Why do pivot rules matter in the simplex method?

Answer First

Pivot rules matter because they determine the path simplex takes through the feasible region. Good pivot rules avoid cycling, reduce the number of iterations, and ensure the algorithm moves toward optimality efficiently.

Without pivot rules, simplex would not know how to choose entering and leaving variables — and could stall, cycle, or take exponentially long paths.

• Pivot rules prevent cycling
• Pivot rules improve convergence speed
• Pivot rules ensure numerical stability
• Pivot rules guarantee correctness of the algorithm

Step-by-Step: Why Pivot Rules Matter

Step 1: Simplex Needs an Entering Variable

For maximization, simplex chooses a variable with a positive reduced cost. Different pivot rules choose differently:

• Largest coefficient rule
• Bland’s rule
• Dantzig’s rule

This choice affects the direction of movement.

Step 2: Simplex Needs a Leaving Variable

The minimum ratio test determines which constraint becomes binding. But ties or degeneracy can cause problems.

Step 3: Pivot Rules Prevent Cycling

In degenerate LPs, simplex can revisit the same bases. Pivot rules like Bland’s Rule guarantee termination.

Step 4: Pivot Rules Improve Efficiency

Some pivot rules lead to fewer iterations, especially in large LPs. Choosing the “best” entering variable can dramatically speed up convergence.

Step 5: Pivot Rules Maintain Numerical Stability

Certain pivot choices avoid extremely small pivots, which can cause rounding errors and unstable solutions.

Numerical Example

Consider the tableau below. Two variables have positive reduced costs.

\[ \begin{array}{c|ccc|c} & x_1 & x_2 & x_3 & \text{RHS} \\ \hline \text{Row 1} & 1 & 2 & 1 & 6 \\ \text{Row 2} & 2 & 1 & 3 & 8 \\ \hline z\text{-row} & 0 & 3 & 2 & 0 \end{array} \]

Step 1: Entering Variable

Dantzig’s rule → choose \( x_2 \) (largest reduced cost = 3). Another rule might choose \( x_3 \).

Step 2: Leaving Variable

Minimum ratio test:

• Row 1: 6 / 2 = 3
• Row 2: 8 / 1 = 8

Row 1 leaves the basis.

Step 3: Different Pivot Rules → Different Paths

Choosing \( x_3 \) instead would lead to a completely different sequence of tableaus. Some paths reach optimality faster; others may stall.