Activity networks and critical pathsAQA A-Level Further Maths: Revision notes

A critical path analysis models a project as a set of activities, each with a duration and immediate predecessors, which are the activities that must be completed before it can start. In an activity-on-node network each activity is drawn as a node, and an arrow from $P$ to $Q$ means that $P$ must finish before $Q$ can start. An activity with several arrows into it can start only when all its predecessors are finished. Activities with no predecessor start at time 0, and a single end point is reached after the final activities. To construct a network, list each activity once, join it from each of its immediate predecessors, and check that the arrows follow the table. Example: if $E$ depends on $B$ and $C$, there are arrows from $B$ and from $C$ into $E$.

The forward pass finds the earliest start time (EST) and earliest finish time (EFT) of every activity, working from the start. EST is 0 for activities with no predecessor. EFT $=$ EST $+$ duration. For any other activity, the EST is the maximum of the earliest finish times of all its immediate predecessors, because it must wait for the last one. The minimum completion time of the project is the largest EFT. Example: $F$ has predecessors $C$ (earliest finish 7) and $D$ (earliest finish 11), so $F$ starts at $\max(7,11)=11$.

The backward pass starts at the minimum completion time and works back to find the latest finish time (LFT) and latest start time (LST) of every activity. For the last activities LFT is the completion time. LST $=$ LFT $-$ duration. For any other activity, the LFT is the minimum of the latest start times of its immediate successors. The float of an activity is the amount of time it can be delayed without delaying the project: float $=$ LFT $-$ EST $-$ duration, or equivalently LST $-$ EST. An activity with float 0 is critical.

A critical activity has zero float: any delay in it delays the whole project. A critical path is a path of critical activities from the start to the end, and its length equals the minimum completion time. A project may have more than one critical path. Non-critical activities have positive float. Worked example: $A$ 3 (none), $B$ 2 (none), $C$ 4 ($A$), $D$ 5 ($A$, $B$), $E$ 3 ($C$, $D$). Forward pass: $A$ 0 to 3, $B$ 0 to 2, $C$ 3 to 7, $D$ 3 to 8, $E$ 8 to 11. Backward pass: $E$ 8 to 11, $C$ latest 4 to 8, $D$ latest 3 to 8, $A$ latest 0 to 3, $B$ latest 1 to 3. The completion time is 11 days and the critical path is $A,D,E$. $C$ has float $4-3=1$ and $B$ has float $1-0=1$.

Real projects change, so the model is refined. A delay to a critical activity delays the project by the same amount. A delay to a non-critical activity delays the project only by the amount it exceeds the float. Shortening an activity helps only if it lies on every critical path, and only until another path becomes critical. Example: if two critical paths each take 22 days, shortening an activity on one of them saves nothing, but shortening an activity common to both saves time. After any change, recalculate the forward and backward passes, or the lengths of all paths. Always describe the effect in context, with units.