Dual Linear Program Calculator

Calculator Input

Primal standard form

Right side vector b

Objective vector c

Optional dual vector y

Optional primal vector x

Decimal places

Feasibility tolerance

Run small vertex solver

Constraint matrix A

Example Data Table

Item	Example Value	Meaning
A	2, 1 1, 2 1, 0	Three primal constraints and two decision variables.
b	100, 80, 40	Available resource limits.
c	40, 30	Objective coefficients for x1 and x2.
Dual result	Min W = 100y1 + 80y2 + 40y3	Dual objective built from b.

Formula Used

For a primal maximization model, use Max Z = cᵀx subject to Ax <= b and x >= 0.

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

For a primal minimization model, use Min Z = cᵀx subject to Ax >= b and x >= 0.

The dual is Max W = bᵀy subject to Aᵀy <= c and y >= 0.

Weak duality checks the difference between the feasible primal and dual objective values.

How to Use This Calculator

Select the standard form that matches your primal model. Enter matrix A with one row per constraint. Enter vector b with one number per row. Enter vector c with one number per decision variable. Optional x and y vectors let you check feasibility, objective values, slack, and the duality gap.

Understanding Dual Linear Programs

A dual linear program gives a second view of a primal model. It turns limits into prices. It turns activity values into requirement tests. This view helps students, analysts, and planners understand the shadow value of each resource.

Why the Dual Matters

The primal model often asks for the best production, mix, route, or schedule. The dual model asks what each resource would be worth. When both models are feasible and bounded, their best objective values match. This idea is called strong duality. It is a useful check on manual work.

Standard Form Used

This calculator uses a common max form. The primal objective is to maximize c times x. The constraints use A x less than or equal to b. Each x value must be nonnegative. The matching dual minimizes b times y. Its constraints use A transpose y greater than or equal to c. Each y value must also be nonnegative.

What the Result Shows

The tool writes the dual objective, every dual constraint, and the nonnegative condition. It can also test a proposed dual vector. If the vector satisfies all dual constraints, the tool marks it feasible. It then computes the dual objective value. When a primal vector is also entered, the tool checks weak duality and reports the gap.

Using the Solver Mode

Small models can be solved by active constraint enumeration. The calculator forms possible corner points. It tests them against the dual rules. Then it keeps the feasible point with the smallest objective value. This approach is helpful for learning. Very large models should still be checked with specialist optimization software.

Good Input Practice

Enter one matrix row per line. Separate numbers with commas or spaces. Keep the vector lengths consistent. Use decimal values when needed. Do not mix different unit systems inside the same model. Review negative resource limits carefully, because the standard form assumes nonnegative variables and clear bounds.

Interpreting the Answer

A high dual value means a resource can strongly affect the objective. A zero dual variable can mean the matching resource has slack. Complementary slackness links unused resources with zero prices. These signs help explain why an optimum is reasonable.

Use it for clearer decisions later.

FAQs

What is a dual linear program?

It is a related optimization model built from a primal model. It changes constraints into variables and variables into constraints. Its objective value helps check and explain the primal result.

Which primal forms does this tool support?

It supports max models with Ax less than or equal to b. It also supports min models with Ax greater than or equal to b. Variables are treated as nonnegative.

Can it solve every linear program?

No. The built-in solver uses vertex enumeration for small models. It is useful for learning and checking. Large models need a dedicated optimization package.

What does vector b become in the dual?

Vector b becomes the coefficient vector of the dual objective. Each b value multiplies the matching dual variable y.

What does vector c become in the dual?

Vector c becomes the right side of the dual constraints. Each c value is compared with a column expression from A transpose y.

What is the duality gap?

It is the difference between feasible primal and dual objective values. At an optimum, the gap should be near zero for feasible and bounded matching models.

Why enter an optional x vector?

The x vector lets the calculator test primal feasibility. It also allows objective and complementary slackness checks when a y vector is entered.

Why enter an optional y vector?

The y vector lets the calculator test dual feasibility. It computes the dual objective and helps confirm weak duality against an entered primal vector.