The critical path with non-zero time-lags: An anomaly or just a mistake?

The critical path is a series of project activities with a zero slack value connected by precedence relations (see “Scheduling projects: How to determine the critical path using activity slack calculations?”). It contains all activities that are critical for the project duration and results in the generally accepted critical path rule:

Critical path rule
“Increasing (decreasing) the duration of a critical activity leads to an increase (decrease) in the critical path length and the project duration”
However, in case more general precedence relations are used in a project network (the four types of relation are start-start, start-finish, finish-start and finish-finish, see “Activity links: How to add precedence relations between activities?”), a discrepancy or deviation from the established critical path rule might occur, leading to counterintuitive and often conflicting results.
This anomaly is illustrated by the three-activity project network as displayed in figure 1. The project network on top contains three activities with a duration equal to 6, 6 and 7 for activities 1, 2 and 3, respectively. Since the critical path length is equal to 10 days, as displayed in the Gantt chart at the top of figure 2, the project manager can decide to reduce the duration of an activity by 50% in order to decrease the total project duration. The new project network is shown in the bottom network of figure 1.
Figure 1. An example project network where the duration of activity 2 will be decreased by 50%
The bottom Gantt chart of figure 2 shows the counterintuitive result of the decrease in the duration of activity 2, which can be summarized in the following anomaly:
Anomaly in the critical path rule
“Increasing (decreasing) the duration of a critical activity leads to a decrease (increase) in the critical path length and the project duration”
In the example of figure 1, increasing the duration of a critical activity results in a decrease of the project duration, as follows:
  • If d2 = 6 then the project duration = 10
  • If d2 = 3 then the project duration = 13
Figure 2. An earliest start schedule for the example project

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