How do we find the shortest time to complete a project of dependent tasks?
Use forward and backward scanning on an activity network to find the critical path, the minimum project time and the float of each activity.
How to draw an activity network, use forward and backward scanning to find earliest and latest times, identify the critical path and minimum project duration, and calculate float.
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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.
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.
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.
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:
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.
Exam-style practice questions
Practice questions written in the style of SACE Board exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.
2019 SACE Stage 21 marksA council playground project involves tasks A to H, with a precedence table giving each task's time and prerequisites. Calculate the minimum completion time for the construction of the playground.Show worked answer →
The minimum completion time is the length of the longest (critical) path through the network, found by a forward scan.
Work out the earliest finish along each chain. From the table, A starts at 0 (2 weeks), C starts at 2 (1 week), then E (start 3, 3 weeks) and F (start 3, 2 weeks). E leads to G (start 6, 2 weeks) finishing at 8. F leads to H (start 5, 4 weeks) finishing at 9.
Comparing the chains, A then C then F then H gives 2 + 1 + 2 + 4 = 9 weeks, which is the longest path.
So the minimum completion time is 9 weeks. Award the mark for stating 9 weeks (the earliest finish at the end node).
2022 SACE Stage 21 marksCompany X's renovation network has a completed forward and backward scan. The company claims that allocating extra resources to task G will reduce the minimum completion time for the project. 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 slack: its earliest start time and latest start time differ, so it is a non-critical activity. Because G is not on the critical path, speeding it up only increases G's float; it does not change the earliest finish time at the end node.
The minimum completion time is determined solely by the critical path, so the claim is incorrect. Shortening a non-critical task such as G has no effect on the overall project duration.
2021 SACE Stage 21 marksOn a garden landscaping network with a completed forward scan, calculate the slack time for task J, where J has duration 8 days, earliest start time 11, and latest start time 15.Show worked answer →
Slack (float) for an activity is the difference between its latest start time and its earliest start time.
Slack = latest start time - earliest start time = 15 - 11 = 4 days.
This means task J can be delayed by up to 4 days without delaying completion of the whole project. A correct answer states the slack as 4 days, with the subtraction shown.