Can the simplex calculator solve minimization problems?

Yes! All seven methods (Simplex, Big-M, Dual Simplex, Two-Phase, Revised Simplex, Interior Point, and Sensitivity Analysis) handle both maximization and minimization problems. The calculator automatically converts the problem internally and adjusts the final answer accordingly.

What is the Dual Simplex Method?

The Dual Simplex Method works on the dual of the original LP problem. It starts with a dual-feasible (but primal-infeasible) tableau and iterates to restore primal feasibility. It is especially efficient for problems with ≥ constraints and minimization objectives.

What is the Two-Phase Simplex Method?

The Two-Phase Method finds an initial basic feasible solution in Phase I by minimizing a sum of artificial variables, then optimizes the original objective in Phase II. It avoids the arbitrary large M penalty of the Big-M method.

What is the Interior Point Method?

The Interior Point (Affine-Scaling) Method traverses the interior of the feasible region instead of moving along edges like simplex. Starting from a strictly feasible point, it uses diagonal scaling and projection to converge to the optimum in polynomial time, making it efficient for very large LP problems.

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: What is the Revised Simplex Method?

The Revised Simplex Method performs the same steps as the standard simplex but explicitly maintains the inverse of the basis matrix (B⁻¹). This is computationally more efficient for large, sparse problems because it avoids updating the entire tableau.

Q: Which LP method should I use?

Use the standard Simplex (Big-M) for general problems. Dual Simplex is best for ≥-heavy or minimization problems. Two-Phase avoids Big-M numerical issues. Revised Simplex is efficient for large sparse problems. Interior Point is ideal for very large problems where polynomial-time guarantees matter.

Simplex

Big-M Method

Dual Simplex

Two-Phase

Revised Simplex

Interior Point

Sensitivity

Examples

Help

Simplex Method

The standard simplex algorithm for linear programming. Enter your LP problem with ≤ constraints only — the solver adds slack variables and iterates through tableau pivots to find the optimal vertex. For problems with ≥ or = constraints, use the Big-M Method or Two-Phase tab.

Decision Variables

Constraints

Optimization Type

Note: Only ≤ constraints are supported in this tab. For ≥ or = constraints, switch to the Big-M Method or Two-Phase tab. All variables are assumed to be non-negative.

Simplex — Solution & Steps

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

Simplex Examples (≤ only)

Maximization (2 var)

Maximize Z = 5x₁ + 4x₂ subject to:

6x₁ + 4x₂ ≤ 24

x₁ + 2x₂ ≤ 6

x₁, x₂ ≥ 0

Minimization (2 var)

Minimize Z = 2x₁ + 5x₂ subject to:

x₁ + x₂ ≤ 6

3x₁ + x₂ ≤ 12

x₁, x₂ ≥ 0

Three Variables

Maximize Z = 3x₁ + 2x₂ + 5x₃ subject to:

x₁ + 2x₂ + x₃ ≤ 430

3x₁ + 2x₃ ≤ 460

x₁ + 4x₂ ≤ 420

x₁, x₂, x₃ ≥ 0

Big-M Method

The Big-M method extends the simplex algorithm to handle ≥ and = constraints by introducing artificial variables with a large penalty coefficient M in the objective function. This forces artificial variables out of the basis, yielding the true optimal solution. Use this tab when your problem has mixed constraint types.

Decision Variables

Constraints

Optimization Type

Note: Supports all constraint types (≤, ≥, =). The solver automatically adds artificial variables with a Big-M penalty for ≥ and = constraints. All variables are assumed to be non-negative.

Big-M Method — Solution & Steps

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

Big-M Method Examples

Minimization with ≥

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

Maximization (≤ only)

Maximize Z = 3x₁ + 2x₂ subject to:

x₁ + x₂ ≤ 4

2x₁ + x₂ ≤ 5

x₁, x₂ ≥ 0

Dual Simplex Method

The dual simplex method maintains dual feasibility (optimality conditions) while working toward primal feasibility. It is useful when the initial solution is optimal but infeasible — e.g., after adding a new constraint to an already‑optimal tableau.

Decision Variables

Constraints

Optimization Type

Note: All constraints are converted to ≥ form internally (multiplying ≤ rows by −1). The algorithm requires the initial objective row to be dual‑feasible (all reduced costs ≥ 0 for maximization). Negative RHS values indicate primal infeasibility.

Dual Simplex — Solution & Steps

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

Dual Simplex Examples

Minimization with ≥ Constraints

Minimize Z = 6x₁ + 8x₂ subject to:

3x₁ + x₂ ≥ 6

x₁ + 2x₂ ≥ 4

x₁, x₂ ≥ 0

Maximization with Mixed Constraints

Maximize Z = 2x₁ + x₂ subject to:

3x₁ + x₂ ≤ 6

x₁ + x₂ ≥ 2

x₁, x₂ ≥ 0

Three Variable Minimization

Minimize Z = 3x₁ + 2x₂ + x₃ subject to:

x₁ + x₂ + x₃ ≥ 6

2x₁ + x₂ ≥ 8

x₁, x₂, x₃ ≥ 0

Two-Phase Simplex Method

Phase I minimises the sum of artificial variables to find a basic feasible solution (BFS). If the minimum is zero a BFS exists and Phase II optimises the original objective using that BFS. This avoids the Big‑M penalty and is numerically more stable.

Decision Variables

Constraints

Optimization Type

Note: The two-phase method handles ≥ and = constraints by introducing artificial variables. Phase I drives artificials to zero; Phase II optimises the real objective. All variables are assumed non‑negative.

Two-Phase — Solution & Steps

No solution yet. Click "Build Input" to set up your problem, then "Solve" to run the two-phase method.

Two-Phase Examples

Minimization with ≥ Constraints

Minimize Z = 4x₁ + 3x₂ subject to:

2x₁ + x₂ ≥ 10

x₁ + 2x₂ ≥ 8

x₁, x₂ ≥ 0

Mixed Constraints (= and ≥)

Maximize Z = 2x₁ + 3x₂ subject to:

x₁ + x₂ = 6

2x₁ + x₂ ≥ 4

x₁, x₂ ≥ 0

Infeasible Problem

Maximize Z = x₁ + x₂ subject to:

x₁ + x₂ ≤ 2

x₁ + x₂ ≥ 5

x₁, x₂ ≥ 0

Revised Simplex Method

The revised simplex method works with the basis inverse B⁻¹ instead of the full tableau. At each iteration it computes only the needed column and performs the pivot, making it more memory-efficient for large problems. This calculator shows B⁻¹, the entering/leaving columns, and updated values at every step.

Decision Variables

Constraints

Optimization Type

Note: Only ≤ constraints are supported directly; ≥ and = constraints use the Big-M method internally. All variables are assumed non-negative.

Revised Simplex — Solution & Steps

No solution yet. Click "Build Input" to set up your problem, then "Solve" to run the revised simplex method.

Revised Simplex Examples

Maximization (2 var)

Maximize Z = 5x₁ + 4x₂ subject to:

6x₁ + 4x₂ ≤ 24

x₁ + 2x₂ ≤ 6

x₁, x₂ ≥ 0

Minimization (2 var)

Minimize Z = 2x₁ + 3x₂ subject to:

x₁ + 2x₂ ≤ 12

2x₁ + x₂ ≤ 12

x₁ + x₂ ≤ 7

x₁, x₂ ≥ 0

Three Variables

Maximize Z = 3x₁ + 2x₂ + 5x₃ subject to:

x₁ + 2x₂ + x₃ ≤ 430

3x₁ + 2x₃ ≤ 460

x₁ + 4x₂ ≤ 420

x₁, x₂, x₃ ≥ 0

Interior Point Method

The primal affine-scaling interior point method traverses the interior of the feasible region rather than moving along edges like simplex. Starting from a strictly feasible point, it iteratively scales and projects the steepest-ascent direction, converging to the optimum in polynomial time. Each iteration shows the current point, projected direction, and step size.

Decision Variables

Constraints

Optimization Type

Step Size γ

Tolerance

Note: Only ≤ constraints with non-negative RHS are accepted. The algorithm starts from an interior point where all slacks are positive. γ controls step aggressiveness (0.01–0.99); smaller values are safer but slower.

Interior Point — Solution & Steps

No solution yet. Click "Build Input" to set up your problem, then "Solve" to run the interior point method.

Interior Point Examples

Maximization (2 var)

Maximize Z = 5x₁ + 4x₂ subject to:

6x₁ + 4x₂ ≤ 24

x₁ + 2x₂ ≤ 6

x₁, x₂ ≥ 0

Minimization (2 var)

Minimize Z = 2x₁ + 3x₂ subject to:

x₁ + 2x₂ ≤ 14

3x₁ + 2x₂ ≤ 18

x₁, x₂ ≥ 0

Three Variables

Maximize Z = 3x₁ + 2x₂ + 5x₃ subject to:

x₁ + 2x₂ + x₃ ≤ 430

3x₁ + 2x₃ ≤ 460

x₁ + 4x₂ ≤ 420

x₁, x₂, x₃ ≥ 0

Sensitivity Analysis

Performs post-optimality sensitivity analysis on a solved LP. After finding the optimal solution, this tab reports shadow prices (dual values), reduced costs, allowable objective-coefficient ranges, and allowable RHS ranges — showing how much each parameter can change before the current basis becomes non-optimal or infeasible.

Decision Variables

Constraints

Optimization Type

Enter your LP problem above and click "Solve & Analyze" to get the optimal solution with full sensitivity report.

Try These Examples

Production Mix

Maximize Z = 5x₁ + 4x₂ subject to:

6x₁ + 4x₂ ≤ 24

x₁ + 2x₂ ≤ 6

x₁, x₂ ≥ 0

Resource Allocation

Maximize Z = 3x₁ + 5x₂ subject to:

x₁ ≤ 4

2x₂ ≤ 12

3x₁ + 5x₂ ≤ 25

x₁, x₂ ≥ 0

Three Variables

Maximize Z = 2x₁ + 3x₂ + x₃ subject to:

x₁ + x₂ + x₃ ≤ 40

2x₁ + x₂ ≤ 60

x₂ + x₃ ≤ 30

x₁, x₂, x₃ ≥ 0

Example Problems

Try these example problems to get started. Clicking an example loads it into the Big-M Method tab (supports all constraint types):

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

Choosing a Method

Switch between the Simplex (pure ≤), Big-M Method (mixed constraints), Dual Simplex, Two-Phase, Revised Simplex, Interior Point, and Sensitivity Analysis tabs. Each method tab works independently with its own inputs, examples, and output. The Sensitivity tab provides post-optimality analysis with shadow prices, reduced costs, and allowable ranges.

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

Note: Interior Point supports ≤ constraints only; other methods support ≤, ≥, and =.

Solving and Results

1. Click "Solve" to run the selected method

2. View the step-by-step solution (tableaux, basis updates, or iteration points)

3. Check the optimal solution and objective value in the solution summary card

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

• For Interior Point, use the γ (gamma) slider: smaller values (0.1–0.3) are safer; larger values (0.5–0.9) converge faster but may overshoot

Problem Formulation

• Ensure all variables are non-negative

• All RHS values should be non-negative

• Use Dual Simplex for problems dominated by ≥ constraints

• Use Two-Phase when you suspect the problem may be infeasible — Phase I will detect it cleanly

• Use Interior Point for very large problems; it accepts ≤ constraints only

Troubleshooting

• If the solution seems incorrect, check your constraint types and RHS signs

• For unbounded problems, review your constraints — try a different method to confirm

• For infeasible problems, check for conflicting constraints or use Two-Phase to verify

• If Interior Point doesn't converge, try reducing γ or increasing the tolerance

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 — along with five additional advanced methods — to help students, educators, and professionals solve LP problems with detailed, 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. For different problem structures, alternative algorithms such as the Dual Simplex, Two-Phase, Revised Simplex, and Interior Point methods can be more efficient or numerically stable.

Seven Methods to Solve LP Problems

This calculator provides seven distinct algorithms, each suited to different types of linear programming problems:

1. Simplex Method (Standard)

The classic simplex algorithm for problems with only ≤ (less-than-or-equal) constraints. It adds slack variables to convert inequalities to equalities and iterates through basic feasible solutions to find the optimum. Best starting point for learning LP and for standard-form problems.

2. Big-M Method

An extension of the simplex that handles mixed constraint types (≤, ≥, =) by introducing artificial variables with a very large penalty coefficient M in the objective. Supports graphical 2D visualization and full tableau display at every iteration. Best for general-purpose LP problems with mixed constraints.

3. Dual Simplex Method

Works on the dual of the original problem. It starts with a dual-feasible (but primal-infeasible) solution and iterates to restore primal feasibility. Particularly efficient for problems dominated by ≥ constraints and for minimization objectives where the dual is naturally feasible.

4. Two-Phase Simplex Method

Separates the problem into two phases: Phase I finds an initial basic feasible solution by minimizing the sum of artificial variables, and Phase II optimizes the original objective. This avoids the numerical issues caused by the arbitrary large M penalty in the Big-M method.

5. Revised Simplex Method

Performs the same logical steps as the standard simplex but explicitly maintains the inverse of the basis matrix (B⁻¹). This is computationally more efficient for large, sparse problems because only the basis inverse and reduced costs are updated — the full tableau is never stored.

6. Interior Point Method (Affine-Scaling)

Unlike simplex-based methods that traverse the boundary of the feasible region, the interior point method moves through the interior. Starting from a strictly feasible point, it uses diagonal scaling and gradient projection to converge to the optimum. It runs in polynomial time and is well-suited for very large LP problems.

7. Sensitivity Analysis (Post-Optimality)

After finding the optimal solution, sensitivity analysis examines how changes in the problem data affect the solution. It reports shadow prices (dual values), reduced costs, and allowable ranges for both objective coefficients and right-hand side values — essential for managerial decision-making and understanding the robustness of an optimal solution.

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.

The Big-M, Dual Simplex, Two-Phase, and Revised Simplex methods follow variations of these steps (with Big-M adding artificial variables for mixed constraints), while the Interior Point method takes a fundamentally different geometric approach — see each tab for detailed, step-by-step walkthroughs.

LP Calculator Features

Seven Solver Algorithms

Choose from Simplex, Big-M Method, Dual Simplex, Two-Phase, Revised Simplex, Interior Point, or Sensitivity Analysis — each with full step-by-step output.

Step-by-Step Solutions

View every iteration with highlighted pivots, basis tracking, reduced costs, and convergence details to fully understand each method.

Maximize & Minimize

Solve both maximization and minimization LP problems. All seven methods handle both objective directions automatically.

Mixed Constraints

Handle all constraint types: less-than-or-equal (≤), greater-than-or-equal (≥), and equality (=) across the Simplex, Dual, and Two-Phase methods.

Pre-Built Examples

Each method includes ready-to-solve example problems — click to load, then hit Solve to see the full solution process instantly.

Free & No Sign-up

Use all seven solvers online without registration. All calculations run entirely in your browser for speed and privacy.

Which LP Method Should I Use?

Method	Best For	Constraint Types	Key Advantage
Simplex (Standard)	Standard-form LP (≤ only)	≤	Pure simplex; clean tableau; ideal for learning
Big-M Method	General LP problems	≤, ≥, =	Handles all constraint types; 2D graph
Dual Simplex	Minimization / ≥-heavy	≤, ≥, =	Starts dual-feasible; efficient for min problems
Two-Phase	Problems needing feasibility check	≤, ≥, =	No Big-M; clean separation of feasibility & optimality
Revised Simplex	Large, sparse problems	≤, ≥, =	Only updates B⁻¹; memory-efficient
Interior Point	Very large LPs	≤ only	Polynomial time; traverses interior
Sensitivity Analysis	Post-optimality analysis	≤, ≥, =	Shadow prices, reduced costs, allowable ranges

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.
Large-Scale Engineering: Network flow, structural optimization, and power grid scheduling — where interior point methods excel due to polynomial-time convergence on massive problem instances.

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! All seven methods handle both maximization and minimization problems. The calculator automatically converts the problem internally and adjusts the final answer.

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. The Two-Phase method offers an alternative that avoids the arbitrary M value.

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.

What is the difference between Standard, Dual, and Two-Phase Simplex?

The standard simplex (Big-M) handles all constraint types in one pass using artificial penalties. The Dual Simplex starts dual-feasible and restores primal feasibility — great for ≥-heavy minimization. The Two-Phase method splits the work: Phase I finds feasibility, Phase II optimizes — avoiding Big-M numerical issues.

Why use the Revised Simplex over the standard Simplex?

The Revised Simplex is algebraically identical but computationally more efficient — it maintains only the basis inverse (B⁻¹) rather than the full tableau. For large, sparse problems this can dramatically reduce memory usage and computation time.

When should I use the Interior Point method?

The Interior Point (Affine-Scaling) method is best for very large LP problems. Unlike simplex-based methods that hop between vertices, it moves through the interior of the feasible region and converges in polynomial time. It currently supports ≤ constraints with non-negative RHS.

Which method should I pick?

Use Simplex (Big-M) for general problems with mixed constraints. Use Dual Simplex for min problems with ≥ constraints. Use Two-Phase to avoid Big-M issues. Use Revised Simplex for large sparse problems. Use Interior Point for very large LPs where polynomial-time matters. Use Sensitivity Analysis after solving to understand how changes in coefficients or RHS values affect the optimal solution.

What is sensitivity analysis in linear programming?

Sensitivity analysis (also called post-optimality analysis) examines how the optimal solution changes when problem parameters are modified. It provides shadow prices (how much the objective improves per unit increase in a constraint's RHS), reduced costs (how much a non-basic variable's coefficient must improve before it enters the solution), and allowable ranges for both objective coefficients and RHS values — the ranges within which the current optimal basis remains valid.

Related Calculators & Tools

Explore more mathematical tools on FujiBit:

Linear Programming Calculator Matrix Calculator Equation Solver Statistics Calculator Numerical Methods Graph Calculator All Calculators FujiBit Home

Simplex Method

Simplex — Solution & Steps

Graphical Visualization (2D)

Simplex Examples (≤ only)

Maximization (2 var)

Minimization (2 var)

Three Variables

What is the Simplex Method?

Seven Methods to Solve LP Problems

1. Simplex Method (Standard)

2. Big-M Method

3. Dual Simplex Method

4. Two-Phase Simplex Method

5. Revised Simplex Method

6. Interior Point Method (Affine-Scaling)

7. Sensitivity Analysis (Post-Optimality)

How Does the Simplex Algorithm Work?

LP Calculator Features

Seven Solver Algorithms

Step-by-Step Solutions

Maximize & Minimize

Mixed Constraints

Pre-Built Examples

Free & No Sign-up

Which LP Method Should I Use?

Applications of Linear Programming

Frequently Asked Questions

Related Calculators & Tools