Linear Programming Dual Calculator

Example Data Table

Formula Used

A standard primal maximization model can be written as Max Z = cTx, subject to Ax <= b and x >= 0.

Its dual is Min W = bTy, subject to ATy >= c and y >= 0.

For minimization, the direction reverses. A primal minimum with Ax >= b and x >= 0 gives a dual maximum with ATy <= c and y >= 0.

Equality constraints create unrestricted dual variables. Unrestricted primal variables create equality constraints in the dual model.

How to Use This Calculator

Choose whether the primal model is a maximization or minimization problem.

Select the number of constraints and variables. Submit once after changing counts.

Enter objective coefficients, matrix coefficients, inequality signs, and right side values.

Select each primal variable sign. This controls each dual constraint direction.

Press the submit button. The dual objective, constraints, signs, and transposed matrix appear above the form.

Use the CSV or PDF buttons to save the generated result.

Linear Programming Dual Calculator Guide

What the Dual Means

A linear programming dual model gives a second view of the same optimization problem. The primal model may describe production, resources, profit, cost, capacity, or demand. The dual model describes shadow prices and limiting values. This calculator converts a primal model into its dual form. It helps students check signs, objective direction, and transposed coefficients without rewriting every step by hand.

Why Dual Conversion Matters

Duality is important because every constraint in the primal creates one dual variable. Every primal variable creates one dual constraint. The coefficient matrix is transposed. Objective coefficients move to the right side of dual constraints. Right side values become coefficients in the dual objective. This structure makes dual models useful for sensitivity analysis and economic interpretation.

Advanced Sign Handling

This tool supports less simple cases. It allows less-than, greater-than, and equality constraints. It also supports nonnegative, nonpositive, and unrestricted primal variables. These choices affect the final dual model. For example, an equality constraint produces an unrestricted dual variable. An unrestricted primal variable produces an equality constraint in the dual. These rules are often missed during manual work.

Interpreting the Output

Read the dual objective first. A primal maximum usually creates a dual minimum. A primal minimum usually creates a dual maximum. Then review each dual constraint. Each one comes from a column of the primal matrix. The displayed transposed matrix shows this movement clearly. Finally, check the dual variable sign list. It comes directly from the original constraint signs.

Best Use Cases

Use this calculator when learning simplex theory, checking homework, preparing operations research notes, or building examples for teaching. It does not replace careful modeling. It helps you avoid structural mistakes before solving. Always confirm that the primal model itself represents the real problem correctly.

FAQs

What is a dual linear programming model?

A dual model is a related optimization problem built from the primal model. Constraints become variables, variables become constraints, and the coefficient matrix is transposed.

Does this calculator solve the final optimum?

This calculator focuses on building the dual model. It prepares the correct objective, constraints, matrix transpose, and variable signs for further solving.

What happens to a primal maximization problem?

A standard primal maximization problem usually becomes a dual minimization problem. The right side values become dual objective coefficients.

What happens to equality constraints?

An equality constraint in the primal creates an unrestricted variable in the dual. This means the dual variable can be positive, negative, or zero.

What does an unrestricted primal variable create?

An unrestricted primal variable creates an equality constraint in the dual. This rule is important when building nonstandard dual models.

Why is the matrix transposed?

Each primal variable becomes one dual constraint. Therefore each primal matrix column becomes one dual constraint row.

Can I use negative coefficients?

Yes. The calculator accepts negative, zero, decimal, and whole number coefficients. Enter them directly in the input fields.

Why should I download CSV or PDF results?

CSV files help with spreadsheet review. PDF files are useful for reports, class notes, assignments, and printed study material.