Simplex Method Calculator

Q: What are slack and surplus variables in the simplex method?

Slack variables are added to less than or equal to (≤) constraints to convert them into equalities. They represent unused resources. Surplus variables are subtracted from greater than or equal to (≥) constraints and represent the amount by which a requirement is exceeded.

Q: Can the simplex calculator solve minimization problems?

Yes! The simplex calculator handles both maximization and minimization problems. For minimization, the calculator automatically converts the problem by negating the objective function coefficients and adjusts the final answer accordingly.

Calculator

Examples

Help

Decision Variables

Constraints

Optimization Type

Note: For minimization problems or constraints with ≥ or =, the calculator will automatically convert to standard form. All variables are assumed to be non-negative.

Solution & Steps

No solution yet. Click "Build Input" to set up your problem, then "Solve" to find the solution.

Example Problems

Try these example problems to get started with the simplex method calculator:

Maximization Problem

Maximize Z = 3x₁ + 2x₂ subject to:

x₁ + x₂ ≤ 4

2x₁ + x₂ ≤ 5

x₁, x₂ ≥ 0

Minimization Problem

Minimize Z = 4x₁ + 3x₂ subject to:

2x₁ + x₂ ≥ 10

x₁ + 2x₂ ≥ 8

x₁, x₂ ≥ 0

Mixed Constraints

Maximize Z = 2x₁ + 3x₂ subject to:

x₁ + x₂ ≤ 6

2x₁ + x₂ ≥ 4

x₁ + 2x₂ = 6

x₁, x₂ ≥ 0

How to Use the Calculator

Setting Up the Problem

1. Enter the number of decision variables and constraints

2. Select whether to maximize or minimize the objective function

3. Click "Build Input" to create the input table

Entering Coefficients

1. Fill in the objective function coefficients

2. Enter constraint coefficients and select constraint type (≤, =, ≥)

3. Enter the right-hand side (RHS) values for constraints

Solving and Results

1. Click "Solve" to run the simplex method

2. View the step-by-step solution process

3. Check the optimal solution and objective value

Tips for Better Results

Numerical Stability

• Use reasonable coefficient values (avoid extremely large or small numbers)

• If possible, scale your problem to have coefficients of similar magnitude

Problem Formulation

• Ensure all variables are non-negative

• For minimization problems, the calculator automatically converts to maximization

• All RHS values should be non-negative

Troubleshooting

• If the solution seems incorrect, check your constraint types

• For unbounded problems, review your constraints

• For infeasible problems, check for conflicting constraints

What is the Simplex Method?

The simplex method is a widely-used algorithm for solving linear programming (LP) problems. Developed by George Dantzig in 1947, it remains one of the most efficient methods for finding optimal solutions to optimization problems with linear constraints. Our free simplex method calculator implements this algorithm to help students, educators, and professionals solve LP problems with step-by-step solutions.

Linear programming involves maximizing or minimizing a linear objective function subject to a set of linear inequality or equality constraints. The simplex algorithm works by moving along the edges of the feasible region (a convex polytope) to find the vertex that optimizes the objective function.

How Does the Simplex Algorithm Work?

The simplex method follows a systematic approach to find the optimal solution:

Convert to Standard Form: Transform all constraints into equations by adding slack variables (for ≤ constraints) or subtracting surplus variables (for ≥ constraints).
Set Up Initial Tableau: Create the initial simplex tableau with the objective function and constraint coefficients.
Identify Pivot Column: Find the most negative coefficient in the objective row (for maximization problems).
Identify Pivot Row: Use the minimum ratio test to determine which row to pivot on.
Perform Pivot Operation: Use row operations to make the pivot element 1 and all other elements in the pivot column 0.
Repeat: Continue iterations until no negative coefficients remain in the objective row.

Simplex Calculator Features

Step-by-Step Solutions

View each iteration of the simplex tableau with highlighted pivot elements to understand the solution process.

Maximize & Minimize

Solve both maximization and minimization linear programming problems with automatic conversion to standard form.

Mixed Constraints

Handle all constraint types including less-than-or-equal (≤), greater-than-or-equal (≥), and equality (=) constraints.

Free & No Sign-up

Use our simplex solver online without registration. All calculations run in your browser for privacy.

Applications of Linear Programming

The simplex method and linear programming have numerous real-world applications across various industries:

Business & Operations Research: Production planning, resource allocation, supply chain optimization, and profit maximization.
Finance: Portfolio optimization, capital budgeting, and risk management.
Manufacturing: Production scheduling, inventory management, and quality control.
Transportation: Route optimization, logistics planning, and fleet management.
Agriculture: Crop planning, feed mix optimization, and land allocation.
Healthcare: Staff scheduling, resource allocation, and diet planning.

Frequently Asked Questions

What is the difference between the simplex method and graphical method?

The graphical method is limited to problems with two decision variables, where you can visualize the feasible region on a 2D graph. The simplex method can handle any number of variables and constraints, making it suitable for real-world problems with many variables.

What does "unbounded solution" mean?

An unbounded solution occurs when the objective function can be increased (for maximization) or decreased (for minimization) indefinitely without violating any constraints. This usually indicates missing or incorrect constraints in the problem formulation.

What are slack and surplus variables?

Slack variables are added to "less than or equal to" (≤) constraints to convert them into equalities. They represent unused resources. Surplus variables are subtracted from "greater than or equal to" (≥) constraints and represent the amount by which a requirement is exceeded.

Can this calculator solve minimization problems?

Yes! Our simplex calculator handles both maximization and minimization problems. For minimization, the calculator automatically converts the problem by negating the objective function coefficients and adjusts the final answer accordingly.

What is the Big M method?

The Big M method is a technique for handling equality constraints and greater-than-or-equal-to constraints by adding artificial variables with very large penalty coefficients (M) in the objective function. This ensures artificial variables are driven to zero in the optimal solution.

How do I interpret the final tableau?

In the final tableau, basic variables have a value of 1 in exactly one row and 0 in all other rows of their column. The values of basic variables are found in the RHS column of the rows where they equal 1. Non-basic variables have a value of 0. The optimal objective value is in the bottom-right cell.

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