Minimum Completion Time
Learn how to find the minimum completion time of a project for NSW Year 12 Mathematics Standard 2. In this critical path analysis topic you read the minimum completion time straight off an activity network as the length of the critical path — the longest chain of activities from start to finish, equal to the earliest start time of the final event.
You will also learn the effect of changing an activity's duration: why speeding up a non-critical task changes nothing, why shortening a critical task can reduce the completion time, and how the critical path can move to a new set of activities — a core Standard 2 skill for planning real builds, events and product launches.
Theory
The minimum completion time of a project is the length of its critical path. This Year 12 Standard 2 (NSW) guide shows how to find it from an activity network, read it as the earliest start time of the final event, and work out the effect of changing an activity's duration on both the completion time and the critical path.
The minimum completion time of a project is the shortest time in which every activity in the project can be finished. In an activity network it is the length of the critical path — the longest chain of activities from the start event to the final event.
It has to be the longest path (not the shortest): the project is only complete once every activity is done, so the slowest chain of dependent tasks controls the finish. Equivalently, the minimum completion time is the earliest start time (EST) of the final event after a forward scan through the network.
Activities on the critical path have no float (spare time); activities off it have float. This decides the effect of changing an activity's duration: shortening a critical activity can reduce the completion time, while shortening a non-critical activity changes nothing.
The minimum completion time \(T\) is read straight from the network:
For a single start-to-finish path, its length is just the sum of its activity durations:
The spare time (float) on an activity from event \(i\) to event \(j\) is:
How to find the minimum completion time
- List every path from the start event to the final event.
- Add the durations along each path (or run a forward scan to get the EST of each event).
- Take the largest total — this is the minimum completion time, and the path that gives it is the critical path.
- To test a duration change, check whether the activity is on the critical path. If it has float, \(T\) is unchanged; if it is critical, re-scan every path to get the new \(T\) and see whether the critical path has moved.
Total the durations along each path; the longest is the minimum completion time.
| \(\text{Path }A\text{-}C\) | \(=\) | \(8+5 = 13\) |
| \(\text{Path }B\text{-}D\) | \(=\) | \(4+6 = 10\) |
| \(\text{Longest path}\) | \(=\) | \(13\text{ days}\) |
| \(\therefore\ T\) | \(=\) | \(13\text{ days}\) |
The critical path is \(A\text{-}C\).
The EST of the final event equals the minimum completion time.
| \(\text{EST}_5\) | \(=\) | \(\max(9{+}5,\ 7{+}6) = 14\) |
| \(\text{EST}_6\) | \(=\) | \(14+2 = 16\) |
| \(\therefore\ T\) | \(=\) | \(16\text{ hours}\) |
Critical path \(A\text{-}B\text{-}D\text{-}F\); the green lower-left number in each event is its EST.
Check whether \(B\) is on the critical path \(A\text{-}C\text{-}E\).
| \(\text{Critical }A\text{-}C\text{-}E\) | \(=\) | \(5+6+4 = 15\) |
| \(\text{New }B\text{-}D\text{-}E\) | \(=\) | \(5+2+4 = 11\) |
| \(\therefore\ T\) | \(=\) | \(15\text{ hours}\) |
Unchanged at \(15\) hours — \(B\) had float, so shortening it only adds more float.
\(D\) is critical, so re-scan every path after the cut.
| \(\text{New }B\text{-}D\) | \(=\) | \(5+3 = 8\) |
| \(\text{Path }A\text{-}C\) | \(=\) | \(7+2 = 9\) |
| \(\therefore\ T\) | \(=\) | \(9\text{ hours}\) |
It falls by only \(2\) hours (not \(3\)): the critical path switches to \(A\text{-}C\).
Common pitfalls
Frequently asked questions
What is the minimum completion time in critical path analysis?
It is the shortest time in which every activity in a project can be finished. On an activity network it equals the length of the critical path, which is the longest chain of dependent activities from the start event to the final event.
How do you calculate the minimum completion time?
List every path from start to finish, add the activity durations along each one, and take the largest total. That largest total is the minimum completion time, and the path that produces it is the critical path. A forward scan gives the same answer as the EST of the final event.
Why is the minimum completion time the longest path, not the shortest?
The project is only complete once every activity is done. The activities on the longest chain of dependencies cannot be overlapped, so that chain sets the earliest possible finish. Any shorter path finishes with time to spare, which is why it is not the binding one.
Does shortening an activity always reduce the completion time?
No. It only helps if the activity is on the critical path. A non-critical activity has float, so speeding it up just adds to that float and the completion time stays the same. Only critical activities control the finish time.
Why can the completion time fall by less than I shortened a critical activity?
Because shortening a critical activity can make a different path the new longest one. Once the original critical path drops below another path, that other path becomes critical and limits how much time you actually save, so always re-scan after the change.
Is the minimum completion time the same as the EST of the final event?
Yes. After a forward scan, the earliest start time of the final event is exactly the length of the critical path, which is the minimum completion time. They are two ways of describing the same value.