Critical Path and Float/Slack Analysis
Two passes, two floats
Every activity carries four dates. The forward pass moves left to right: ES = latest EF among predecessors (plus lag); EF = ES + duration. The project finish date is the largest EF. The backward pass moves right to left from that finish date: LF = earliest LS among successors (minus lag); LS = LF − duration.
Total float TF = LS − ES = LF − EF: how long an activity can slip without delaying the project finish. Free float FF = earliest ES of successors − EF: how long it can slip without disturbing any successor. Free float is always less than or equal to total float.
The critical path is the longest path through the network; it sets project duration and normally has zero total float. Several critical paths can exist. Near-critical paths (small float) carry almost as much delivery risk and matter in risk-oriented questions, not just schedule questions.
Traps. Confusing critical with "shortest" or "most activities"; assuming float belongs to one activity when it is shared along a chain (using it on one activity removes it from the others); adding floats along a path (they are not additive); assuming a critical activity always has zero free float (it can have positive free float where a non-critical branch merges into the same successor); and forgetting that an imposed external finish date makes float negative when the plan runs late.
Worked example. A(3) precedes B(4) and C(2); B and C both precede D(2). Forward: A 0–3, B 3–7, C 3–5, D 7–9. Backward: D LF 9 / LS 7; B LF 7 / LS 3; C LF 7 / LS 5; A LF 3 / LS 0. Total float: A 0, B 0, C 2, D 0. Free float of C = 7 − 5 = 2. Critical path A–B–D, duration 9 days.
Rule of thumb. Compute float as LS − ES every time; if it disagrees with LF − EF, you have an arithmetic error.