Simplex Method Pivot Calculator
Simplex method pivot calculator that identifies the pivot column and pivot row and performs each pivot operation step by step.
Simplex Calculator
How Simplex Method Calculator Works
Enter the LP Problem
Type the objective function coefficients and every constraint row with its right-hand-side value.
Choose Maximize or Minimize
Pick your optimization goal. The tool builds the initial tableau with slack variables automatically.
Run the Pivot Iterations
The calculator identifies pivot column by Cj-Zj, computes ratios, performs elementary row operations until optimal.
Read the Optimal Solution
Final tableau displays optimal variable values, Zj row, and the maximum/minimum objective value.
Sample Simplex Tableau Output
Example tableau iteration for a 2-variable maximization problem
| Basis | x1 | x2 | s1 | s2 | RHS | Cj-Zj |
|---|---|---|---|---|---|---|
| x1 | 14 | 0 | 0 | 1 | 14 | 0 |
| x2 | 7 | 1 | 0 | 0 | 7 | 5 |
| Zj | 35 | 5 | 0 | 0 | 35 |
How the Pivot Works
A pivot is the core operation of the simplex method. The pivot column (entering variable) is the column with the most favourable Cj minus Zj value, and the pivot row (leaving variable) is found by the minimum-ratio test. This simplex method pivot calculator highlights both, then performs the elementary row operations so the pivot element becomes 1 and the rest of the pivot column becomes 0.
Following Each Pivot
After every pivot the basis changes and the objective improves. The calculator shows the pivot element, the entering and leaving variables, and the updated tableau for each iteration until no improving pivot remains and the solution is optimal.
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Frequently Asked Questions
What is a pivot in the simplex method?
A pivot is one iteration that swaps a non-basic entering variable for a basic leaving variable using elementary row operations, improving the objective value.
How do you find the pivot column?
For maximization, the pivot column is the one with the most positive Cj minus Zj value; for minimization, the most negative.
How do you find the pivot row?
Divide each right-hand-side value by the positive entry in the pivot column and choose the smallest non-negative ratio - that row is the pivot (leaving) row.
What is the pivot element?
The pivot element is the number at the intersection of the pivot column and pivot row; the row is divided by it so it becomes 1.
When do pivots stop?
Pivoting stops when no Cj minus Zj value can improve the objective, meaning the current tableau is optimal.