Solve Simplex Method Calculator

Build simplex tableaus for clear linear programming decisions. Track pivots, constraints, objective value, and feasibility. Export results for homework, business planning, and optimization reviews.

Calculator Input

All decision variables are treated as nonnegative. Use Big M rows for greater-than and equality constraints.


Objective Function Coefficients

Enter coefficients for Z = c1x1 + c2x2 + ... + cnxn.


Constraint Matrix

Constraint 1

Constraint 2

Constraint 3

Example Data Table

Part x1 x2 Sign RHS
Objective Max Z 3 5
Constraint 1 2 3 <= 8
Constraint 2 2 1 <= 4
Constraint 3 0 1 <= 2
Expected Result x1 = 1 x2 = 2 Z 13

Formula Used

Objective: Maximize or minimize Z = c1x1 + c2x2 + ... + cnxn.

Constraints: ai1x1 + ai2x2 + ... + ainxn ≤, ≥, or = bi.

Reduced cost: Cj - Zj = Cj - Σ(Cbaij).

Ratio test: Ratio = RHS ÷ positive entering-column value. The smallest valid ratio leaves the basis.

Pivot update: Divide the pivot row by the pivot element. Then eliminate the entering column from all other rows.

Big M: Artificial variables receive a large negative objective cost in the transformed maximization model.

How to Use This Calculator

  1. Choose maximization or minimization.
  2. Set the number of variables and constraints.
  3. Click Build Matrix when you change the matrix size.
  4. Enter the objective coefficients and all constraint values.
  5. Choose each constraint sign and enter the right side value.
  6. Press Solve Simplex Method to view the answer and tableaus.
  7. Use CSV or PDF export for reports and class notes.

Why Use a Simplex Calculator

Linear programming helps choose the best result under limits. A simplex calculator turns that task into clear steps. It is useful for production plans, diet mixes, shipping choices, staffing, and classroom work. The method checks corner points without drawing every graph. It also shows why one corner beats another.

What the Calculator Solves

This tool handles maximization and minimization models. You can enter several variables, many constraints, and mixed signs. Less than, greater than, and equal constraints are supported. The solver adds slack, surplus, and artificial variables when needed. Big M handling makes difficult starting bases easier to inspect.

Reading the Results

The result area shows the final objective value first. It also lists each decision variable. Iteration tables show basis variables, coefficients, ratios, and reduced costs. The entering column improves the objective. The leaving row is chosen by the smallest valid ratio. After each pivot, a new table starts.

Why Tableaus Matter

A tableau keeps the calculation organized. Each row represents a converted constraint. The bottom row measures improvement. Positive reduced cost in a maximization model means more gain is possible. When no positive value remains, the current basic feasible solution is optimal. For minimization, the model is converted before solving.

Practical Uses

Business users can test product mixes with limited labor or materials. Students can verify homework and learn each pivot. Managers can compare capacity plans before making commitments. Operations teams can review bottlenecks. Export buttons help save a report for notes, records, or later review.

Tips for Better Input

Write every constraint in numeric form. Keep variables nonnegative unless your model is reformulated. Use consistent units across all rows. For example, do not mix minutes and hours in one column. Very large coefficients can create rounding issues. Increase precision when values are close. Check feasibility when artificial variables remain positive.

Common Modeling Mistakes

Do not place constants on the left side. Move them into the right side value. Check signs before solving for accuracy. A reversed inequality can change the answer. When a resource is limited, use less than or equal. When a requirement must be met, use greater than or equal.

FAQs

What is the simplex method?

The simplex method solves linear programming problems. It moves from one feasible corner point to another. Each move improves the objective until no better move remains.

Can this calculator solve minimization problems?

Yes. The calculator converts minimization into a related maximization form. It then solves the transformed tableau and reports the original objective value.

What does Big M mean?

Big M is a large penalty used for artificial variables. It helps the solver start with a basis when constraints use greater-than or equality signs.

Why do I need slack variables?

Slack variables convert less-than constraints into equations. They show unused capacity, such as remaining material, time, budget, or machine hours.

What is the ratio test?

The ratio test selects the leaving variable. It divides each positive entering-column value into the right side. The smallest valid ratio leaves the basis.

What does unbounded mean?

Unbounded means the objective can improve without a limiting constraint. The entering column has no valid positive pivot row to stop growth.

Why can a problem be infeasible?

A problem is infeasible when constraints conflict. No set of nonnegative decision variable values can satisfy every condition at the same time.

Can I export the result?

Yes. Use the CSV button for spreadsheet records. Use the PDF button to save the visible result and iteration tables for review.

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Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.