## Slack

Thus far in this discussion we have focused mostly on the events in the network. We found the EOTs foT the project milestones. It is now helpful to focus on the activities by finding their earliest starting times (EST) and latest possible starting times (LST). As noted in the previous section, the EST for an activity is equal to the EOT for the event from which the activity emanates. Activity i cannot start until event 5 has occurred. Event 5 has an EOT of 24 days, and so activity I has an EST of 24 days. An important question for the PM is this: What is the latest time (LST) activity 1 could start without making the entire project late?

Refer again to Figure 8-15. The project has a critical time of 43 days. Activity i requires 18 days to be accomplished. Therefore, 1 must be started no later than day 25 (43 - 18 = 25) if the project is to be complete on day 43. The LST for activity i is day 25. Because i cannot begin until event 5 has occurred, the latest occurrence time (LOT) for event 5 is also day 25. The difference between the LST and the EST for an activity is called its slack or float. In the case of activity i, it must be started no later than day 25, but could be started as early as day 24, so it has one day of slack. It should be immediately obvious that all activities on the critical path have zero slack. They cannot be delayed without making the project late.

For another example, consider activity f. Its EST is day 20, which is equal to the EOT for event 3 from which it emanates. The LST for activity f is 43 - 14 = 29. If f is started later than day 29, it will delay the entire project. Activity f has slack of LST -EST = 29 - 20 = 9 days.

To find the slack for any activity or the LOT for any event, we make a backward pass (right to left) through the network just as we made a forward pass (left to right) to find the critical path and time and the EOTs for all events (which are also the

ESTs for successor activities). There is one simple convention we must adopt: When there are two or more noncritical activities on a path, it is conventional to calculate the slack for each : activity as if it were the only activity in the path. Thus, when finding the slack for activity t, for example, we assume that none of I's predecessors are delayed, and that event 5 occurred on its EOT of day 24. Of course, if some activity, x, had six days of slack (given a specific EOT for the immediate preceding event), and if an earlier activity was late, causing the event to be delayed say two days, then activity x would have only four days of slack, having lost two days to the earlier delay.

It is simple to calculate slack for activities that are immediate predecessors of' the final node. As we move to earlier activities, it is just a bit more complicated. Consider activity g. Remembering our assumption that the other activities in the> same path use none of the available slack, we see that activity i must follow g, and that g emanates from event 3. Starting with the network's critical time of 43 days, we subtract 18 days for activity i and four more days for g (43 - 18 — 4 = 21). Thus g „ can begin no later than day 21 without delaying the network. The EST for g (EOT for* event 3) is day 20, so g has one day of slack.

To find the LOT for event 3, we must investigate each path that emanates from,1 it. We have already investigated two paths, one with activity g and one with activity f. Recall that f could start as late as day 29. For f not to delay the network, event 3* would have to.be complete not later than day 29. But activity g must start no later-than day 21, so event 3 must be complete by day 21 or the g-1 path will cause a delay. Now consider activity e, the only remaining activity starting from event 3 -Activity e must be completed by day 35 or event 6 will be late and the network will, be delayed. (Note that we do not have to work backward from the end of the network to find the slack for any activity that ends at a node on the critical path. All events and activities on the critical path have zero slack, so any activity ending on this path must arrive at event 6 not later than day 35.) The LST for e is 35 - 10 = 25 Its EST is day 20, so activity e has five days of slack.

We now can see that the LOT for event 3 is day 21, the most restrictive (earliest), time required, so that no activity emanating from it will cause the network to be[ late. Table 8-3 shows the LST, EST, and slack for all activities, and the LOT, EOT and slack for all events.

On occasion, the PM may negotiate an acceptable completion date for a pro] which allows for some slack in the entire network. If, in our example, an acceptab] date was 50 working days after the project start, then the network would have a totak. of 50 - 43 = 7 days of slack. This is the latest occurrence time minus the earliest occurrence time for the ending node, 7, of the network. /