## The concept of free float

Students often find it difficult to understand the concept of free float. The mathematical definitions are unhelpful, and the graphical representation on page 108 can be confusing. The easiest way to understand the difference between total float and free float is to inspect the end node of the activity in question. As stated earlier, free float can only occur where two or more activities enter a node. If the earliest end times (i.e. the forward pass) for each individual activity are placed against the node, the free float is simply the difference between the highest number of the earliest time on the node and the number of the earliest time of the activity in question.

In the example given in Figure 21.5 the earliest times are placed in squares, so following the same convention it can be seen from the figure (which is a redrawing of Figure 21.1 with all the earliest and latest node times added) that

HA BA

Figure 21.5

HA BA

Figure 21.5

 0 3 3 A 0 3 3
 3 5 8 B 3 0 8 11 FF -5 6 3 2 5 D 11 8 13
 8 2 10 C 8 0 10 8 3 11 E 10 2 13
 10 4 14 F 10 0 14
 11 1 12 G 13 2 Figure 21.6 shows the equivalent precedence (AoN) diagram from which the free float can be easily calculated by subtracting the early finish time of the preceding node from the early start time of the succeeding node. Free float of activity D = 11 - 5 = 6 Free float of activity G = 14 - 12 = 2 Activity E, because it is not on the critical path has total float of 13 - 11 = 2 but has no free float. The check of the free float by the formal definition is as follows: Free float = TEe — TEB — D For activity D = 11 — 3 — 2 = 6 For activity G = 14 — 11 — 1 = 2 The check of the total float by the formal definitions is as follows: Total float = TLE — TEb — D For activity E = 13 — 8 — 3 = 2 D = 13 — 3 — 2 = 8 G = 14 — 11 — 2 = 2 It was stated earlier that total float is the same as beginning slack. This can be shown by rewriting the definition of total float = TLE — TEb — D as total float = TLE — D — TEb but TLE — D = TLb. Therefore = Beginning slack To show that free float = total float - end slack, consider the following definitions: Subtracting equation (21.3) from equation (21.2) = TLE - TEB - D - (TLE - TEE) = TLE - TEb - D - TLE + TEE = Free float Therefore equation (21.1) = equation (21.2) - equation (21.3) or free float = total float - end slack.