Critical path analysis (SACE Stage 2 General Mathematics)
“Use forward and backward scanning on an activity network to find the critical path, the minimum project time and the float of each activity.”
Critical path analysis schedules a project of dependent tasks. Forward scanning gives the earliest start times, backward scanning gives the latest start times, and the critical path is the longest path through the network; its length is the minimum completion time.
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What this dot point is asking
You must build an activity network, run forward and backward scans, find the critical path and minimum duration, and calculate the float of non-critical activities.
Activity networks and dependencies
A project is broken into activities, each with a duration. Some activities cannot start until others finish; these are immediate predecessors. A network diagram shows activities as edges (or nodes) with durations, respecting the order set by the predecessors.
The critical path is the longest path through the activity network from start to finish. Its total duration is the minimum possible project completion time, because every activity on it must be done in sequence with no spare time.
Forward scanning: earliest start times
Working left to right, the earliest start time (EST) of an activity is the latest of the earliest finish times of all its immediate predecessors. The earliest finish time is the earliest start plus the duration. The project's minimum completion time is the largest earliest finish time at the end.
Forward scan
A project has activities: A (3 days, no predecessor), B (4 days, after A), C (2 days, after A), D (5 days, after B and C). Find the minimum completion time.
A starts at 0 and finishes at 3. B starts at 3, finishes at 7. C starts at 3, finishes at 5. D needs both B and C finished, so it starts at the later finish, , and finishes at .
The minimum completion time is 12 days.
Backward scanning: latest start times
Working right to left from the project finish, the latest finish time (LFT) of an activity is the smallest of the latest start times of the activities that follow it. The latest start time is the latest finish minus the duration. An activity is critical when its earliest and latest start times are equal.
On the critical path, earliest start equals latest start for every activity, so there is no spare time. Delaying any critical activity delays the whole project. The critical path is the chain of these zero-float activities from start to finish.
Identifying the critical path
For the project above (finish at 12), back-scan to find the critical path.
D must finish by 12, so its latest start is . B must finish by 7 (when D starts), latest start . C must also finish by 7, latest start . A must finish by 3, latest start .
Comparing earliest and latest starts: A (0 = 0), B (3 = 3) and D (7 = 7) are critical, but C (earliest 3, latest 5) is not. The critical path is A to B to D, length days.
Float: the slack in non-critical activities
The float (or slack) of an activity is how long it can be delayed without delaying the whole project:
Calculating float
Find the float of activity C above.
C has earliest start 3 and latest start 5, so:
Activity C can be delayed by up to 2 days without pushing back the 12-day completion.
Crashing and interpretation
Reducing the project time ("crashing") means shortening activities on the critical path; shortening a non-critical activity does nothing for the overall duration until its float is used up. Examiners often ask which activity to speed up, and the answer must be a critical one.
After the forward scan, the largest earliest finish is the project duration; the backward scan should start from that same value. If the start node does not return to time 0 on the backward scan, you have an arithmetic error somewhere.
Do not take the smallest path as critical; the critical path is the longest path through the network, because every task on it must be completed in order. Also, when an activity has several predecessors, its earliest start is the latest (maximum) of their finishes, not the earliest; choosing the minimum understates the schedule.
Exam-style questions
Questions in the style of SACE Board exam questions on this dot point, each with a worked answer. They are written by ExamExplained unless tagged "Past paper"; the year shows the paper a question is modelled on.
SACE 2022-style2 marksCalculator-assumed. A council playground project has tasks A to H with given durations and prerequisites. A forward scan gives chain A then C then F then H with durations 2, 1, 2 and 4 weeks. Calculate the minimum completion time and name the critical path.Show worked answer →
The minimum completion time is the length of the longest (critical) path, found by a forward scan.
The chain A then C then F then H totals weeks, the longest path through the network.
So the minimum completion time is 9 weeks and the critical path is A-C-F-H. Marks: one for the duration of 9 weeks, one for naming the critical path.
SACE 2023-style1 markCalculator-assumed. A renovation network has a completed forward and backward scan. The company claims that allocating extra resources to task G, which has positive float, will reduce the minimum completion time. State why this claim is incorrect.Show worked answer →
Reducing a task's duration only shortens the project if that task lies on the critical path.
Task G has float (its earliest and latest start times differ), so it is non-critical. Speeding it up only increases G's float and does not change the earliest finish at the end node.
The minimum completion time is set solely by the critical path, so the claim is incorrect. The mark is for this reasoning.
SACE 2021-style1 markCalculator-assumed. On a landscaping network, task J has earliest start time 11 and latest start time 15. Calculate the float (slack) for task J.Show worked answer →
Float is the latest start time minus the earliest start time: days.
So task J can be delayed by up to 4 days without delaying the whole project. The mark is for the subtraction giving 4 days.