The final lesson names the network interpretation. The plan is a min-cost flow, and the reduced-cost table is the exact certificate that the displayed flow is optimal.
highlighted = computed this step
Min-Cost Flow View
Transportation is min-cost flow on a bipartite network: supplies send flow, demands absorb flow, and arcs carry unit costs. Motivation: conservation is the same row-sum and column-sum arithmetic.
transportation is a bipartite min-cost flow
Certificate View
The optimal shipment cost is 11 and the dual certificate also has value 11. Interpretation: the graph gives the plan, and the certificate table proves it.
zflow=11=11
Recomputed Comparison
If demands change to X equals 3 and Y equals 2, the exact recompute ships 3 on A-X and 2 on B-Y for cost 8. Why: the same potential certificate machinery follows the changed balanced instance.
comparison cost=8
Diagram note
Certificate note: This certificate is exact for the pinned balanced transportation instance; uncertain demand, stochastic travel time, and capacity expansion are outside this model. Pixel positions are rounded for layout; every number shown is exact.