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Chapter 4 of 10

Network analysis — CPM & PERT

In the TNPSC AE Civil syllabus under Project Management & Estimation · 2 parts

Network Analysis — CPM · PERT

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📑 Contents (17 sections)

Part 1 of 2

Network Analysis — CPM

Last reviewed 16 Sept 2026 · 8 min read

Critical Path Method (CPM)

CPM was developed in the late 1950s (by DuPont with Remington Rand) for planning and scheduling projects with deterministic (known) activity durations. It is activity-oriented and widely used in construction, where durations can be estimated from experience; it also forms the basis of time–cost trade-off (crashing).

Network terminology

Term	Meaning
Activity	A task consuming time and resources (shown by an arrow in AOA)
Event (node)	The start or completion of activities — a point in time, consumes no time or resources
Dummy activity	Fictitious activity of zero duration and no resources, drawn as a dashed arrow — used to show logical dependency or to give unique identification to parallel activities
Predecessor / successor	Activities immediately before / after an activity
Burst event	An event from which more than one activity starts
Merge event	An event at which more than one activity ends
Duration	Time estimated for an activity
Critical path	The longest path through the network; determines project duration

Types of network diagrams

Activity on arrow (AOA)

Activities on arrows, events as nodes (circles); activity identified by $(i, j)$ — tail and head events.
Needs dummies for correct logic.

Activity on node (AON) / precedence diagram

Activities in nodes (boxes); arrows show dependencies only.
No dummies needed; easier to draw and modify; used by most software.
Allows relationships FS, SS, FF, SF with lags/leads.

Rules for drawing AOA networks

Each activity is represented by one arrow only.
No two activities may have the same tail and head events — use a dummy.
There should be one initial and one final event (in a basic network).
Arrows flow from left to right; no looping (circular logic).
No dangling — every activity except the last must have a successor, and every event except the first must have a predecessor.
Avoid redundant dependencies.
The length of the arrow has no significance (not to scale).

Fulkerson's rule for numbering events

Number the initial event 1.
Delete all arrows emerging from numbered events; this creates new initial events — number them 2, 3, …
Repeat until the final event is numbered.

Result: for every activity $(i, j)$ , $i < j$ (head number greater than tail number).

Common errors

Looping (cycling) — endless loop of activities.
Dangling — an activity disconnected from the end event.
Redundancy — unnecessary dummy or link where dependency is already implied.

Time computations

FormulaEvent times (AOA)

Forward pass — earliest event time:

E_{1} = 0, E_{j} = i max (E_{i} + t_{ij})

Backward pass — latest event time:

L_{n} = E_{n} (or scheduled date), L_{i} = j min (L_{j} - t_{ij})

Event slack $= L - E$

FormulaActivity times for activity (i, j) of duration t

Earliest start $E S T = E_{i}$ ; earliest finish $E F T = E_{i} + t$
Latest finish $L F T = L_{j}$ ; latest start $L S T = L_{j} - t$

Floats

FormulaFloats of activity (i, j)

Total float

T F = L S T - E S T = L F T - E F T = L_{j} - E_{i} - t

The maximum time an activity can be delayed without delaying the project.

Free float

F F = E_{j} - E_{i} - t = T F - (L_{j} - E_{j})

Time an activity can be delayed without delaying the earliest start of any succeeding activity.

Independent float

I F = E_{j} - L_{i} - t = F F - (L_{i} - E_{i})

Delay possible when predecessors finish as late as possible and successors start as early as possible (if negative, taken as zero).

Interfering float $= T F - F F =$ head event slack $(L_{j} - E_{j})$

Relationship: $I F \leq F F \leq T F$ .
Critical activities have zero total float (when the project's scheduled completion equals its earliest completion). Negative float indicates the schedule cannot meet a target date.
The critical path joins critical activities from start to end; there may be more than one critical path.

Worked example 1 — AOA network

Worked ExampleCPM computations

Activity	(i–j)	Duration (days)	Predecessor
A	1–2	3	—
B	2–3	4	A
C	2–4	2	A
D	3–5	5	B
E	4–5	3	C
F	5–6	2	D, E

Forward pass: $E_{1} = 0$ ; $E_{2} = 3$ ; $E_{3} = 7$ ; $E_{4} = 5$ ; $E_{5} = max (7 + 5, 5 + 3) = 12$ ; $E_{6} = 14$

Backward pass: $L_{6} = 14$ ; $L_{5} = 12$ ; $L_{3} = 7$ ; $L_{4} = 12 - 3 = 9$ ; $L_{2} = min (7 - 4, 9 - 2) = 3$ ; $L_{1} = 0$

Activity	t	EST	EFT	LST	LFT	TF	FF
A	3	0	3	0	3	0	0
B	4	3	7	3	7	0	0
C	2	3	5	7	9	4	0
D	5	7	12	7	12	0	0
E	3	5	8	9	12	4	4
F	2	12	14	12	14	0	0

Check for C: $F F = E_{4} - E_{2} - 2 = 5 - 3 - 2 = 0$ ; interfering float $= 4 - 0 = 4$ (slack of event 4). For E: $F F = 12 - 5 - 3 = 4$ ; $I F = E_{5} - L_{4} - 3 = 12 - 9 - 3 = 0$ .

Critical path: A–B–D–F; project duration = 14 days.

Worked example 2 — using a dummy

Worked ExampleDrawing logic with a dummy

Activities: A and B start the project; C depends on A; D depends on both A and B.

Solution (AOA). Draw A (1–2) and B (1–3). C starts from node 2. D must follow both A and B: draw D from node 3 and add a dummy from node 2 to node 3 (so D waits for A), while C remains dependent only on A. Without the dummy, C would wrongly depend on B, or D would not depend on A.

Precedence diagram relationships (AON)

Relationship	Meaning
Finish-to-start (FS)	Successor starts after predecessor finishes (most common)
Start-to-start (SS)	Successor starts after predecessor starts (with lag) — e.g. plastering starts 3 days after brickwork starts
Finish-to-finish (FF)	Successor finishes after predecessor finishes
Start-to-finish (SF)	Successor finishes after predecessor starts (rare)
Lag / lead	Waiting time (e.g. curing) / overlap

AON node boxes typically show ES, EF, LS, LF, duration and total float.

Advantages of CPM

Shows interdependencies; identifies critical activities needing close control.
Computes floats — flexibility for resource allocation.
Basis for crashing, resource levelling, cost control and updating.
Effect of delays on the project can be analysed.

Frequently tested points

CPM: deterministic durations, activity-oriented, used for time–cost trade-off; PERT: probabilistic, event-oriented.
Dummy: zero time, zero resources, dashed arrow — logic or unique identification.
Fulkerson's rule numbers events so that $i < j$ for every activity.
Errors: looping, dangling, redundancy.
$E_{j} = max (E_{i} + t)$ ; $L_{i} = min (L_{j} - t)$ .
TF = L_j − E_i − t; FF = E_j − E_i − t; IF = E_j − L_i − t; interfering float = TF − FF = head slack.
IF ≤ FF ≤ TF; critical activities TF = 0; critical path = longest path; may be several.
AON needs no dummies; FS, SS, FF, SF relationships with lags.

Common MistakeCommon mistakes

Taking the minimum instead of the maximum at merge events in the forward pass (and vice versa in the backward pass).
Calling the shortest path the critical path.
Assuming free float equals total float for every activity.

Revision SummaryChapter summary

CPM is a deterministic, activity-oriented network technique identifying the critical path and floats.
Networks may be activity-on-arrow (with dummies and Fulkerson numbering) or activity-on-node (precedence diagrams).
Forward and backward passes give earliest and latest event and activity times.
Total, free, independent and interfering floats measure scheduling flexibility; critical activities have zero total float.
Precedence relationships (FS, SS, FF, SF) with lags model real construction logic.

Part 2 of 2

PERT

Last reviewed 16 Sept 2026 · 5 min read

Programme Evaluation and Review Technique (PERT)

PERT was developed in 1958 for the US Navy's Polaris missile programme (with Booz Allen Hamilton and Lockheed). It is used for projects with uncertain activity durations — research, development and first-of-a-kind projects.

Features

Probabilistic — each activity has three time estimates.
Event-oriented — emphasis on events (milestones).
Estimates the probability of completing the project by a scheduled date.
Uses slack (for events) and the critical path.

Three time estimates

Estimate	Meaning
Optimistic time $t_{o}$	Shortest possible time if everything goes exceptionally well
Most likely time $t_{m}$	Time required under normal conditions (mode)
Pessimistic time $t_{p}$	Longest time if things go badly (excluding major catastrophes)

Activity durations are assumed to follow a beta (β) distribution, which can be skewed either way and has finite end points.

FormulaExpected time and variance of an activity

t_{e} = \frac{t _{o} + 4 t _{m} + t _{p}}{6}

σ = \frac{t _{p} - t _{o}}{6}, σ^{2} = (\frac{t _{p} - t _{o}}{6})^{2}

If $t_{p} - t_{m} > t_{m} - t_{o}$ , the distribution is skewed to the right and $t_{e} > t_{m}$ .
A larger range $t_{p} - t_{o}$ means more uncertainty.

Project duration and probability

FormulaProject completion (central limit theorem)

Expected project duration $T_{E} = \sum t_{e}$ along the critical path.
Project variance $σ_{T}^{2} = \sum σ^{2}$ of critical activities; standard deviation $σ_{T} = \sum σ^{2}$ (standard deviations are not added).
The project duration is assumed normally distributed.

Probability of completion by scheduled time $T_{S}$ :

Z = \frac{T _{S} - T _{E}}{σ _{T}}, P (T \leq T_{S}) = Φ (Z)

Time for a desired probability:

T_{S} = T_{E} + Z σ_{T}

Useful normal distribution values

Z	−2	−1	0	0.5	1.0	1.28	1.5	1.645	2.0	3.0
Φ(Z)	0.0228	0.1587	0.500	0.6915	0.8413	0.90	0.9332	0.95	0.9772	0.9987

$T_{S} = T_{E}$ → probability 50%.
$T_{E} \pm 1 σ$ covers ≈ 68%, $\pm 2 σ$ ≈ 95.4%, $\pm 3 σ$ ≈ 99.7% of outcomes.

Slack

Event slack $= T_{L} - T_{E}$ (latest allowable minus earliest expected time).
Positive slack — ahead of schedule/resources to spare; zero slack — critical; negative slack — behind schedule (when the scheduled date is earlier than the expected date).

When two paths have close durations, the path with higher variance may govern the probability; standard practice uses the critical path, but near-critical paths should be checked.

Worked example

Worked ExamplePERT network

Activity	Events	$t_{o}$	$t_{m}$	$t_{p}$
A	1–2	2	4	6
B	2–3	3	5	13
C	2–4	2	3	4
D	3–5	4	6	8
E	4–5	5	8	11

(Durations in weeks.)

Step 1 — expected times and variances

Activity	$t_{e}$	$σ^{2}$
A	(2 + 16 + 6)/6 = 4	(4/6)² = 0.444
B	(3 + 20 + 13)/6 = 6	(10/6)² = 2.778
C	(2 + 12 + 4)/6 = 3	(2/6)² = 0.111
D	(4 + 24 + 8)/6 = 6	(4/6)² = 0.444
E	(5 + 32 + 11)/6 = 8	(6/6)² = 1.000

Step 2 — paths

A–B–D: 4 + 6 + 6 = 16 weeks; variance = 0.444 + 2.778 + 0.444 = 3.667
A–C–E: 4 + 3 + 8 = 15 weeks; variance = 1.556

Critical path A–B–D, $T_{E} = 16$ weeks, $σ_{T} = 3.667 = 1.915$ weeks.

Step 3 — probability of completion in 18 weeks $Z = \frac{18 - 16}{1.915} = 1.04$ → $P \approx 0.85$ (about 85%)

Step 4 — probability of completion in 14 weeks $Z = \frac{14 - 16}{1.915} = - 1.04$ → $P \approx 0.15$ (about 15%)

Step 5 — duration for 95% confidence $T_{S} = 16 + 1.645 \times 1.915 =$ 19.15 weeks

Worked ExampleSingle activity

An activity has $t_{o} = 4$ , $t_{m} = 7$ , $t_{p} = 16$ days. Find $t_{e}$ and $σ$ .

Solution. $t_{e} = \frac{4 + 28 + 16}{6} = 8$ days; $σ = \frac{16 - 4}{6} = 2$ days (variance 4)

PERT vs CPM

Aspect	PERT	CPM
Nature	Probabilistic	Deterministic
Time estimates	Three ( $t_{o}$ , $t_{m}$ , $t_{p}$ )	One
Orientation	Event-oriented	Activity-oriented
Distribution	Beta (activities), normal (project)	None
Used for	R&D, new, uncertain projects	Construction, repetitive projects with experience
Cost	Time is the main concern	Time–cost trade-off (crashing)
Critical measure	Slack (events)	Float (activities)
Origin	US Navy, Polaris (1958)	DuPont and Remington Rand (late 1950s)

Frequently tested points

PERT: probabilistic, event-oriented, three time estimates, beta distribution.
$t_{e} = (t_{o} + 4 t_{m} + t_{p}) /6$ ; $σ = (t_{p} - t_{o}) /6$ .
Project variance = sum of variances on the critical path; $σ_{T} = \sum σ^{2}$ .
$Z = (T_{S} - T_{E}) / σ_{T}$ ; $T_{S} = T_{E}$ gives 50% probability.
95% → Z = 1.645; 84% → Z ≈ 1; 97.7% → Z = 2.
Slack (PERT) vs float (CPM); negative slack → behind schedule.
CPM for construction and crashing; PERT for R&D.

Common MistakeCommon mistakes

Adding standard deviations instead of variances.
Using the most likely time instead of the expected time for the critical path.
Forgetting that probability is 50% when the scheduled time equals the expected time.

Revision SummaryChapter summary

PERT handles uncertain durations using optimistic, most likely and pessimistic estimates with a beta distribution.
Expected time is $(t_{o} + 4 t_{m} + t_{p}) /6$ and standard deviation $(t_{p} - t_{o}) /6$ .
Project expected duration and variance are sums along the critical path, with a normal distribution assumed.
$Z = (T_{S} - T_{E}) / σ_{T}$ gives completion probabilities and required durations.
PERT (probabilistic, event-oriented, slack) complements CPM (deterministic, activity-oriented, float, crashing).

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Activity	t	EST	EFT	LST	LFT	TF	FF
A	3	0	3	0	3	0	0
B	4	3	7	3	7	0	0
C	2	3	5	7	9	4	0
D	5	7	12	7	12	0	0
E	3	5	8	9	12	4	4
F	2	12	14	12	14	0	0

Activity	t	EST	EFT	LST	LFT	TF	FF
A	3	0	3	0	3	0	0
B	4	3	7	3	7	0	0
C	2	3	5	7	9	4	0
D	5	7	12	7	12	0	0
E	3	5	8	9	12	4	4
F	2	12	14	12	14	0	0

Activity	t	EST	EFT	LST	LFT	TF	FF
A	3	0	3	0	3	0	0
B	4	3	7	3	7	0	0
C	2	3	5	7	9	4	0
D	5	7	12	7	12	0	0
E	3	5	8	9	12	4	4
F	2	12	14	12	14	0	0