Operations & Logistics
Maximum Flow & Min-Cut Network Model (Node Conservation, Arc Slack & Bottleneck Cut)
Model directed logistics and pipeline networks by auditing SUMIFS transshipment flow conservation (Outflow - Inflow = 0), residual arc capacity slack, and Max-Flow = Min-Cut optimality.
Verifying feasible flow conservation and the Max-Flow Min-Cut certificate
In a directed network from source S to sink T through intermediate hubs A and B, a flow vector is feasible if every arc satisfies 0 <= Flow <= Capacity and every intermediate hub conserves mass: SUMIFS(Flow, From, Hub) - SUMIFS(Flow, To, Hub) = 0.
By the Ford-Fulkerson Max-Flow Min-Cut theorem, total throughput leaving S reaches its theoretical maximum as soon as it equals the capacity of at least one S-T cut—for example, when both sink-entering arcs A -> T (cap 8) and B -> T (cap 13) are 100% saturated at 8 + 13 = 21.
| Directed Arc | Capacity | Assigned Flow | Residual Slack | Bottleneck Status |
|---|---|---|---|---|
| S -> A | 12 | 12 | 0 | Saturated |
| S -> B | 10 | 9 | 1 | Has spare capacity |
| A -> B | 5 | 4 | 1 | Redirects A surplus to B |
| A -> T | 8 | 8 | 0 | Min-Cut Sink Bottleneck |
| B -> T | 13 | 13 | 0 | Min-Cut Sink Bottleneck |
Intermediate Hub Flow Conservation Formula (SUMIFS)
Subtract total inflow entering hub node A from total outflow leaving A; any non-zero residual indicates leaked or phantom inventory.
=SUMIFS($D$2:$D$6, $A$2:$A$6, "A") - SUMIFS($D$2:$D$6, $B$2:$B$6, "A")Max-Flow = Min-Cut Optimality Check
Verify simultaneously that intermediate balances equal 0, min arc slack >= 0, and Source Outflow equals Sink Cut Capacity.
=AND(H2 = 0, H3 = 0, MIN(E2:E6) >= 0, H4 = H5)Try the example
Use the original values or change them to test your understanding.