Special Cases in the Simplex Method: How to identify Degeneracy, Unbounded Solutions, and Multiple Optima

Introduction

While the standard Simplex method reliably finds the optimal solution, you will sometimes encounter specific structural characteristics within the problem. These special cases manifest distinctly in the Simplex tableau.

1. Degeneracy

Degeneracy occurs when there is a tie for the minimum non-negative ratio during the pivot row selection. As a result, in the next iteration, one or more basic variables will take a value of zero.

While degenerate problems can theoretically cause the Simplex algorithm to cycle infinitely, modern software (and careful pivot rules, like Bland's Rule) prevents this. Visually, degeneracy implies redundant constraints at a single vertex.

2. Unbounded Solutions

An unbounded solution happens when the objective function can be increased indefinitely without violating any constraints.

You can identify this in the Simplex tableau if you select an entering variable (a negative value in the objective row for maximization), but all elements in the pivot column are zero or negative. Because you cannot calculate a valid minimum ratio, no variable can leave the basis, proving the feasibility region is unbounded in the direction of optimization.

3. Multiple Optima

Multiple Optima (or Alternative Optimal Solutions) occur when the objective function is parallel to a binding constraint, resulting in an infinite number of optimal solutions lying along a line segment.

In the optimal Simplex tableau, this is identified when a non-basic variable has a coefficient of exactly zero in the objective row. If you pivot on that column, you will reach a new optimal basic feasible solution with the exact same objective value.