Critical Path Analysis is used to schedule project activities without delay. It is useful in variety of fields such as construction, software engineering and facility expansion. Such analysis usually displayed as a 202204112118.
To find the critical path, that is an order of scheduled activities that should not be delayed to achieve the completion of the project, we need to find out the earliest completion, \(EC_w\), and the latest completion, \(LC_w\), for each activity.
The earliest completion time for each activity could be found by compute them in order, that is from the start of the event to the end of it. If there is a parallel (more than one event pointing to an event), choose the one that has the longest earliest completion time after addition of the duration. The mathematical equation to compute it is shown below:
$$ \begin{align} EC_1 &= 0 \\ EC_w &= \mathop{\text{max}}_{(v,w) \in E} (EC_v + c_{v,w}) \end{align} $$
For latest completion time, it should be computed in reverse order in contrast to the earliest completion time. If there is a parallel (an event point to more than one event), choose the one that has the shortest latest completion time after subtraction of the duration. The mathematical equation to compute it is shown below:
$$ \begin{align} LC_n &= EC_n \\ LC_v &= \mathop{\text{min}}_{(v,w) \in E} (LC_w - c_{v,w}) \end{align} $$
Slack Time for each activity, which determine how much time could an activity be delayed, could be found after the subtraction of its \(LC\) and \(EC\). The formula is shown as below:
$$ \text{Slack}_w = LC_w - EC_w $$
A critical path could be realised if a simple path of activities that does not have any slack time could be formed.