Corner Point Calculator

Linear programming relies on the Fundamental Theorem of Linear Programming, which states that if an optimal solution exists for a bounded linear programming problem, it will occur at one or more of the corner points (vertices) of the feasible region. The process begins by translating real-world constraints into linear inequalities. When graphed, these inequalities intersect to create a multi-sided polygon known as the feasible region. The calculator sets up the equations of boundary lines, identifies their intersection points (the corners), tests whether each point satisfies all active constraints, and finally plugs these coordinate pairs into the objective function (P = Px*x + Py*y) to discover the absolute maximum or minimum value.

Imagine you manage a small manufacturing facility making two types of widgets. You want to maximize your profit function given by P = 40x + 30y, subject to three strict constraints: Constraint 1 (2x + y ≤ 100), Constraint 2 (x + 2y ≤ 80), and non-negativity rules (x ≥ 0, y ≥ 0). First, we find the intersection points of the boundary lines. The origin (0, 0) yields a profit of $0. The intersection of the axes gives points like (0, 40) where profit is $1,200, and (50, 0) where profit is $2,000. Next, we solve the system for the intersection of the two main constraint lines (2x + y = 100 and x + 2y = 80), which yields the vertex coordinate (40, 20). Evaluating the objective function at this corner point gives P = 40(40) + 30(20) = $1,600 + $600 = $2,200. Comparing all verified corner points, we find that producing 40 units of x and 20 units of y yields the maximum profit of $2,200.

Always double-check your inequality signs before initiating the calculation to avoid turning a feasible problem into an infeasible one. Ensure your objective function coefficients match the exact units of your variables (e.g., matching dollars per unit). Finally, remember to always verify that boundary intersections fall strictly within the non-negativity domain (x ≥ 0, y ≥ 0) unless your specific real-world model explicitly permits negative values.