Shadow Prices

Shadow Prices from the Certificate

Shadow prices are dual prices attached to RHS resources. This lesson reconnects the simplex and duality books by reading the prices from a recomputed primal-dual certificate.

highlighted = computed this step

Base optimum

The base LP optimum is (4/3, 4/3) with objective value 8/3. Motivation: sensitivity starts from a certified optimum, not from an isolated slope.

z∗=8/3z^*=8/3
Base shadow-price certificateThe primal optimum and dual prices are recomputed from the flagship LP.base certificatePrimal114/34/3Dual441/31/3cx*by*primal z=8/3 = dual z=8/3

Shadow prices

The two shadow prices are 1/3 and 1/3. Why: these are the dual prices attached to the two RHS resources.

y∗=1/3,1/3y^*=1/3,1/3
Base shadow-price certificateThe primal optimum and dual prices are recomputed from the flagship LP.base certificatePrimal114/34/3Dual441/31/3cx*by*primal z=8/3 = dual z=8/3

Certificate equality

The dual value is also 8/3. Interpretation: the prices are useful because they certify the same value as the primal plan.

zP=zD=8/3z_P=z_D=8/3
Base shadow-price certificateThe primal optimum and dual prices are recomputed from the flagship LP.base certificatePrimal114/34/3Dual441/31/3cx*by*primal z=8/3 = dual z=8/3

Certificate note

RHS sensitivity is local: the shadow-price prediction is exact while the same basis remains primal feasible and dual feasible. Outside that range, the LP must be recomputed. Pixel positions are rounded for layout; every number shown is exact.

shadow prices come from the exact certificate\text{shadow prices come from the exact certificate}
Base shadow-price certificateThe primal optimum and dual prices are recomputed from the flagship LP.base certificatePrimal114/34/3Dual441/31/3cx*by*primal z=8/3 = dual z=8/3