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Project Planning and Critical Path Analysis

Key Terms

An activity network represents a project as a directed graph of activities
EST (Earliest Start Time)
: the earliest an activity can begin (forward scan)
LST (Latest Start Time)
: the latest an activity can begin without delaying the project (backward scan)
Float
= LST − EST: how much an activity can be delayed without affecting the project
Critical activity
float = 0 (cannot be delayed)
Critical path
the longest path through the network = minimum project duration
The minimum completion time = sum of durations along the critical path
Forward scan (EST):
EST(activity) = max EST of all immediate predecessors + their duration

Backward scan (LST):
LST(activity) = min LST of all immediate successors − duration of this activity

Float: Float = LST − EST

Critical path: path where every activity has float = 0
Worked Example: Project with activities A(3), B(5), C(2), D(4), E(3). Predecessors: A–none; B–none; C–A; D–B; E–C,D.

Forward scan: EST(A)=0, EST(B)=0, EST(C)=3, EST(D)=5, EST(E)=max(3+2, 5+4)=max(5,9)=9
Project duration = 9+3 = 12

Backward scan: LST(E)=9 (must end at 12), LST(D)=9−4=5, LST(C)=9−2=7, LST(B)=5−5=0, LST(A)=7−3=4

Float: A=4−0=4, B=0−0=0, C=7−3=4, D=5−5=0, E=9−9=0
Critical path: B → D → E (all zero float), duration = 5+4+3 = 12
Hot Tip: The critical path is the longest path through the network (not the shortest). Speeding up a non-critical activity saves no time at all. To shorten a project, you must reduce the duration of a critical activity.

What Is Critical Path Analysis?

Critical Path Analysis (CPA) is a project management technique that identifies the most time-sensitive sequence of tasks in a project. It answers: “What is the shortest possible time to complete this project, and which tasks cannot be delayed?”

Setting Up an Activity Network

A project is broken into activities, each with a duration and a list of predecessor activities that must be completed first. This information is usually presented as a table.

The network can be drawn as an Activity on Node (AoN) diagram, where vertices represent activities and directed edges show the sequence.

Forward Scan

Work left to right through the network. For each activity, EST = maximum (EST + duration) of all predecessors. This gives the earliest the activity can begin.

Backward Scan

Work right to left. Start with LST = EST for the final activity (or a target completion time). For each activity, LST = minimum (LST of successors − duration of this activity). This gives the latest it can begin without delaying the project.

Float and the Critical Path

Float = LST − EST. Activities with float = 0 are critical — they cannot be delayed without extending the project. The sequence of critical activities forms the critical path, which gives the minimum project duration.

Strategy: In a table, write EST and LST side by side for each activity. Activities where EST = LST are on the critical path. After finding the critical path, read off the minimum project duration from the last activity’s EST + duration.

Mastery Practice

See Answers ➔
  1. Fluency A project has activities A(4), B(6), C(3). B must follow A; C must follow A. Find the EST of B and C, and the minimum project completion time.
  2. Fluency An activity has EST = 8 and LST = 12. What is its float? Is it a critical activity?
  3. Fluency A project network has these activities and predecessors:
    A(5) — no predecessor; B(3) — predecessor A; C(7) — predecessor A; D(2) — predecessor B,C.
    Perform a forward scan to find the EST of each activity and the minimum project duration.
  4. Fluency For the network in Question 3, perform a backward scan and find the LST and float for each activity. State the critical path.
  5. Understanding A construction project has the following activities:
    ActivityDuration (days)Predecessors
    A4—
    B5—
    C3A
    D6B
    E4C, D
    F2E

    Find all ESTs, LSTs, floats, and the critical path. What is the minimum project duration?
  6. Understanding A project manager wants to finish the project from Question 5 two days early. Which activities can be shortened (and by how much) to achieve this, and which cannot?
  7. Understanding A project network has 7 activities. The critical path takes 20 days. Activity G is not on the critical path and has float 4. What happens to the project completion time if:
    • (a) Activity G is delayed by 3 days?
    • (b) Activity G is delayed by 5 days?
    • (c) A critical activity is sped up by 2 days?
  8. Understanding A project has 8 activities with durations and predecessors given below. Find the critical path and minimum project duration.
    ActivityDurationPredecessors
    A3—
    B5—
    C4A
    D2A
    E6B, C
    F3D
    G4E, F
    H2G
  9. Problem Solving A software project has activities A–H. Activity G is on the critical path. The project manager has extra budget to reduce one activity’s duration:
    • Option X: reduce activity G from 6 days to 4 days (cost: $2000)
    • Option Y: reduce activity F from 4 days to 2 days (F is NOT on the critical path, float = 3 days)
    Which option should the project manager choose and why?
  10. Problem Solving A manufacturing process has 10 activities. Two parallel critical paths exist, each taking 25 days: Path 1: A → C → F → H (durations 5+8+7+5=25) Path 2: B → D → G → H (durations 6+9+5+5=25)
    • (a) What is the minimum project duration?
    • (b) Can the project be completed in 22 days if activity C is reduced to 5 days? Explain.
    • (c) What activities must be reduced (and by how much) to achieve a 22-day completion?

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