Dual Linear Programming Calculator

Feasible Region and Corner Points

The graph is drawn for two-variable primal models. It shows constraint boundaries and feasible corner candidates.

Sample Linear Program

Dual Linear Programming Rules

For a maximization primal, a ≤ constraint creates a nonnegative dual variable. A ≥ constraint creates a nonpositive dual variable. An equality constraint creates a free dual variable. For a minimization primal, the ≤ and ≥ sign rules reverse.

Step-by-Step Use

1. Select whether the primal model is a maximization or minimization problem.
2. Choose the number of decision variables and constraints.
3. Enter the objective coefficients for x1, x2, and any extra variables.
4. Enter each constraint row, inequality sign, and right hand side value.
5. Press Calculate Dual to create the dual model and solve supported two-variable cases.
6. Review the chart, slack table, dual signs, and objective comparison.
7. Use the CSV or PDF buttons to save the result.

Understanding Dual Linear Programming

Dual linear programming is a useful way to read any linear model from another side. A primal model uses activities, resources, limits, and profit or cost coefficients. The dual model turns those limits into values. These values are often called shadow prices.

This calculator helps you build both models in one workspace. You enter the primal objective coefficients, constraint coefficients, relation signs, and right hand side values. The tool writes the matching dual structure automatically. It also solves two variable cases by checking feasible corner points. That makes it practical for classroom work, planning examples, and sensitivity checks.

The main idea is simple. Every primal constraint becomes one dual variable. Every primal variable becomes one dual constraint. A maximization model with less than or equal constraints becomes a minimization dual with greater than or equal constraints. A minimization model uses the reverse pattern. Equality constraints create free dual variables, because their value can move above or below zero.

The graphical solver is based on the corner point method. It draws the original constraint lines, finds boundary intersections, removes infeasible points, and evaluates the objective at each remaining point. The best point is reported with slack or surplus values. When the dual also has two variables, the same method can compare the dual result.

Use the output as a guide, not as a replacement for formal proof. Linear programs can be unbounded, degenerate, infeasible, or affected by unusual sign rules. Large models should be checked with a professional optimization solver. Still, this page gives a clear learning path. You can see how each coefficient moves from the primal table into the dual table. You can also export the summary for notes, reports, or worksheets.

For best results, start with a clean standard form. Keep units consistent. Use positive right hand side values when possible. Review each inequality sign before solving. Then compare the primal value, dual value, slacks, and chart. These pieces show how the model behaves, where resources bind, and which limits have the strongest economic meaning.

Finally, test changed coefficients often today with care. Small changes reveal tradeoffs and hidden bottlenecks. They support safer decisions before real planning work.

Frequently Asked Questions