## Float

This is the name given to the spare time of an activity, and is one of the more important by-products of network analysis. The four types of float possible will now be explained.

### Total float

It can be seen that activity 3-6 in Figure 21.3 must be completed after 13 time units, but can be started after 8 time units. Clearly, therefore, since the activity itself takes 3 time units, the activity could be completed in 8 + 3 = 11 time units. Therefore there is a leeway of 13 - 11 = 2 time units on the activity. This leeway is called total float, and is defined as latest time of end event minus earliest time of beginning event minus duration, or TLE - TEB - D.

Figure 21.3 shows that total float is, in fact, the same as beginning slack. Also, free float is the same as total float minus end slack. The proof is given at the end of this chapter.

### Free float

Some activities, e.g. 5-6, as well as having total float have an additional leeway. It will be noted that activities 3-6 and 5-6 both affect activity 6-7. However, one of these two activities will delay 6-7 by the same time unit by which it itself may be delayed. The remaining activity, on the other hand, may be delayed for a period without affecting 6-7. This leeway is called free float, and can only occur in one or more activities where several meet at one event, i.e. if x activities meet at a node, it is possible that x - 1 of these have free float. This free float may be defined as earliest time of end event minus earliest time of beginning event minus duration, or TEE - TEB - D.

For a more detailed discussion on the use of floats, and a rapid manual method for calculating total float, see Chapter 24.

Interfering float

The difference between the total float and the free float is known as interfering float. Using the previous notation, this can be expressed as

(TLe - TEB - D) - (TEE - TEB - D) = TLE - TEB - D - TEE + TEB + D

i.e. as the latest time of the end event minus the earliest time of the end event. It is, therefore, the same as the end slack.

Independent float

The difference between the free float and the beginning slack is known as independent float:

since free float = TEE - TEB - D and beginning slack = TLB - TEB

independent float = TEE - TEB - D - (TLB - TEB) = TEe - TEB - D

Thus independent float is given by the earliest time of the end event minus the latest time of beginning event minus the duration.

In practice neither the interfering float nor the independent float find much application, and for this reason they will not be referred to in later chapters. The use of computers for network analysis enables these values to be produced without difficulty or extra cost, but they only tend to confuse the user and are therefore best ignored.

Summarizing all the above definitions, Figure 21.4 and the following expressions may be of assistance.

### Notation

D = duration of activity TEb = earliest time of beginning event TEE = earliest time of end event TLb = latest time of beginning event TLE = latest time of end event

Definitions

 beginning slack = TLB - TEB end slack = = TLE - TEE total float = = TLe - TEB

free float = TEE - TEB - D interfering float = TLE - TEE(= end slack)

Beginning event

End event

Earliest Latest time time

TEb TLB

Beginning , slack

Duration of activity

Duration of activity

Earliest Latest time time

TEe TLe End slack

Total float

Free float

Independent float

Interfereing float

Late free float

Figure 21.4 Floats independent float = TEe - TLB - D last free float = TLE — TLB — D