The Critical Path Method (CPM) - Construction Project Scheduling and Control - page 61

In mathematical terms, the late finish (LF) time for activity j ( LF j ) is as
follows:

LF j = min ( LS k ) (4.3)

where ( LS k ) represents the late start times for all succeeding activities.
Likewise, the late start (LS) time for activity j ( LS j ) is as follows:

LS j = LF j − Dur j (4.4)

The backward pass is defined as the process of navigating through a
network from finish to start and calculating the late dates for all activities.
This pass, along with the forward-pass calculations, helps identify the critical
path and the float for all activities.
If you refer to Figure 4.5b, you can see that for some activities (light lines),
the late dates (shown under the boxes) are later than their early dates (shown
above the boxes). For other activities (thick lines), late and early dates are
the same. For the second group, we can tell that these activities have strict
start and finish dates. Any delay in them will result in a delay in the entire
project. We call these activities critical activities . We call the continuous chain
of critical activities from the start to the end of the project the critical path .
Other activities have some leeway. For example, activity C can start on
day 5, 6, 7, 8, 9, or 10 without delaying the entire project. As mentioned
previously, we call this leeway float .
There are several types of float. The simplest and most important type of
float is total float (TF) :

TF = LS − ES or TF = LF − EF or TF = LF − Dur − ES (4.5)

We tabulate the results in the following table (boldface activities are
critical):

Activity Duration ES EF LS LF TF

A 5 05050

B 8 5 13 5 13 0

C 6 5 1 0 6 5

D 9 13 22 13 22 0

E 6 13 19 16 22 3

F 3 11 14 19 22 8

G 1 22 23 22 23 0

With the completion of the backward pass, we have calculated the late
dates for all activities. With both passes completed, the critical path is now
defined and the amount of float for each activity is calculated.