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3.8 · Paper 2B
Activity networks, early and late times, critical paths and float — how to plan a project and find the activities that cannot slip.
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A project is split into activities, each with a duration and a list of immediate predecessors (activities that must finish first). A precedence table records this, and an activity network draws it: a box for each activity with arrows from each immediate predecessor. Following an algorithm — a clear step-by-step procedure — is the same skill used to build these networks.
The forward pass finds each activity's early time (earliest start): take the largest of (early time + duration) over its predecessors. The backward pass finds each late time (latest finish): take the smallest of (late time of a successor − its duration). The minimum completion time is the longest route through the network.
Float = late time − early time − duration. Activities with zero float are critical; they form the critical path — a delay to any of them delays the whole project. A Gantt chart shows each activity as a bar with its float dashed. Shortening (crashing) only helps if you shorten a critical activity, and a new critical path may then appear.
A project has activities A (3 days), B (5 days), C (4 days, after A), D (3 days, after B) and E (3 days, after C and D). Find the minimum completion time and the critical path.
Forward pass: A and B start at 0. C starts at 3. D starts at 5. E must wait for both C (3+4 = 7) and D (5+3 = 8), so E starts at max(7, 8) = 8. Minimum completion time = 8 + 3 = 11 days.
Backward pass: E's late time is 11, so C and D must finish by 11 − 3 = 8. B must finish by 8 − 3 = 5; A by 8 − 4 = 4.
Float: A = 4−0−3 = 1; B = 5−0−5 = 0; C = 8−3−4 = 1; D = 8−5−3 = 0; E = 11−8−3 = 0. Critical path: B → D → E.
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