Verify a delivery flow with a capacity-cut witness

Name the network rather than only its headline capacity

Use one clearly defined class of interchangeable demonstration kits in a fictional steady-state transfer model. The source is s, the destination is t and A and B are intermediate transfer points. All quantities use the same invented rate unit. Travel time, storage changes, handling loss and competing products are excluded. A real operation needs those assumptions reviewed before this model can support a planning decision.

Require nonnegative flows no greater than capacities, with balanced inflow/outflow at intermediate nodes. Directed capacities: s→A 4, s→B 3, A→B 2, A→t 2, B→t 4.

Ask the model owner to supply a named edge list, not just a visual diagram with unlabeled arrows. A connection in one direction does not establish that the reverse route is available. Give each intermediate location a stable identity and retain the input version. If several records describe one shared facility, resolve that representation before interpreting the reported total.

Write the operating exclusions beside the proposed use. A convenient numerical example does not establish that real kits can be transferred without delay, that every link is simultaneously available or that shared labor imposes no additional restriction. If an omitted condition determines feasibility, revise the specification with the responsible owner rather than adding a reassuring disclaimer after the plan has been accepted.

Check one attainable edge-flow record

In the same edge order, flows (3, 3, 1, 2, 4) give total 6. At A, 3 = 1 + 2; at B, 3 + 1 = 4. Each flow respects its edge capacity.

Keep the proposed flow beside each edge’s limit. A review should reveal which condition failed rather than returning only a single pass or fail. Ask a second reader to reconstruct the intermediate balances from the named edges without consulting the solver output. This is especially useful when a copied report has changed arrow labels or reordered rows.

SciPy maximum_flow accepts a directed integer-capacity CSR graph and returns a flow value and flow graph. The local SciPy 1.17.0 execution returns 6; the opened reference is 1.18.0. No untested version-specific feature is claimed.

Preserve the actual edge record alongside the library result. A returned total is not enough evidence that the inputs matched the intended network. The local fixture uses only the stated invented graph. It does not connect to a warehouse, inspect a live transportation service or establish measured delivery performance. Those would be separate evidence requests under their own scope.

Show why a larger result is impossible in this model

For a set containing s but not t, outgoing capacity bounds the total. Choose {s,A,B}: A→t and B→t contribute 2 + 4 = 6. The feasible flow attains this limit, proving optimality.

All four eligible sets provide a second check: {s}, {s,A}, {s,B}, {s,A,B} have capacities 7, 7, 8, 6. The incoming A→B edge is excluded for {s,B}.

Retain the partition and the actual crossing edges in the report. Coloring a region of a diagram is insufficient when its legend or arrow direction is ambiguous. A reviewer should be able to reconstruct the claimed limit using the edge list alone. The witness belongs to this input version, so a later routing change requires a newly checked explanation.

Independent enumeration finds 33 balanced assignments among 900 integer combinations; maximum 6. This checks a tiny fixture, not large-network scalability.

Separate the two parts of the argument during review. The proposed record answers whether the stated quantity can be attained; the partition answers whether anything larger could fit the supplied limits. If either part is missing or disagrees with the other, keep the result unresolved. Do not resolve the discrepancy by changing a displayed total while leaving the underlying witness unchanged.

Turn the verified model result into the right question

Deliver the input edge list, the attainable record, the partition, the independent checks and the operating exclusions together. Have the owner state what practical decision the result is intended to inform. It might justify a more detailed routing study, but it does not on its own justify a shipment promise, a capacity purchase or a revised supplier commitment.

If the receiving implementation reports a different result, first compare the complete input graph and the meaning of absent edges. Then inspect the returned transfer record and reproduce the discrepancy in a controlled fixture. Keep an alternative valid record separate from an incorrect total: two implementations may provide different explanations of the same supported outcome without either explanation being a defect.

For a real deployment, ask qualified operations staff to assess the omitted timing, availability and shared-resource conditions. Avoid converting a steady-state calculation into an arrival-time guarantee. An operational plan also needs evidence that the approved routes are usable for the actual goods and period. The teaching model provides no such evidence and authorizes no external action.

Close the worksheet when another reviewer can reproduce both the feasible transfer record and its limiting witness from the supplied inputs. Preserve the version and the scope with any later presentation. The deliverable should explain the model’s result plainly while leaving the separate operational decision with the owner who can assess the real constraints.

Make the reasoning concrete.

A fictional event-logistics analyst supplies a throughput headline from a routing model. The coordinator needs to understand whether the figure follows from the proposed connections before requesting a real operating assessment.

The coordinator requests the complete model record before asking for an authorized operational recommendation.

Take the questions into your next brief.

Use these prompts in your own notes. Nothing is submitted from this page.

1. Scope

   What item and rate unit does the graph represent?

2. Edges

   Which directed connections and limits are explicitly present?

3. Feasibility

   Can every intermediate balance be reproduced?

4. Bound

   Which partition and outgoing edges support the claimed limit?

5. Use

   Which omitted operating conditions prevent a delivery promise?

Watch for the wrong conclusion.

• Treating an arrow as a bidirectional connection without evidence.
• Approving a solver total without its input and transfer record.
• Presenting a steady-state model as an arrival-time guarantee.

Sources & scope

All business examples and numerical fixtures are original and fictional. Sources support the identified mathematical or implementation distinctions. Qualified analysts own model specification and inference. No causal result, model fit, actual customer data or operational change is claimed.

• Network flows →Primary university notes, Michel Goemans, 2013 · Supports: Supports conservation and the directed-cut bound.
• SciPy maximum_flow →Primary implementation documentation · Supports: Supports the integer-capacity solver interface.