Slack (also called float) measures how much an activity can be delayed without delaying the project. The critical path is the longest-duration path through the project network and determines the minimum project completion time. This page explains how to compute earliest and latest times, calculate slack, and identify the critical path using diagrams and numerical examples.

1. The Project Network Structure

A project network consists of activities connected by precedence relationships.

   (Start)
      |
     A (3)
      |
     B (5) ----\
      |         \
     C (4)       >--> E (6) --> (Finish)
      |         /
     D (2) ----/

• Letters = activities
• Numbers in parentheses = durations
• Arrows = precedence constraints

2. Step 1 — Forward Pass (Earliest Times)

Compute the earliest start (ES) and earliest finish (EF) for each activity.

\[ EF = ES + \text{duration} \]

For activities with multiple predecessors: \[ ES = \max(EF_{\text{all predecessors}}) \]

Forward Pass Example

Earliest project completion time = EF(E) = 20 days.

3. Step 2 — Backward Pass (Latest Times)

Compute the latest start (LS) and latest finish (LF) without delaying the project.

\[ LS = LF – \text{duration} \]

For activities with multiple successors: \[ LF = \min(LS_{\text{all successors}}) \]

Backward Pass Example

4. Step 3 — Compute Slack (Float)

Slack measures schedule flexibility:

\[ \text{Slack} = LS – ES = LF – EF \]

Slack Example

All activities have zero slack → they are all critical.

5. Step 4 — Identify the Critical Path

The critical path is the sequence of activities with zero slack.

In this example:

\[ A \rightarrow B \rightarrow C \rightarrow D \rightarrow E \]

Total duration = 3 + 5 + 4 + 2 + 6 = 20 days.

6. Visual Diagram of Critical Path

   (Start)
      |
   [A:3] *
      |
   [B:5] *
      |
   [C:4] *
      |
   [D:2] *
      |
   [E:6] *
      |
   (Finish)

* = critical activity (slack = 0)

7. Common Mistakes

• Using the minimum instead of maximum EF when computing ES for converging paths.
• Using the maximum instead of minimum LS when computing LF for diverging paths.
• Assuming the longest-duration activity is always critical (it depends on the network).
• Forgetting that slack = 0 is the defining feature of critical activities.

8. Summary

• Forward pass → earliest times.
• Backward pass → latest times.
• Slack = LS − ES.
• Critical path = activities with zero slack.