Simplex Method Calculator

What Is the Simplex Method?

The simplex method is the most widely used algorithm for solving linear programming (LP) problems. Developed by George Dantzig in 1947, it finds the optimal value of a linear objective function subject to a set of linear constraints. A simplex method calculator automates this process: you enter your objective function and constraints, and the calculator performs every pivot operation until it reaches the optimal solution, showing each tableau along the way.

Linear programming models appear everywhere - maximizing profit, minimizing cost, allocating resources, planning production, and solving transportation and diet problems. Whenever the relationships are linear, the simplex method efficiently moves from one corner point of the feasible region to the next, improving the objective at each step until no further improvement is possible.

Free Online Simplex Solver

This page is a free online simplex solver - nothing to install and no sign-up needed. Enter your objective function and constraints, choose maximize or minimize, and the solver returns the optimal value with every tableau iteration shown. It works as a simplex tableau solver, a linear programming solver, and a maximization or minimization solver in one tool.

Unlike many simplex solvers online, every step is free here: the full pivot process, the Zj and Cj-Zj rows, and the final optimal solution are all displayed, and results can be shown as exact fractions or as decimals.

Whether you call it a simplex calculator, a simplex solver or a simplex algorithm calculator, the tool works the same way: it builds the initial simplex tableau, applies the simplex algorithm one pivot at a time, and stops at the final optimal tableau. You can use it as an online simplex tableau calculator to check homework, verify an exam answer, or solve a real linear programming model in seconds.

Simplex Method Formula and Standard Form

Before applying the algorithm, a problem is written in standard form. For a maximization problem, every constraint is converted to an equation by adding a slack variable (for less-than constraints), subtracting a surplus variable (for greater-than constraints), and, where needed, adding an artificial variable. The objective is written as Maximize Z = c1x1 + c2x2 + ... + cnxn, subject to the constraint equations and all variables greater than or equal to zero.

These coefficients populate the first simplex tableau. The tableau calculator builds this table automatically, computing the Zj and Cj minus Zj rows for every iteration so you can follow the math exactly.

How to Solve the Simplex Method Step by Step

Take the example Maximize Z = 3x1 + 5x2 subject to x1 + 2x2 less than or equal to 14 and x1 + x2 less than or equal to 8. First, set up the initial tableau by adding slack variables to the two constraints. Second, compute Cj minus Zj and pick the most positive value as the entering variable (the pivot column). Third, apply the ratio test by dividing each right-hand-side value by the positive pivot-column entry and choose the smallest non-negative ratio to find the leaving variable (the pivot row). Fourth, pivot using elementary row operations. Finally, repeat until every Cj minus Zj value is less than or equal to zero.

For this example the optimal solution is x1 = 2, x2 = 6, with Z = 36. The step-by-step calculator on this page shows every iteration so you can reproduce it on paper.

Maximization vs Minimization

The simplex method handles both directions. In a maximization problem the algorithm increases Z until no positive Cj minus Zj remains. A minimization problem is solved either by converting it to an equivalent maximization (minimize Z is the same as maximize negative Z) or by selecting the most negative Cj minus Zj. Minimization problems often involve greater-than constraints, which require the Big M or two-phase technique.

Big M Method and Two-Phase Method

When a problem contains greater-than or equal constraints, artificial variables are introduced. The Big M method assigns these artificial variables a very large penalty so the algorithm drives them out of the basis. The two-phase simplex method achieves the same goal in two stages: Phase 1 minimizes the sum of artificial variables to find a feasible starting point, and Phase 2 optimizes the real objective. Both reach the same optimal solution.

Dual and Revised Simplex Methods

The dual simplex method starts from an optimal but infeasible tableau and restores feasibility, which is efficient when constraints are added to an already-solved problem. The revised simplex method stores only the inverse of the basis matrix instead of the full tableau, making it far more memory-efficient for large problems while producing identical results.

Simplex Method on a Graphing or TI-84 Calculator

You can run the simplex algorithm by hand on a graphing calculator using matrix row operations. On a TI-84, store the tableau as a matrix, then use rowSwap, multiply-row, and multiply-row-add from the MATRIX MATH menu to pivot. Casio scientific calculators have no built-in simplex function, but their Matrix mode can perform the same row operations manually. For instant results without manual pivoting, the online calculator on this page does every step for you.

Simplex Method vs Graphical Method

The graphical method works only for two-variable problems, where the feasible region can be drawn and the optimum read off a corner point. The simplex method has no such limit - it solves problems with any number of variables. Use the two-variable calculator for textbook examples, or the three-variable and four-variable calculators when a problem grows beyond what you can graph.

Linear Programming Problems (LPP)

A linear programming problem (LPP) combines a linear objective with linear constraints. The LPP simplex method calculator and the general linear programming calculator solve maximization and minimization LPPs online, showing the full tableau process so you can learn the method while you solve real problems.