objective z is a plane over that region, or, seen from above, a family of parallel lines c₁x + c₂y = constant. Sliding such a line in the improving direction, the last point it touches before leaving the region is always a corner (or a whole edge). That is why only the corners need to be checked, which turns an infinite search into a finite one.
Worked example
Default inputs: maximize 40x + 30y subject to x + y ≤ 12, 2x + y ≤ 16, x ≥ 0, y ≥ 0.
- Constraints used, including x, y ≥ 0: 4.
- Corners: (0, 0), (0, 12), (4, 8) where x + y = 12 meets 2x + y = 16, and (8, 0): 4 corners.
- z values: (0, 0) → 0; (0, 12) → 360; (4, 8) → 160 + 240 = 400; (8, 0) → 320.
- Optimal corner (4, 8) with optimal value 400.
| Objective: coefficient of y | Optimal value of the objective |
|---|---|
| 7.5 | 320 |
| 15 | 320 |
| 22.5 | 340 |
| 30 | 400 |
| 37.5 | 460 |
| 45 | 540 |
| 60 | 720 |
How to read the result
In the default story, x and y are hours at two jobs paying 40 and 30 per hour, with at most 12 hours of work and 16 hours of preparation. The best plan is 4 hours at the first job and 8 at the second. Constraints that pass through the optimal corner are the binding ones; relaxing them would improve the result, while the others have slack.
Common mistakes
- Forgetting x ≥ 0 and y ≥ 0 when the problem implies them.
- Keeping intersection points that break another constraint.
- Typing ≥ as ≤, which flips the region.
- Assuming an optimum exists for an unbounded region.
Frequently asked questions
What is the corner point theorem?
If a linear program has an optimum, it is reached at a corner of the feasible region.
What if two corners give the same best value?
Then every point on the edge between them is also optimal.
Can I minimize instead of maximize?
Yes, choose "Minimize"; the smallest corner value is then reported.
Sources: R. Sekhon & R. Bloom, Applied Finite Mathematics (LibreTexts), §3.1 Maximization Applications