Pivot Element, Pivot Row and Pivot Column in the Simplex Method (Minimum Ratio Test)

Every iteration of the simplex method comes down to one operation: the pivot. You choose a pivot column, a pivot row and the pivot element where they cross, then use row operations to produce the next tableau. If you can pivot correctly, you can solve any simplex problem.

This guide explains each part with a worked example, including the exact row operations. You can check every step with the simplex pivot calculator.

Definitions

Term	What it is	How to find it (maximization)
Pivot column	The column of the entering variable	Largest positive Cj − Zj
Pivot row	The row of the leaving variable	Smallest ratio RHS ÷ positive pivot-column entry
Pivot element	The number where the pivot row and column cross	Read from the tableau
Minimum ratio test	The rule used to choose the pivot row	Compare ratios, ignore entries ≤ 0

For a minimization problem solved directly, the pivot column is the one with the most negative Cj − Zj. The ratio test is the same.

The Example

Maximize Z = 4x1 + 3x2 subject to 2x1 + x2 ≤ 10, x1 + x2 ≤ 8, x1, x2 ≥ 0.

Initial tableau with slack variables s1 and s2:

Basis	Cb	x1	x2	s1	s2	RHS
s1	0	2	1	1	0	10
s2	0	1	1	0	1	8
Cj − Zj		4	3	0	0	Z = 0

Step 1: Find the Pivot Column

The Cj − Zj values are 4 for x1 and 3 for x2. The largest is 4, so x1 enters and the x1 column is the pivot column. Each unit of x1 raises Z by 4 at this point, more than any other variable.

Step 2: Find the Pivot Row (Minimum Ratio Test)

For each row, divide the RHS by the entry in the pivot column, using only positive entries:

• Row s1: 10 ÷ 2 = 5
• Row s2: 8 ÷ 1 = 8

The smallest ratio is 5, so s1 leaves and its row is the pivot row.

Why the smallest ratio? Each ratio is how far x1 can increase before that row’s basic variable hits zero. Row s1 allows x1 = 5, row s2 allows x1 = 8. If x1 went to 8, s1 would become 10 − 2(8) = −6, which is not allowed. The smallest ratio is the largest step that keeps every variable non-negative.

Why ignore zero and negative entries? A zero entry means that row is not affected by x1 at all. A negative entry means increasing x1 makes that basic variable larger, so it can never block the step.

Step 3: Identify the Pivot Element

The pivot row (s1) and pivot column (x1) cross at 2. That is the pivot element.

Step 4: Perform the Row Operations

Make the pivot element 1. Divide the whole pivot row by 2:

• New x1 row: 2/2, 1/2, 1/2, 0/2 | 10/2 = 1, 1/2, 1/2, 0 | 5

Make the other entries in the pivot column 0. Row s2 has a 1 in the x1 column, so subtract 1 times the new x1 row from it:

• s2 row: (1 − 1), (1 − 1/2), (0 − 1/2), (1 − 0) | (8 − 5) = 0, 1/2, −1/2, 1 | 3

The general formula for any non-pivot row is: new row = old row − (its pivot-column entry) × new pivot row.

Recompute Zj and Cj − Zj with the new Cb (4 for x1):

Basis	Cb	x1	x2	s1	s2	RHS
x1	4	1	1/2	1/2	0	5
s2	0	0	1/2	−1/2	1	3
Zj		4	2	2	0	20
Cj − Zj		0	1	−2	0

The x1 column is now a unit column (1 in its own row, 0 elsewhere), as every basic variable’s column must be. Z has risen from 0 to 20.

Second Pivot

x2 still has a positive Cj − Zj of 1, so pivot again.

• Pivot column: x2.
• Ratio test: row x1 gives 5 ÷ (1/2) = 10; row s2 gives 3 ÷ (1/2) = 6. s2 leaves.
• Pivot element: 1/2.

Row operations:

• Divide the s2 row by 1/2 (multiply by 2): new x2 row = 0, 1, −1, 2 | 6
• x1 row minus 1/2 × new x2 row: 1, 0, 1, −1 | 2

Basis	Cb	x1	x2	s1	s2	RHS
x1	4	1	0	1	−1	2
x2	3	0	1	−1	2	6
Zj		4	3	1	2	26
Cj − Zj		0	0	−1	−2

All Cj − Zj ≤ 0, so this is optimal: x1 = 2, x2 = 6, Z = 26. Check: 2(2) + 6 = 10 ✓ and 2 + 6 = 8 ✓.

Tie-Breaking Rules

• Tie in the pivot column (two equal Cj − Zj values): either can enter. Many textbooks pick the lower-numbered variable.
• Tie in the ratio test: either row can leave, but the next tableau will contain a basic variable equal to zero. This is called degeneracy and is explained in special cases in the simplex method.
• No positive entry in the pivot column: there is no pivot row, and the problem is unbounded.

Pivoting Mistakes to Avoid

• Dividing the wrong row. Only the pivot row is divided by the pivot element.
• Using the old pivot row in the elimination step. Always subtract multiples of the new (already divided) pivot row.
• Taking a ratio with a negative entry. A negative ratio is never a valid choice.
• Forgetting to update the Basis and Cb columns after the pivot.

Practise Pivoting

Enter 4x1 + 3x2 with the two constraints in the simplex pivot calculator. It shows the entering and leaving variable at each iteration, highlights the pivot element, and prints the tableau after every pivot, so you can compare each row operation with the steps above. For the theory behind the Zj row, see how to calculate Zj and Cj − Zj.