Critical Paths / Activities (Leaving Cert Applied Maths): Revision Notes

Critical Paths / Activities

Understanding early and late times

In project management, we need to understand two important timing concepts that help us schedule activities effectively.

The forwards and backwards pass method

To calculate these times in an activity network, we use a systematic two-step approach:

1. Forwards pass: We work our way through the network from start to finish, calculating early times as we go
2. Backwards pass: We work backwards from the end of the project, determining late times for each event

Let's see how this works with a practical example.

When we start our forwards pass, we begin at the source node (the starting point) and work through each activity systematically.

Calculating early times (forwards pass)

During the forwards pass, we follow these steps:

• Start at the source node with an early time of 0
• For each activity, add its duration to the early time of the preceding event
• When multiple activities converge at a node, take the largest early time value (this ensures all preceding activities are genuinely finished)

For example, if activities B and C both lead to the same event, and they give early times of 8 and 9 respectively, we choose 9 because this is when both activities will definitely be complete.

Calculating late times (backwards pass)

After completing our forwards pass, we work backwards to find late times:

• Start at the sink node (project end) where late time equals early time
• Work backwards, subtracting activity durations
• When multiple activities diverge from a node, take the smallest late time value to prevent project delays

The key principle is that late times represent deadlines - we must choose the most restrictive deadline to ensure the project stays on schedule.

Understanding total float

Total float measures how much an activity can be delayed without affecting the project's completion date. It represents the flexibility we have in scheduling that particular activity.

The formula for calculating total float is:

$\text{Total Float} = d - a - t$

Where:

• $d$ = late time of the ending event
• $a$ = early time of the starting event
• $t$ = duration of the activity

Let's apply this formula to calculate the total float for each activity in our example:

Identifying critical activities and paths

A critical activity is any activity with a total float of zero. This means there is no flexibility in its timing - any delay to a critical activity will delay the entire project.

A critical path is a continuous sequence of critical activities that runs from the project start to finish. This path determines the minimum project duration.

Practical applications

Understanding critical paths helps project managers:

• Focus attention on activities that could delay the project
• Allocate resources more effectively
• Identify where schedule compression efforts should be concentrated
• Plan for contingencies by knowing which activities have flexibility

Remember that if any critical activity is delayed, the entire project timeline shifts unless other adjustments are made elsewhere.