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Discrete mathematics
Lesson 64 of 75
Challenge

Critical path analysis

Objective. Calculate earliest, latest and float times.

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Learn: the key idea

Critical path analysis models a project as a network of activities with durations and dependencies, used to find the minimum time to complete the whole project and identify which activities cannot be delayed without delaying the project. Early start times are calculated by working forwards through the network taking the latest of all preceding early finish times, while late finish times are found working backwards from the project end, taking the earliest of all succeeding late start times. Activities where early start equals late start (zero float) lie on the critical path, the longest path through the network.

Worked example

In a project network, activity X finishes with an early time of 8, and activity Y (which depends on both X and another activity finishing at early time 5) begins immediately after. Find the early start time of activity Y.

8 … 5
Illustrated card showing 8 … 5
  1. 1The early start of Y depends on all activities that must finish before it begins
  2. 2Two preceding early finish times are given: 8 (from X) and 5 (from the other activity)
  3. 3The early start of Y must be the later of these two times, since both must have finished
  4. 4The early start time of Y is 8

Answer: 8

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Common mistake. Students calculate early event times by taking the earliest incoming time rather than the latest, which understates how long preceding activities actually take to complete. At each event, the early time must be the maximum of all routes arriving there, not the minimum.

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