What Is the Simplex Method? A Complete Beginner's Guide

The simplex method is the most widely used algorithm for solving linear programming problems. It was developed by mathematician George Dantzig in 1947 and remains the backbone of operations research today.

Linear Programming in One Minute

A linear programming (LP) problem asks you to optimize a linear objective function - maximize profit or minimize cost - subject to a set of linear constraints. The set of points that satisfy every constraint is called the feasible region, and the optimal solution always lies at one of its corners (vertices).

How the Simplex Method Works

Instead of checking every corner, the simplex method moves intelligently from one vertex to a neighbouring one, improving the objective at each step until no further improvement is possible. Each move is called an iteration and is recorded in a simplex tableau.

The core steps are:

1. Convert the problem to standard form by adding slack, surplus, or artificial variables.
2. Build the initial tableau and compute the Cj − Zj row.
3. Choose the entering variable (pivot column) and leaving variable (pivot row) using the minimum-ratio test.
4. Pivot using elementary row operations.
5. Repeat until the tableau is optimal.

You can watch all of these steps happen automatically in our simplex method calculator with steps.

Maximization and Minimization

The method handles both directions. See the maximization calculator and the minimization calculator for worked examples of each.

Beyond Two Variables

The graphical method only works for two variables. The simplex method has no such limit - try the 3-variable or 4-variable calculators to see it in action. For any general problem, the linear programming calculator solves it online with full tableau output.