The critical path in relation to a series of inter-dependent projects or tasks is the set of those projects, which if delayed will cause the sequence as a whole to be delayed.
This is usually calculated using Program Evaluation Review Technique, or PERT, and shown on logic network diagrams with project interconnections and durations. Another common representation is the Activity-on-Node (AON) diagram which shows float and other variables from the calculations.
Lets look at how this is worked out.
Each project will have an expected duration (in AlchemyWorks Projects this is the remaining work) and there are a number of other variables which we are going to calculate;
There is also a potential lag between activities, as there is usually an overhead in switching between activities. In AlchemyWorks Projects this is set by the scheduling granularity in system settings.
The first thing we do is work forwards through the graph calculating the Earliest Start (ES) and Earliest Finish (EF) times for each activity. The ES we get from the latest EF from any preceding activity. The EF for any activity is simply it's ES plus the activity duration.
The second phase is to work backwards; for any activity which does not have a successor we can assign the Latest Finish (LF) to be the latest Earliest Finish (EF) across the graph. This is effectively the projected end time for the project sequence as a whole. The Latest Start (LS) for each activity can be set to it's LF less the duration. If the activity does have successors, then it's LF is the earliest of the LS values from it's immediate successors.
Once we have done this we can calculate the float values. The Free Float (FF) value is the lowest ES of an activity successors less it's EF. The Total Float (TF) value is just the difference between LS and ES (or LF and EF - it should be the same).
The critical path is simply the set of activities that have no float.
AlchemyWorks Projects show this critical path on Gantt charts with projects outlined in red. On PERT charts (example above) the critical path is indicated by the vectors, and on AON charts (Activity On Node) the project information boxes are coloured to indicate status or inclusion on a critical path.