What is the simplex method?

An algorithm for linear programming that moves along vertices of the feasible region, improving the objective at each step until finding the optimal solution.

What is a pivot operation?

Exchanging a basic variable with a non-basic variable to move to an adjacent vertex. It involves row operations to make the pivot element 1 and eliminate the column.

What are slack variables?

Variables added to convert ≤ inequalities to equations. For ax ≤ b, add slack s ≥ 0: ax + s = b. Slack represents unused resources.

How do you know when simplex is optimal?

When all coefficients in the objective row are non-negative (for maximization). No improving direction exists.

Simplex Method Calculator

Problem Setup

Number of Variables

Number of Constraints

Optimization Type

Objective Function: Maximize Z

Coefficients (comma-separated): c1, c2

Z = 3x1 + 5x2

Constraints

Enter coefficients and RHS for each constraint (comma-separated)

C1:

C2:

Understanding the Simplex Method

The Simplex Method is an algorithm for solving linear programming problems. It systematically examines vertices of the feasible region to find the optimal solution.

Standard Form

Problems are converted to standard form with:

All constraints as equations (using slack/surplus variables)
All variables non-negative
RHS values non-negative

Algorithm Steps

Convert to standard form and create initial tableau
Check optimality (no negative coefficients in objective row)
Select entering variable (most negative coefficient)
Select leaving variable (minimum ratio test)
Perform pivot operation
Repeat until optimal or detect unbounded/infeasible

Special Cases

Optimal Solution

All objective row coefficients are non-negative

Unbounded

No valid pivot row exists (all ratios negative/undefined)

Infeasible

Artificial variables remain in the basis with positive values

About Simplex Method Calculator - Linear Programming Solver

Solve linear programming problems using the simplex algorithm. View tableaus, pivot operations, and find optimal solutions step-by-step.

Our **Simplex Method Calculator** solves linear programming optimization problems using the classic simplex algorithm. Enter your objective function and constraints, then watch the algorithm find the optimal solution through pivot operations. See also our Linear Programming Calculator.

The simplex method systematically examines vertices of the feasible region to find the optimal solution. Our calculator shows each tableau, identifies pivot elements, and explains every step of the iteration process.

Handle maximization and minimization problems with equality, ≤, and ≥ constraints. The calculator uses the Big-M method for artificial variables when needed. Essential for operations research and optimization studies.

Key Features

Step-by-step tableaus

Pivot identification

Big-M method support

Slack variables

Optimal solution display

Unbounded/infeasible detection

Why Use This Tool?

Learn simplex algorithm

Operations research

Optimization problems

Resource allocation

Production planning

Common Use Cases

Manufacturing: Maximize profit with resource constraints.

Transportation: Minimize shipping costs.

Scheduling: Optimal workforce allocation.

Portfolio: Investment optimization.

Related Tools

Linear Programming Calculator

Graphical LP method.

LU Factorization Calculator

Matrix decomposition.

How to Use

Enter number of variables and constraints

Define objective function coefficients

Enter constraint coefficients and RHS

Select constraint types (≤, ≥, =)

Click Solve