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.
Reviewed by: AI editorial process; not yet individually human-reviewed
Have a quick question? Jump to the Q&A page
Jump to a section
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.
SACE 20222 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 20231 marksCalculator-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 20211 marksCalculator-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.