Linear Programming Calculator

What Is Linear Programming?

Linear programming (LP) is a mathematical optimization technique used to find the best outcome (maximum profit or minimum cost) given a set of linear constraints. It is widely used in business, economics, engineering, and operations research.

Standard Form:
Maximize/Minimize: Z = ax + by (objective function)
Subject to: constraints (linear inequalities)
x ≥ 0, y ≥ 0 (non-negativity)

Graphical Method (2 Variables)

Graph each constraint as a line
Identify the feasible region (where all constraints are satisfied)
Find the corner points (vertices) of the feasible region
Evaluate the objective function at each corner point
The optimal solution is at the corner with the best value

Applications

Business: Production planning, resource allocation
Logistics: Transportation and routing optimization
Finance: Portfolio optimization
Agriculture: Crop planning, feed mixing

How to Use

Enter the coefficients of your objective function (Z = ax + by) and one constraint. The calculator evaluates the objective at key points.

Limitations

This calculator handles simple 2-variable problems. For larger problems with many variables and constraints, use the Simplex Method or specialized software.

Corner Point Theorem

The optimal solution to a linear programming problem always occurs at a corner point (vertex) of the feasible region, never in the interior.

Frequently Asked Questions

A mathematical method for finding the optimal solution (max/min) to a problem with linear constraints.

At a corner point (vertex) of the feasible region, according to the Corner Point Theorem.

The function you want to maximize or minimize, expressed as Z = ax + by in 2 variables.

The set of all points that satisfy all constraints simultaneously. The optimal solution must be in this region.

Yes, but you need the Simplex Method or software. The graphical method only works for 2 variables.

The requirement that x ≥ 0 and y ≥ 0. Most real-world quantities cannot be negative.