Understanding Simplex Tableau Pivoting
A simplex tableau pivot is the core move inside the simplex method. It changes one basic variable into another. The goal is to move toward a better corner point without rebuilding the whole linear program. This calculator focuses on that single pivot step. It also explains why the selected entry is safe.
What the Calculator Checks
The tool reads a rectangular tableau. The last column is treated as the right hand side. You may let the calculator choose the entering column, or you may set the pivot row and column yourself. In automatic mode for maximization, the most negative objective coefficient is selected. Then the ratio test finds the limiting row. Only positive pivot column entries are valid for this test. This avoids moving past a feasible boundary.
Why the Pivot Matters
The pivot value becomes one after normalization. Every other entry in the pivot column becomes zero. This creates a new basis. The transformed tableau keeps the same feasible solution region, but it represents a new vertex. If the objective row has no improving coefficient, the tableau is already optimal for that rule. If no valid leaving row exists, the model can be unbounded in that direction.
Useful Study Features
The calculator shows the entering column, leaving row, pivot value, ratio list, and row operations. It also lists possible basic variables after the pivot. These details help students find arithmetic errors. They also help analysts document model changes. The CSV export is helpful for spreadsheets. The PDF export is useful for homework records, peer review, and quick reporting.
Practical Notes
Simplex tableaus depend on correct model setup. Constraints should be converted before entry. Slack, surplus, and artificial columns should already be included when needed. Keep units consistent. Use decimals or fractions with care. A pivot result is only as reliable as the tableau supplied. For difficult problems, compare several pivots and watch degeneracy. A zero ratio can indicate a degenerate move. Degeneracy is valid, but cycling rules may be needed in advanced cases.
Use the precision setting to control displayed decimals. Higher precision reveals small rounding effects. Lower precision gives cleaner classroom tables. Always keep an untouched copy of the starting tableau for checking later.