What is the distinction between primal degeneracy and dual degeneracy in an optimal Simplex tableau, and how does primal degeneracy lead to non-unique shadow prices in sensitivity analysis?
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Primal degeneracy occurs when a basic variable equals zero at an optimal vertex, whereas dual degeneracy means a non-basic variable has a zero reduced cost. Primal degeneracy leads to non-unique shadow prices because multiple bases can represent the exact same vertex, resulting in different basis inverses that cause left- and right-hand marginal rates of change for constraint right-hand sides to diverge.
Primal degeneracy occurs when a basic variable equals zero, creating an over-determined vertex with redundant binding constraints, whereas dual degeneracy occurs when a non-basic variable has a zero reduced cost, indicating multiple optimal solutions; primal degeneracy causes non-unique shadow prices because shifting a degenerate constraint's limit in opposite directions (+\Delta b_i vs. -\Delta b_i) changes which set of constraints defines the optimal vertex, creating asymmetric marginal rates of return and multiple valid dual bases.
Primal degeneracy happens when a basic variable is zero, while dual degeneracy occurs when the optimal solution can have more than one set of shadow prices. Primal degeneracy can lead to non-unique shadow prices because different optimal bases may produce different shadow prices for the same solution
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