Amogh N P
✦ In loving memory of Amogh N P — Architect · Designer · Visionary ✦
A hand-drawn project network diagram on paper — event circles joined by activity arrows: the logic behind the schedule that a bar chart cannot show.
Unit IIProject Management

Project Programming & the Critical Path Method

Activities, events, dummies — and the forward and backward pass.

≈ 50 min + studio task

Replace the bar chart with a network that carries logic. Learn its elements — the activity, the event, and the zero-duration dummy that shows dependency without consuming time; the rules of construction and Fulkerson's numbering; AOA vs AON and the WBS behind them; and the heart of the course — CPM time analysis. Run the forward and backward pass, compute total, free and independent float, and find the critical path. Try the live CPM network explorer.

Learning objectives

By the end of this lesson, you will be able to — mapped to the course outcomes for Project Management:

1
CO2 · Understand

Identify network elements — activity, event, dummy — and the rules of network logic.

2
CO2 · Apply

Number events by Fulkerson's rule and build AOA / AON networks from a WBS.

3
CO2 · Apply

Run the forward and backward pass to find earliest and latest times.

4
CO2 · Analyse

Compute total, free and independent float and identify the critical path.

Activity, event, dummy

The network and its rules

An activity consumes time, an event marks an instant, and a dummy carries logic alone — drawn under rules (one start, one end, no loops) that make the diagram an honest map of what must precede what.[1]

Network elements (AOA) 12 A (5 d) activity = arrow (time + resource) event = node 34 dummy (0 d) logic only — no time, no resource A dummy is dotted and zero-duration — it carries a dependency or keeps two parallel activities uniquely identified. It never consumes time or resources. No loops, one start and one end, arrows left to right — the rules that make the diagram honest.
DiagramNetwork elements — an activity arrow, an event node, and a dotted zero-duration dummy showing logic only

The three elements

In an AOA (activity-on-arrow) network an ACTIVITY is a time- and resource-consuming task, drawn as an ARROW; an EVENT (node) is an instantaneous point in time — the start or finish of activities — drawn as a CIRCLE, consuming no time. A DUMMY is a dotted, ZERO-duration, ZERO-resource arrow used only to show a logical dependency or to keep two parallel activities uniquely identified. MISCONCEPTION→correct: 'a dummy activity consumes time or resources' — it does not; it carries logic alone.[1]

AOA vs AON AOA — activity on arrow AB needs dummies AON — activity on node (PDM) ABC no dummies Arrows in AON carry dependency only. MS Project and Primavera use AON / precedence diagramming.
DiagramAOA puts activities on arrows and needs dummies; AON puts activities in boxes and needs none, as software uses
Forward, backward, float

CPM time analysis

The forward pass finds earliest times, the backward pass the latest; the difference is float — total, free and independent — and zero float marks the critical path.[1, 3]

Earliest times

The FORWARD PASS finds how early things can happen. Start at time 0: an activity's EARLIEST START (EST) = the latest EARLIEST FINISH of its predecessors; its EARLIEST FINISH (EFT) = EST + duration. Carry the maxima forward through the network; the largest EFT at the end event is the PROJECT DURATION. In the worked example the forward pass gives a 20-day project.[1]

The three floats activity (earliest) total float → project end (LFT) activity (earliest) free float → successor's early start activity (earliest) independent float (preds late, succs early) Always: IF ≤ FF ≤ TF Total float is SHARED along a path — spend it on one activity and the successors lose it.
DiagramTotal, free and independent float — total float runs to the project end, free float to a successor's early start
Critical path B → D → F → G

The worked network

A seven-activity build gives a 20-day project with the critical path B → D → F → G; A, C and E carry float and can slip without delaying handover.

The worked network — critical path B → D → F → G (20 d) A · 4TF 2 C · 3TF 7 E · 4TF 7 B · 6TF 0 D · 5TF 0 F · 7TF 0 G · 2TF 0 Orange = zero float (critical). 6 + 5 + 7 + 2 = 20 days. A, C and E carry float and can slip. The longest, zero-float path sets the project duration — and can shift if you crash it.
DiagramThe worked network — activities A to G with the critical path B, D, F, G traced in orange for a twenty-day project
Interactive

Compute the critical path

Edit each activity's duration; the explorer reruns the forward and backward pass, lists every activity's float, and re-traces the critical path live. Stretch a floating activity past its float and watch the critical path shift.

CPM network · edit the durations

Dependencies are fixed (A,B → … → G); only durations change.

ActESTEFTLSTLFTTF
A04262
B06060
C4711147
D6116110
E71114187
F111811180
G182018200
Project duration20 days
Critical pathB → D → F → G

Activities with TF = 0 (orange) are critical. Stretch a floating activity and watch the critical path shift.

AOA vs AON

At a glance

AspectAOAAON
Activity shown asAOA: an arrowAON: a node / box
Dummies needed?AOA: yes (for logic/identity)AON: no
Events / nodesAOA: numbered circles (Fulkerson)AON: the activity boxes themselves
Used byAOA: classic hand methodsAON: MS Project, Primavera (PDM)
Float = 0 meansBoth: critical activityOn the critical path
Vocabulary

Key terms

Activity / event

A time-consuming task (arrow in AOA) / an instant in time (node).

Dummy activity

A zero-duration, zero-resource arrow that only shows a logical dependency.

Fulkerson's rule

Event-numbering so every arrow runs from a lower to a higher number.

Forward / backward pass

Computes earliest (EST/EFT) and latest (LST/LFT) times.

Total float

LFT − EST − duration: slack before the PROJECT end is affected.

Critical path

The continuous zero-float, longest path fixing the project duration.

Apply it

Studio task

For the seven-activity network in this lesson (A 4, B 6, C 3, D 5, E 4, F 7, G 2; C after A, D after A and B, E after C, F after D, G after E and F), compute EST, EFT, LST and LFT for every activity by hand, then the total float of each. State the critical path and the project duration, and confirm your answer against the explorer. Which activity has the largest float, and is it free or interfering?

Check your understanding

Self-assessment

1. A dummy activity in an AOA network —

2. Total float of an activity is —

3. The critical path is —

In a nutshell

Recap

A network carries logic the bar chart can't: activities, events, and zero-duration dummies for dependency.
Number events by Fulkerson's rule (lower → higher); AON needs no dummies and is what software uses.
Forward pass gives earliest times (EST/EFT); backward pass gives latest times (LST/LFT).
Total float = LFT − EST − duration and is shared along a path; free and independent float are stricter.
Critical activities have zero float; their chain is the longest, project-determining critical path — which can shift.
The evidence

References & further reading

  1. [1]B.C. Punmia & K.K. Khandelwal, Project Planning and Control with PERT and CPM — networks, Fulkerson's rule, floats, critical path.
  2. [2]PMI, PMBOK Guide & Practice Standard for Scheduling — WBS, precedence diagramming (AON/PDM).
  3. [3]J.D. Wiest & F.K. Levy, A Management Guide to PERT/CPM — critical-path computation and interpretation.
  4. [4]K.K. Chitkara, Construction Project Management — network analysis in Indian construction practice.

Further reading

  • B.C. Punmia & K.K. Khandelwal — Project Planning and Control with PERT and CPM.
  • J.D. Wiest & F.K. Levy — A Management Guide to PERT/CPM.
  • K.K. Chitkara — Construction Project Management.

Sources gathered and fact-checked June 2026. Published values vary by source, sample and method — treat as indicative and confirm against the cited standard before structural use.