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Simplex Method Iteration Calculator

Enter your simplex tableau (last row is the objective, last column is RHS) to perform one pivot.

How it is calculated

Type the tableau for a maximization problem in standard form: rows separated by semicolons, numbers within a row separated by commas or spaces. The last row is the objective row of reduced costs and the last column is the right-hand side, and every row must have the same length. The tool performs one pivot. The entering column is the one with the most negative number in the objective row. The leaving row is the one with the smallest ratio RHS / a among rows where the entry a in that column is positive. The pivot row is divided by the pivot element, and a multiple of it is subtracted from every other row. Pivot is reported as row, column, counted from 1. If no objective entry is negative the tableau is already optimal and the pivot shows 0, 0. If no row has a positive entry in the entering column, the pivot shows 0 and the column number: the problem is unbounded in that direction.

Worked example

Take maximize 3x + 5y with x ≤ 4, 2y ≤ 12, 3x + 2y ≤ 18. The tableau has columns x, y, s1, s2, s3 and RHS, and its objective row is −3, −5, 0, 0, 0, 0. The most negative entry is −5, so column 2 enters. The ratios are 12 / 2 = 6 and 18 / 2 = 9 (the first row has a 0 in that column and is skipped), so row 2 leaves. The tool reports pivot 2, 2. Row 2 becomes 0, 1, 0, 0.50, 0, 6, and the objective row becomes −3, 0, 0, 2.50, 0, 30, so the objective value is 30.00.

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Common mistakes

  • Entering a minimization problem as is. This tool chooses the most negative objective entry, which is the maximization rule; for a minimization you must first negate the objective.
  • Putting the objective row anywhere but last, or the right-hand side anywhere but the last column. The tool takes both positions literally.
  • Leaving out slack variable columns. The tableau needs the identity columns for the constraints already added, otherwise the pivot is meaningless.
  • Reading a pivot of 0, 0 as an error. It means no objective entry is negative, so the tableau is already optimal. A pivot like 0, 3 means column 3 has no positive entry: the problem is unbounded there.
  • Counting rows and columns from 0. The reported pivot position starts at 1, and the objective row is not a candidate for the leaving row.

About this calculator

Dantzig's simplex method solves linear programming problems by walking along the vertices of the feasible region's polytope, improving the objective value at each step until no further improvement is possible. One pivot iteration on the tableau consists of picking the entering variable (usually the column with the most negative coefficient in the objective row, for a maximization problem) and the leaving variable (found via the minimum-ratio test on the RHS column, to keep all values feasible), then performing Gauss-Jordan elimination around that pivot element.

This calculator takes a simplex tableau — with the objective in the last row and the right-hand-side values in the last column — and executes exactly one pivot step, returning the updated tableau so you can see the entering/leaving variable choice and the arithmetic explicitly. Operations research and linear programming students use it to check a single hand-worked iteration of a larger simplex problem, rather than a solver that jumps straight to the optimal solution.

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