Calculator Inputs
Example Data Table
This sample model maximizes profit under two resource limits.
| Item | x1 | x2 | Right Side | Meaning |
|---|---|---|---|---|
| Objective | 3 | 5 | Maximize | Profit contribution |
| Constraint 1 | 2 | 1 | 10 | Resource A limit |
| Constraint 2 | 1 | 2 | 8 | Resource B limit |
Formula Used
Primal maximization model:
Maximize Z = cTx
Subject to Ax ≤ b, x ≥ 0
Dual minimization model:
Minimize W = bTy
Subject to ATy ≥ c, y ≥ 0
Slack is calculated as s = b - Ax.
Dual surplus is calculated as r = ATy - c.
At an ideal optimum, Z = W.
Complementary slackness checks yi × si = 0 and xj × rj = 0.
How to Use This Calculator
- Select the number of variables and constraints.
- Enter objective coefficients for the primal maximization model.
- Enter each constraint coefficient and right side value.
- Keep the model in the standard ≤ maximization form.
- Choose decimal precision and tolerance.
- Press the calculate button.
- Review primal values, dual values, slack, and gap.
- Use the CSV or PDF buttons to save the result.
Article: Understanding LP Primal and Dual Models
What the Calculator Solves
Linear programming helps choose the best result under limits. The primal model usually describes the main business question. It may represent production, staffing, transport, blending, or budget use. The objective shows what should be maximized. The constraints show what resources cannot be exceeded. This calculator uses a common standard form. It maximizes a linear objective with less-than-or-equal limits.
Why the Dual Matters
The dual model gives a second view of the same problem. It prices the limits instead of the activities. Each dual variable belongs to one primal constraint. A high dual value can show a scarce resource. A zero dual value often means unused capacity. This is useful for planning. It helps managers see which limits affect the final result.
Slack and Surplus Insight
Slack measures unused resource in the primal model. If slack is zero, that constraint is binding. If slack is positive, the resource remains available. Dual surplus measures extra strength in each dual inequality. These two measures support sensitivity thinking. They also help check whether the solution is consistent.
Strong Duality and Checks
Strong duality says both objective values should match at optimum. The primal maximum should equal the dual minimum. A small gap usually appears because of rounding. A large gap may show an input issue. Complementary slackness gives another quality check. A binding constraint may have a positive dual value. An unused constraint should have a zero dual value. An active variable should normally have zero dual surplus.
Practical Use
Use this tool when you need a quick LP review. It is helpful for small models. It shows the primal answer and the dual interpretation together. It also exports reports for records. Always confirm that your model follows the stated form. Convert greater-than constraints before using this version. Check units before trusting decisions. Clean data gives better optimization insight.
FAQs
1. What is an LP primal dual calculator?
It solves a small linear programming model and builds its dual. It compares objective values, variables, slack, and complementary slackness conditions.
2. What model form does this calculator use?
It uses a maximization primal with less-than-or-equal constraints and nonnegative variables. The matching dual is a minimization model.
3. What does the dual value mean?
A dual value shows the shadow price of a constraint. It estimates how much the objective may improve if that limit increases slightly.
4. What is slack in linear programming?
Slack is unused capacity. It equals the right side value minus the used amount for a less-than-or-equal constraint.
5. Why should primal and dual objectives match?
Strong duality says both optimal objective values match when feasible bounded optima exist. Small differences can appear from rounding tolerance.
6. What does complementary slackness show?
It checks whether primal slack and dual values agree. Binding limits may have value. Unused limits should have zero dual value.
7. Can this handle negative coefficients?
Yes, coefficients may be negative. However, the model should still follow the stated standard form for correct interpretation.
8. Why was no solution found?
The model may be infeasible, unbounded, singular, or outside the supported form. Check signs, limits, and active variable counts.