Advanced Dual LP Calculator

Example Data Table

This example is a maximization model with two constraints and two nonnegative variables.

The generated dual is: Minimize W = 10y1 + 12y2, subject to 2y1 + y2 ≥ 3 and y1 + 3y2 ≥ 5.

Formula Used

For a primal maximization model:

Maximize Z = c^Tx, subject to Ax ≤ b and x ≥ 0.

The standard dual is:

Minimize W = b^Ty, subject to A^Ty ≥ c and y ≥ 0.

For a primal minimization model:

Minimize Z = c^Tx, subject to Ax ≥ b and x ≥ 0.

The standard dual is:

Maximize W = b^Ty, subject to A^Ty ≤ c and y ≥ 0.

Equality constraints create free dual variables. Free primal variables create equality dual constraints. Reversed inequalities also reverse sign restrictions.

How to Use This Calculator

1. Select whether the primal model is a maximization or minimization problem.
2. Choose the number of constraints and variables.
3. Click Update Input Size when you change those counts.
4. Enter objective coefficients in the c fields.
5. Enter the constraint matrix, inequality signs, and right side values.
6. Select each primal variable sign rule.
7. Optionally enter candidate x and y values for feasibility checks.
8. Click Calculate Dual LP to show the generated result above the form.
9. Use CSV or PDF export buttons to save the report.

Understanding Dual Linear Programs

A dual linear program rewrites a primal optimization model from another viewpoint. Each primal constraint becomes a dual variable. Each primal variable becomes a dual constraint. This switch helps analysts study limits, prices, and resource pressure without rebuilding the whole model.

Why Dual Forms Matter

The dual model is useful in planning, allocation, and sensitivity review. In a production plan, primal variables may represent product quantities. The dual variables can represent the implied value of materials, labor, or capacity. When both models are feasible and bounded, their optimal objective values match. This is the strong duality idea. It gives a direct check on solver results.

Input Structure

This calculator accepts objective coefficients, a constraint matrix, right side values, constraint signs, and variable sign rules. You can enter a maximization or minimization primal form. The tool then applies standard dual rules. It also builds readable expressions for the dual objective, dual constraints, and dual variable restrictions.

Advanced Review

Optional primal and dual value fields help you test candidate solutions. The calculator checks whether the entered primal values satisfy the original restrictions. It checks whether the entered dual values satisfy the generated restrictions. It also reports objective values, so you can compare them. This is helpful when reviewing homework, spreadsheets, solver output, or manual simplex work.

Export and Documentation

CSV export is useful for spreadsheet records. PDF export is useful for sharing a compact report. The example table shows the input pattern before you start. The formula section explains the transformation rules, so the output remains understandable.

Best Practice

Use clean units and consistent signs. Check every inequality direction before calculating. Avoid mixing costs, profits, and capacities without labels. If your source model uses equality constraints or free variables, review the generated dual carefully. These cases are valid, but they need close attention. A dual model is not just a copied table. It is a structured economic interpretation of the original problem.

Common Output Checks

Read the dual objective first. Then inspect every generated constraint. The number of dual variables must equal the number of primal constraints. The number of dual constraints must equal the number of primal variables. This simple count often catches setup mistakes early quickly.