Calculator Input
Example Data Table
| Item | x1 | x2 | Sign | RHS |
|---|---|---|---|---|
| Objective profit | 3 | 5 | Max | Z |
| Machine hours | 2 | 3 | <= | 12 |
| Labor hours | 1 | 1 | <= | 5 |
Formula Used
The model uses a linear objective: Z = c1x1 + c2x2 + ... + cnxn.
Constraints are written as Ax <= b, Ax >= b, or Ax = b with x >= 0.
The tableau reduced cost is rj = Zj - Cj. For primal simplex, the entering column is the most negative rj.
The primal leaving row uses the smallest positive ratio bi / aij.
The dual pivot method first chooses the row with the most negative RHS. It then selects a valid negative row coefficient by the dual ratio rj / |aij|.
The pivot operation divides the pivot row by the pivot value. It then clears the pivot column in every other row.
How to Use This Calculator
- Select maximize or minimize.
- Enter objective coefficients in one line.
- Enter one constraint coefficient row per line.
- Enter one sign per constraint row.
- Enter right side values in the same order.
- Choose auto, primal, or dual pivot mode.
- Press the submit button and review the result above the form.
- Use CSV or PDF export for reporting.
Simplex Online Dual Pivot Calculator Guide
What the Calculator Does
A simplex calculator helps you solve linear programming models. It turns an objective and constraints into a structured tableau. This page supports dual pivot work. That helps when the starting tableau has negative right side values but acceptable reduced costs.
Linear programming is used when resources are limited. A factory may choose product quantities. A planner may compare labor, budget, and storage limits. A transport team may minimize cost. Each case needs a best mix under fixed rules.
Primal and Dual Steps
The primal simplex method moves from one feasible corner to a better corner. It selects an entering variable from the objective row. It then uses a ratio test to choose the leaving row. The process stops when no improving column remains.
The dual simplex method works differently. It starts with a row that is not feasible. Usually, that row has a negative right side value. The calculator selects that row first. It then picks a pivot column that restores feasibility while keeping the objective test valid.
This tool is built for study, checking, and planning. You can enter objective coefficients, a constraint matrix, signs, and right side values. You can also switch between maximize and minimize models. Slack, surplus, and artificial handling is simplified for common classroom forms.
Why Steps Matter
Tableau steps are important. They show why a result is reached. A final answer without steps may hide an error. The step list helps you review pivots, basis changes, ratios, and stopping rules.
Use clean data for best results. Put one constraint row per line. Keep the same number of coefficients in each row. Match each sign and right side value to the same row. Avoid empty rows. Use decimals when needed.
Optimization results should be checked in context. A mathematically optimal answer may still need business review. Some models ignore setup time, demand limits, or risk. Treat the output as a decision aid, not as the only decision.
Exports and Review
The export buttons help with records. CSV is useful for spreadsheets. PDF is useful for notes, reports, and class submissions. Save the inputs with the final values. That makes the calculation easy to audit later.
FAQs
What is a simplex calculator?
It solves linear programming problems by moving through tableau pivots. It finds the best objective value under stated constraints.
What does dual pivot mean?
Dual pivoting starts with an infeasible right side row. It selects a pivot that restores feasibility while preserving the objective condition.
Can this handle minimization?
Yes. Select minimize before calculating. The solver converts the objective internally and reports the original objective value.
Which signs can I enter?
You can enter <=, >=, or =. Put one sign per line so each sign matches the same constraint row.
Why do artificial variables appear?
Artificial variables help start models with >= or equality constraints. They should be zero in a feasible final answer.
What if the answer says unbounded?
An unbounded message means the objective can improve without a limiting leaving row. Review your constraints and missing bounds.
Can I export the solution?
Yes. Use the CSV button for spreadsheet work. Use the PDF button for notes, reports, or class records.
Why should I review tableau steps?
The steps show each pivot and basis change. They help you check data entry, ratios, and final optimality.