Enter Primal Model
Example Data Table
This sample shows a primal maximization model and its dual structure.
| Part | x1 | x2 | Sign | Right Side |
|---|---|---|---|---|
| Constraint 1 | 1 | 2 | ≤ | 8 |
| Constraint 2 | 3 | 2 | ≤ | 12 |
| Constraint 3 | 1 | 1 | ≤ | 10 |
| Objective | 3 | 5 | Max | Z |
Formula Used
Primal max standard form:
Max Z = cᵀx
Ax ≤ b, x ≥ 0
Dual form:
Min W = bᵀy
Aᵀy ≥ c, y ≥ 0
The calculator transposes the coefficient matrix. The right side values become coefficients in the dual objective. The primal objective coefficients become the right side values of the dual constraints. Constraint signs control the sign rules of dual variables. Variable restrictions control the direction of dual constraints.
How to Use This Calculator
Choose whether the primal model is a maximization or minimization problem. Enter the number of constraints and variables. Add objective coefficients in one line. Then enter the right side values in one line. Add the matrix coefficients row by row. Select each constraint sign and variable rule. Press the calculate button. The dual model appears above the form and below the header section.
Understanding the LP Dual Calculator
Purpose of the Tool
A linear programming dual model gives another view of the same optimization problem. It changes resources, constraints, prices, and shadow values into a related model. This calculator helps students, analysts, and planners build that related model without doing every transpose step by hand.
Why Duality Matters
Duality is important because it explains the hidden value of scarce resources. A primal model often describes production, allocation, scheduling, or blending. Its dual can describe resource prices, limits, or marginal worth. When both models have feasible optimal solutions, their objective values match. This idea is called strong duality.
Matrix Based Conversion
The core operation is matrix transposition. Rows of the primal coefficient matrix become columns in the dual. Columns become rows. The primal right side vector becomes the dual objective vector. The primal objective vector becomes the dual right side vector. This clean structure makes the calculator useful for checking homework and validating model design.
Handling Signs
Sign direction is a common source of mistakes. A less than or equal constraint in a maximization model normally creates a nonnegative dual variable. A greater than or equal constraint may create a nonpositive dual variable. An equality constraint usually creates an unrestricted dual variable. Variable restrictions also change the dual constraint direction.
Practical Use
Use this tool before solving a model with a larger optimizer. It can reveal whether the formulation is balanced. It can also show whether a resource price interpretation makes sense. The optional two variable estimate is helpful for simple classroom examples. It checks corner points for standard maximization cases.
Better Model Review
The export options make reporting easier. You can download a table for spreadsheets or create a simple document for records. The example table also shows the input style. With careful entries, the calculator gives a fast and readable dual model for many general linear programming problems.
FAQs
1. What is an LP dual?
An LP dual is a related optimization model formed from a primal linear program. It swaps rows and columns, changes objective direction, and gives useful resource value interpretation.
2. Does the calculator solve every linear program?
No. It mainly builds the dual model. It also gives a simple estimate for two variable standard maximization problems with nonnegative variables and less than or equal constraints.
3. What does matrix transposition mean?
Matrix transposition means turning rows into columns and columns into rows. In duality, this changes primal constraint coefficients into dual constraint coefficients.
4. Why do signs change in the dual?
Signs change because each primal constraint and variable restriction creates a matching dual rule. These rules preserve the mathematical relationship between both models.
5. What are unrestricted variables?
An unrestricted variable can be positive, negative, or zero. In duality, equality constraints often produce unrestricted variables, depending on the primal objective type.
6. Can I export my result?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a simple report containing the calculated dual model and matrix summary.
7. What should I enter in the matrix box?
Enter one constraint per row. Separate numbers with spaces or commas. The number of rows should match your selected number of constraints.
8. Why is duality useful in planning?
Duality shows the implied value of resources. It helps managers understand limits, shadow prices, and whether constraints are worth relaxing or tightening.