Operations research
Explainer
Degeneracy: When Simplex Stalls at a Corner
Your simplex pivot worked, yet Z did not improve at all. Here's why.
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Lesson notes
Key points
- Degeneracy happens when a basic variable takes the value zero in a basic feasible solution.
- Geometrically, more constraint lines than needed pass through the same corner point, so several bases describe one vertex.
- In the tableau you spot it as a tie in the minimum ratio test, and the next pivot brings a zero step, leaving Z unchanged.
- Rarely, simplex can cycle through the same bases forever, which rules like Bland's rule prevent by always picking the lowest-index variable.
Definitions
- Degenerate solution
- A basic feasible solution in which at least one basic variable equals zero.
Takeaway: Degeneracy: a basic variable at zero, so pivots may not improve Z.
Quick check: What tableau sign warns you a degenerate solution is coming?
- Negative right-hand side
- Tie in minimum ratio test
- All reduced costs zero
A tie means two basic variables hit zero together, so one stays basic at zero.
DegeneracyOvercrowded cornerTied ratio testCycling
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