The Minimum-Cost Plan

The Minimum-Cost Plan

Once feasibility is fixed, optimization is exact arithmetic: multiply shipment by unit cost and minimize the sum. This lesson pins the optimum cost.

highlighted = computed this step

Cost Terms

The four arc cost terms are 4, 5, 0, and 2. Motivation: each term is flow times unit cost on that arc, so the objective is fully auditable.

arc terms are recomputed from xijcij\text{arc terms are recomputed from }x_{ij}c_{ij}
Cost arithmeticEach cost term is recomputed as shipped flow times unit cost.shipment cost arithmeticAXAYBXBYtotalcost450211

Minimum Cost

The recomputed minimum transportation cost is 11. Interpretation: the exact solver enumerates feasible balanced shipment matrices and picks the least cost.

z∗=11z^*=11
minimum-cost shipment2/c21/c50/c42/c1A supply 3B supply 2X demand 2Y demand 3

Diagram note

Certificate note: the graph is the same plan as the table, with labels showing flow over unit cost. Pixel positions are rounded for layout; every number shown is exact.

the displayed plan is recomputed, not authored\text{the displayed plan is recomputed, not authored}
minimum-cost shipment2/c21/c50/c42/c1A supply 3B supply 2X demand 2Y demand 3