The Critical Path Method (CPM) - Construction Project Scheduling and Control - page 60

For this example, we have calculated two types of dates:

1. The expected completion date of the project: day 23
2. The earliest date when each activity can start and finish

These dates are called the early start (ES) and the early finish (EF) dates
for each activity. As you will soon learn, an activity cannot start earlier than
its ES date and cannot finish earlier than its EF date, but it may startorinish
later than these dates.
In mathematical terms, the ES time for activity j ( ES j )isasfollows:

ES j = max ( EF i ) (4.1)

where ( EF i ) represents the EF times for all immediately preceding activities.
Likewise, the EF time for activity j ( EF j ) is as follows:

EF j = ES j + Dur j (4.2)

where Dur j is the duration of activity j .
The forward pass is defined as the process of navigating through a network
from start to finish and calculating the early dates for each activity and the
completion date of the project. See Figure 4.5.

Solution: The Backward Pass

Now let us start from the end of the project and work our way back to the
start. We already know the end-of-project date 5 : day 23. Activity G must finish
by day 23. Its duration is only 1 day, so it must start no later than day 22
(23 − 1) so that it does not delay the project. Similarly, activities D, E, and F
must finish no later than day 22 so that they will not delay activity G. Through
simple computations, we can find their late start dates: activity F: 22 − 3 =
19; activity E: 22 − 6 = 16; and activity D: 22 − 9 = 13. Activity C must
finish before activities E and F can start. Their late start dates are 16 and 19,
respectively. Clearly, activity C must finish by the earlier of the two dates, day
16, so that it will not delay the start of activity E. Thus, its late start date is day
10 (16 − 6). Similarly, activity B must finish by the earlier of its successors'
late start dates: day 13 for D and day 16 for E. Therefore, the late finish date
for activity B is day 13, and its late start date is day 5 (13 − 8). The last
activity (from the start) is A: It must finish by the earlier of the late start dates
for activities B and C, which are day 5 for B and day 10 for C. Consequently, the
late finish date for activity A is day 5, and its late start date is day 0 (5 − 5).

5 For now, we'll start the backward pass with the project's completion date, as calculated in the forward pass .
Later on in this chapter, we will discuss other cases when an imposed finish date is implemented.