The critical path is conditional on the network

The forward pass asks when each task could begin and finish if its entered predecessors and duration assumptions govern. A root task starts at the relative origin. A successor starts after the latest finish among its predecessors. Its finish is that start plus its estimated mean duration. The project finish is the latest task finish. This is a conditional calculation. It assumes the task can actually start when the network permits, which is why resource and calendar exclusions matter. A computed early start is not a mobilisation authorisation.

The reference study begins with design release. Equipment supply and receiving works then proceed on parallel branches. Assembly waits for the later branch, followed by integrated testing and acceptance. The engine reports project duration 19.00 weeks and selected critical path A, B, D, E, F. These identifiers refer to the fictional tasks listed in the appendix. The path is a consequence of the entered topology and means; its membership can change when either changes.

The mean duration of equipment supply is 8.00 weeks. Its early start is 4.00 and early finish is 12.00, measured in relative weeks from the model origin. Receiving works finishes at 10.00. Assembly therefore waits for supply in this study. This identifies the binding branch under these assumptions. It does not prove supply is the only practical threat to delivery. Receiving can become critical if its duration changes, and omitted constraints can make both branches infeasible.

The critical-path result should be interpreted as a sensitivity of this network to its task means. A task with positive float can absorb some delay without moving the calculated finish, provided the rest of the model remains fixed. That proviso is essential. A delay can change resource availability, disrupt a review window or increase uncertainty on a successor. The static calculation does not capture those effects automatically. Float is a conditional network property, not a transferable reserve that one team can spend without affecting others.

The forward pass also exposes incomplete integration. If a readiness decision is omitted, the engine cannot wait for it. The interface-gate study inserts a separate decision between assembly and integrated testing. Expected completion becomes 22.00 weeks. The additional activity reveals a previously excluded condition. Calling this result worse than the shorter reference obscures the control improvement. The relevant question is which network represents the actual obligation; the more attractive output does not deserve preference simply because it is smaller.

The backward pass starts from the calculated project finish and asks how late each task can finish without moving that finish under the entered logic. A terminal task's late finish is the project duration. Other tasks use the earliest late start among their successors. Late start is late finish minus estimated duration. Float is late start less early start, subject to the engine's numerical safeguards. The reference receiving task has float 2.00 weeks. That is a property of the illustrative network, not a contractual allowance.

Float ownership is outside the calculation. A contract may specify how it is used, a sponsor may reserve it for integration, or teams may agree a control policy. The engine does not decide entitlement. A planner should avoid presenting computed float as permission to delay a deliverable. The delay could consume a shared margin, interfere with another resource or move an acceptance window that the simple network omits. The float register should state the model basis and the authority that may authorise reliance on it.

The duration-switch study reduces only the equipment supply assumptions. Its project duration is 17.00 weeks. Receiving works becomes critical, while supply gains float 2.00 weeks. The result demonstrates path switching. Improving the original critical task does not create unlimited improvement in project completion because another branch becomes binding. Recovery analysis should therefore recompute the complete network after each meaningful intervention rather than extrapolating savings from a single task in isolation.

Illustrative. Tied branches create another interpretive issue. In the tied-branches study the engine counts 2 critical paths. Both branches govern the same finish under the entered means. Selecting one path for display does not make the other irrelevant. A review that focuses only on the selected list can miss a concurrent route to delay. The output provides a path count and bounded path enumeration to disclose this. The uncertainty approximation still uses a selected path, which is a narrower result discussed in the uncertainty chapter.

A task can have zero float because it is part of a critical route, because a zero-duration milestone lies on that route, or because multiple equal branches connect to a shared successor. These cases deserve different management attention. The model's label does not rank operational difficulty or consequence. A trivial acceptance recording milestone and a long supply activity can both be critical. The action should follow the mechanism and evidence, not merely the presence of a coloured bar.

A fictional receiving owner sees positive float in a conditional network and postpones work to accommodate another assignment. The owner treats the displayed margin as a personal allowance. The integrated manager discovers that the accepting authority has a reserved review window and may be unavailable at the later finish. The network had omitted that calendar condition. Its float was mathematically correct within the entered model, but the proposed use was not operationally supported.

The manager preserves the calculation and adds the missing evidence to the decision. The owner can either retain the original readiness date, secure another valid review window or propose a different executable arrangement. The authority governing the shared margin should decide within the actual project rules. The planner cannot resolve entitlement by pointing to a float number. A resource or acceptance constraint can make a local postponement consequential even when the simple network's project duration does not move.

The control improvement is to pair reported float with its basis and permitted use. Where a margin depends on fixed review or resource windows, the operational programme should represent them appropriately. Where it is genuinely available, the authority can permit bounded use. The case does not imply that float must always be centrally reserved. It demonstrates why conditional network flexibility and actual permission to delay are separate questions. The result is a more executable forecast rather than a claim that the original arithmetic was wrong.

A critical-path review should trace each reported critical activity back to its scope, estimate basis and release evidence. Exact arithmetic on unsupported means produces an unsupported forecast. The review should also inspect near-critical branches and missing constraints. A receiving activity with modest float may be a greater operational concern than supply if its prerequisites are uncertain or its accepting party unavailable. The calculation helps locate conditional sensitivities; judgment determines which evidence and interventions deserve priority.

The app's dates are relative and its links are finish-to-start. Real programmes may use working calendars, start-to-start relations, lags, resources and external milestones. Those features can be valuable, but they complicate interpretation. The small model is useful as an inspectable reference precisely because it keeps the vocabulary narrow. It should not be presented as a replacement for an executable field programme when excluded features govern. A discrepancy between the app and another scheduling system should prompt a basis reconciliation, not an automatic judgment that one must be wrong.

Schedule targets should be shown alongside calculated outcomes without editing the network to match them. In the earlier-target study, only the deadline changes. Project duration remains 19.00 weeks. The target can change the selected-path deadline reading, but it does not accelerate work. This is a useful check against a common reporting error: a new desired date is announced and the programme is relabelled as if its logic now supported it. The forecast should retain its basis until an executable intervention changes that basis.

An action recommendation should name the affected mechanism. If supply is binding, investigate its release completeness, supplier capacity, inspections and logistics. If receiving becomes binding after supply improvement, investigate its own readiness and acceptance. If an omitted gate governs testing, resolve the gate's evidence and authority. General instructions to work harder do not follow from the network result. The model is most useful when it changes the question from which task is red to what credible intervention changes the accepted finish.

Retain the exact input and output with the review. This allows someone to reproduce the path, test a claimed intervention and distinguish model changes from display changes. A screenshot of a Gantt chart cannot establish the dependency topology or duration basis. The frozen studies in this paper provide input, output and scalar bindings so the published values can be checked against the canonical engine. Their reproducibility is evidence of calculation integrity, while their fictional nature remains an explicit limit on real-world inference.