Corner Point Calculator
Solve linear programming problems using the corner point method
Linear Programming Problem Setup
Objective Function
Maximize P = 30x + 40y
Constraints
Constraint 1: a1x + b1y ≤ c1
2x + 3y ≤ 18
Constraint 2: a2x + b2y ≤ c2
x + y ≤ 9
Constraint 3: a3x + b3y ≤ c3
x + 2y ≤ 16
Quick Examples
Solution
Optimal Solution
Corner Points Analysis
| Corner Point | x-coordinate | y-coordinate | Objective Value | Status |
|---|---|---|---|---|
| (9, 0) | 9 | 0 | 270 | Optimal |
| (-12, 14) | -12 | 14 | 200 |
Linear Programming Guide
Decision Variables
Variables to optimize (usually x and y in 2D problems).
Objective Function
Function to maximize or minimize (profit, cost, etc.).
Constraints
Limitations that define the feasible region.
Corner Points
Intersection points where optimal solution occurs.
Corner Point Method
Convert constraints to equalities
Find intersection points of constraint lines
Test which points satisfy all constraints
Evaluate objective function at corner points
Select optimal solution
Key Formulas
P = p₁x + p₂y
Objective function
aᵢx + bᵢy ≤ cᵢ
Constraint inequalities
x = (c₁b₂ - c₂b₁)/(a₁b₂ - a₂b₁)
Intersection point (Cramer's rule)
Understanding Linear Programming & Corner Points
What is Linear Programming?
Linear programming is a mathematical optimization technique used to find the best solution from a set of linear constraints. It's widely used in business, economics, and engineering to maximize profits, minimize costs, or optimize resource allocation.
Key Components
- •Decision Variables: Variables we want to optimize (x, y)
- •Objective Function: Function to maximize or minimize
- •Constraints: Linear inequalities that limit our choices
Corner Point Method
The corner point method is based on the fundamental theorem that if a linear programming problem has an optimal solution, it will occur at a corner point (vertex) of the feasible region.
Feasible Region
The set of all points that satisfy all constraints simultaneously.
Corner Points
Intersection points of constraint boundaries where optimal solutions are found.
Example: Production Optimization
Problem Setup
Objective: Maximize profit P = 30x + 40y
Constraints:
2x + 3y ≤ 18 (Material constraint)
x + y ≤ 9 (Labor constraint)
x + 2y ≤ 16 (Machine constraint)
x ≥ 0, y ≥ 0 (Non-negativity)
Solution
Corner Points:
(0, 0): P = 0
(0, 6): P = 240
(6, 2): P = 260
(9, 0): P = 270 ← Optimal
Optimal Solution: x = 9, y = 0, P = 270
Real-World Applications
Manufacturing
Optimize production mix to maximize profit while respecting resource constraints.
Transportation
Minimize shipping costs while meeting demand and supply constraints.
Diet Planning
Minimize cost while meeting nutritional requirements and taste preferences.