Simplex Method Input
Example Data Table
| Part | x1 | x2 | x3 | RHS |
|---|---|---|---|---|
| Objective Max Z | 3 | 5 | 0 | - |
| Constraint 1 | 1 | 0 | 0 | 4 |
| Constraint 2 | 0 | 2 | 0 | 12 |
| Constraint 3 | 3 | 2 | 0 | 18 |
Formula Used
The calculator uses the standard linear programming form:
Maximize or Minimize Z = c1x1 + c2x2 + c3x3
Subject to:
a1x1 + a2x2 + a3x3 ≤ b
Slack variables convert each inequality into an equation:
a1x1 + a2x2 + a3x3 + s = b
The entering column is selected from the most negative objective row value. The leaving row is selected by the smallest positive RHS divided by pivot column value. The pivot row is normalized. Other rows are then reduced to form the next tableau.
How to Use This Calculator
Enter the objective coefficients first. Select maximize or minimize. Then enter each constraint coefficient and right hand side value. This calculator assumes less-than-or-equal constraints with non-negative decision variables. Press calculate to see the optimal value, final variable values, slack values, final tableau, and pivot steps. Use the export buttons to save the result.
Linear Programming Simplex Method Guide
What This Calculator Does
This simplex method calculator helps solve linear programming models with three decision variables and three constraints. It is designed for Maths study, classroom checking, business planning, and operations research practice. The tool builds a simplex tableau from your objective function and constraints. It then performs pivot operations until an optimal tableau appears.
Why the Simplex Method Matters
Linear programming is used when limited resources must be assigned carefully. A factory may want maximum profit. A planner may want minimum cost. A student may need each tableau step for homework. The simplex method gives a structured path from a starting feasible solution to an improved solution.
Important Input Rules
This calculator uses less-than-or-equal constraints. It also assumes non-negative variables. These are common rules in introductory simplex problems. If a constraint has a negative right side, the program adjusts the row before solving. For advanced cases with mixed signs, artificial variables may be required.
How Pivoting Works
The objective row shows whether improvement is still possible. A negative value in that row means a variable can enter the basis. The most negative value is selected as the entering column. Then the ratio test chooses the leaving row. The smallest positive ratio keeps the solution feasible.
Reading the Results
The result section shows the objective value first. It also lists decision variables and slack variables. A positive slack means unused resource remains. A zero slack usually means the related constraint is binding. The final tableau shows the completed matrix after all pivots.
Export Options
The CSV export is useful for spreadsheets. The PDF export creates a clean summary for records. Both options use the same submitted input. This makes it easy to compare models, store reports, and review simplex work later.
FAQs
1. What is the simplex method?
The simplex method is an algorithm for solving linear programming problems. It moves between feasible corner points and improves the objective value until no better move exists.
2. What type of constraints does this calculator support?
It supports less-than-or-equal constraints. The calculator adds slack variables to convert those inequalities into equations for the simplex tableau.
3. Can I solve minimization problems?
Yes. Choose the minimization option. The calculator internally converts the objective direction and returns the adjusted final objective value.
4. What are slack variables?
Slack variables measure unused resource in each constraint. A positive slack value means that limit is not fully used in the final solution.
5. What is a pivot?
A pivot is the key value used to update the tableau. It changes the basis and creates the next improved solution.
6. What does unbounded mean?
Unbounded means the objective can improve without a finite limit. In that case, no final maximum or minimum exists under the given constraints.
7. Why is the ratio test needed?
The ratio test selects the leaving row. It protects feasibility by keeping all right hand side values non-negative after pivoting.
8. Can I download my result?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a short printable summary of the simplex result.