Pivoting

Choosing the Entering Variable

A pivot begins by choosing a nonbasic variable that can improve the objective. This lesson reads that choice from the reduced costs in the z-row. The highlighted column is recomputed from the same rule simplex uses.

highlighted = computed this step

Reduced costs

The decision entries in the z-row are -1 for x and -1 for y. Why: a negative reduced cost says increasing that variable can raise z.

negative reduced costs can enter\text{negative reduced costs can enter}
Choosing the entering columnThe highlighted column is recomputed from the most-negative reduced cost rule.Entering column2110412014-1-1000xys1s2rhss1s2z

Tie rule

Both decision columns tie, so the lowest column index chooses x. Why: a fixed tie rule makes the pivot path deterministic.

tie goes to the leftmost eligible column\text{tie goes to the leftmost eligible column}
Choosing the entering columnThe highlighted column is recomputed from the most-negative reduced cost rule.Entering column2110412014-1-1000xys1s2rhss1s2z

Diagram note

The highlighted entering column is recomputed from the z-row, not authored by the lesson. Pixel positions are rounded for layout; every number shown is exact.

entering column comes from the z-row\text{entering column comes from the z-row}
Choosing the entering columnThe highlighted column is recomputed from the most-negative reduced cost rule.Entering column2110412014-1-1000xys1s2rhss1s2z