The special case is degeneracy, which can occur when one or more basic variables are zero, allowing pivots that change the basis without changing the solution and may cause the algorithm to return to the same basic feasible solution.

The special case is cycling, which is usually caused by degeneracy. It can occur when one or more basic variables become zero, so a pivot can change the basis without improving the objective value. If the pivot rule repeatedly selects these non-improving pivots, the Simplex algorithm can return to a previously visited basic feasible solution and continue repeating the same sequence indefinitely, even though there is no mathematical error.

The special case is degeneracy. When basic variables take on a value of zero, the algorithm can pivot to a new basis without moving the solution point, potentially causing it to return to an earlier basic feasible solution.

The special case is cycling due to degeneracy. It can occur when one or more basic variables have a value of zero, causing the Simplex Method to make pivots without improving the objective value. As a result, the algorithm can return to the same basic feasible solution repeatedly.

Yes, that explanation is correct. A simpler version is:

Cycling can occur because of degeneracy, where one or more basic variables are zero. During the Simplex method, the algorithm may pivot between different bases that represent the same basic feasible solution and eventually return to a previous basis. This creates a cycle, even though the calculations are correct.

This is degeneracy — it happens when a basic variable equals zero, so pivoting swaps one zero-valued basic variable for another without changing the solution point or objective value. The Simplex method stays at the same corner because more constraints intersect there than needed, creating redundant directions that lead nowhere new. Cycling — repeatedly revisiting the same tableaus — can occur if poor pivot choices keep rotating these zero-value variables, even though no calculation mistake was made. Anti-cycling rules (like Bland’s rule) prevent this by enforcing a consistent tie-breaking order when multiple variables qualify to enter or leave the basis.