Let’s take a closer look at the network that we obtained after completing the forward and backward scan in 9.09:
Remember that the numbers above each vertex represent the earliest possible start time for the activity, and the numbers below represent the latest possible start time.
Notice that there is a single path from start to finish through vertices where the earliest start time and the latest start time are the same. If we highlight this path, we have what is called the critical path.
The same network, with the critical path highlighted. There is only one critical path in this network, but some may have more.
The difference between the top and bottom numbers is called the float time, and it tells you how much flexibility there is on the task. Tasks that have a float time of $0$0 lie on a critical path. Delays in these tasks will either result in an overall delay for the whole project, or additional expenditure to get things back on track (called crashing).
For the other tasks, you can see how much “extra time” there is available. The float times for tasks not on the critical path have been written in the vertex to show the extra time.
The float times for activities off the critical path have been indicated.
Interpreting these results can help with planning and managing a project that involves multiple tasks. For example, having the earliest and latest start times for each task, and available delay times, allows us to make some observations:
- The tasks $B$B, $F$F, $J$J, and $L$L cannot have any delay.
- Task $M$M can take up to $13$13 ($8+5$8+5) days even though it only needs $5$5 ($41-36$41−36).
- Task $G$G can take $16$16 ($6+10$6+10) days even though it only needs $10$10 ($31-21$31−21).
- The tasks lying above the critical path ($A$A-$C$C-$D$D-$E$E-$G$G) have only $6$6 days of leeway between all of them. If task $A$A takes $7$7 days instead of $1$1, all subsequent tasks must be started on their latest start date in order to avoid delays.
What else can you observe?
Practice questions
Question 1
The following network has the earliest and latest starting times marked at each vertex.
Determine the critical path through the network by listing the activities in order.
Write all activities on the same line, separated by commas.
Question 2
Given the following network with the critical path highlighted in red.
List all non-critical activities.
Write each activity on the same line, separated by commas.
Complete the table by determining the float time of each activity.
Activity Float time $A$A $\editable{}$ $C$C $\editable{}$ $E$E $\editable{}$
Question 3
In order to insert a window, several activities need to be performed to complete the project. The following table displays the project’s activities and their descriptions, dependencies, and durations.
| Activity | Description | Dependencies | Duration (hours) |
|---|---|---|---|
| $A$A | Buy handtools | - | $2$2 |
| $B$B | Buy raw material | - | $4$4 |
| $C$C | Cut a hole in the wall | $A$A | $3$3 |
| $D$D | Mix up cement | $B$B | $2$2 |
| $E$E | Lay thin film of cement on the hole's borders. | $C,D$C,D | $3$3 |
| $F$F | Insert window frame | $E$E | $1$1 |
| $G$G | Insert window | $F$F | $1$1 |
| $H$H | Insert sealant and clean smudges. | $G$G | $2$2 |
Which network correctly represents the information in the activity table?
A
B
C
DFind the earliest starting time (EST) for each vertex.
Vertex Start 1 2 3 4 5 6 Finish EST $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ Now find the latest starting time (LST) for each vertex.
Vertex Start 1 2 3 4 5 6 Finish EST $0$0 $2$2 $4$4 $6$6 $9$9 $10$10 $11$11 $13$13 LST $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ Determine a critical path through the network by listing the activities in order.
Write all activities on the same line, separated by commas
What is the duration of the critical path?
If the sealant arrived one hour after that activity's earliest start time, how would that affect the project finish time?
There would be no effect on the finish time for the project.
AThe project would be delayed by one hour.
BThe project would finish one hour earlier.
C