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Study notes

Qwestrum Engineering360 · Civil Engineering · Construction Management

CPM and PERT

Do a forward pass for earliest times and a backward pass for latest times, identify the critical path (zero-float activities) that fixes the project duration, and use PERT’s three time estimates for probabilistic scheduling.

Exam tip: keep SI units consistent end-to-end, write the governing relation symbolically before substituting, and sanity-check magnitude and sign.

Key formulas & points

Skim these first — then read the full notes below.

Forward pass: max EF; backward pass: min LS
Crash cost slope for time-cost trade-off on critical activities
$PERT probability: z = \frac{(D - T_{E})}{\sigma_{project}}$

Topic details

Introduction

CPM (Critical Path Method) and PERT (Program Evaluation and Review Technique) are network scheduling techniques that show activity dependencies and identify the project duration. CPM uses single, deterministic durations while PERT uses three estimates to handle uncertainty.

Scope in B.Tech and GATE syllabus

The forward pass computes the earliest start and finish of each activity; the backward pass computes the latest start and finish. The difference gives the total float — the slack an activity has without delaying the project. Activities with zero float form the critical path.

Why this topic matters in practice

The critical path is the longest path through the network and fixes the minimum project duration; any delay on it delays the whole project. Time-cost trade-off (crashing) shortens the project by expediting critical activities at extra cost, and PERT quantifies the probability of meeting a target date.

Key relations & formulas

Formulas (Indian textbook notation)

$CPM: ES, EF, LS, LF; Total float TF = LS - ES = LF - EF$

Formulas (Indian textbook notation)

$Critical path: activities with TF = 0$

Formulas (Indian textbook notation)

$PERT: t_{e} = \frac{(t_{o} + 4t_{m} + t_{p})}{6}; \sigma^{2} = (\frac{(t_{p} - t_{o})}{6})^{2}$

Notation and sign conventions

Relation 1 —

CPM: ES, EF, LS, LF; Total float TF = LS - ES = LF - EF

Formulas (Indian textbook notation)

$CPM: ES, EF, LS, LF; Total float TF = LS - ES = LF - EF$

Write this relation with symbols exactly as in Construction Management — PS Ghai before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.

Relation 2 —

Critical path: activities with TF = 0

Formulas (Indian textbook notation)

$Critical path: activities with TF = 0$

Write this relation with symbols exactly as in Construction Management — PS Ghai before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.

Relation 3 —

PERT: t_{e} =

Formulas (Indian textbook notation)

$PERT: t_{e} = \frac{(t_{o} + 4t_{m} + t_{p})}{6}; \sigma^{2} = (\frac{(t_{p} - t_{o})}{6})^{2}$

Write this relation with symbols exactly as in Construction Management — PS Ghai before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.

Fundamentals and definitions

The forward pass moves left to right taking the maximum earliest finish of predecessors as the earliest start of a successor; the backward pass moves right to left taking the minimum latest start of successors as the latest finish of a predecessor. These two passes bracket each activity’s scheduling window.

Governing relations in practice

Total float TF = LS − ES = LF − EF measures how much an activity can slip without affecting the project end; free float is the slack without affecting the next activity. Zero-float activities cannot slip at all — they are critical.

Design and analysis considerations

The critical path, being the longest chain of dependent activities, determines the project duration; there may be more than one critical path, and shortening the project requires shortening critical activities (crashing) considering the cost slope of each.

Advanced theory and extensions

PERT treats activity duration as uncertain, using optimistic, most-likely and pessimistic estimates to compute an expected time t_e = (t_o + 4t_m + t_p)/6 and a variance; summing variances along the critical path gives the project standard deviation, allowing the probability of finishing by a target date to be found from the normal distribution.

Assumptions and validity limits

State assumptions explicitly before using any relation for cpm and pert — steady state, uniform properties, linear elastic material, ideal gas, incompressible flow, etc., as applicable.

Wrong assumptions invalidate the entire solution even when the formula is correct. In Construction Management viva and GATE descriptive questions, listing valid assumptions often earns separate marks.

Step-by-step problem approach

1. Read the question and list given data with SI units (common in Construction Management papers).
2. Draw a neat labelled diagram where applicable — examiners in Indian universities award diagram marks even when arithmetic slips.
3. Identify which relation from this topic applies to cpm and pert.
4. Use equation 1:

CPM: ES, EF, LS, LF; Total float TF = LS - ES = LF - EF

.
5. Use equation 2:

Critical path: activities with TF = 0

.
6. Substitute values, compute, and verify units and sign (direction).
7. State conclusion in one line — e.g. safe/unsafe, stable/unstable, feasible/infeasible.

Applications & exam relevance

CPM and PERT appears in EPC and infrastructure projects. In Indian civil curricula this topic is tested because it connects theory to planning, scheduling, and contracts.

GATE and semester exams often combine cpm and pert with earlier units — revise prerequisites before attempting mixed problems.

Industry interview panels sometimes ask: "Where did you use cpm and pert?" — answer with a lab, mini-project, or plant visit example if possible.

Common mistakes in exams

• Taking the minimum (not maximum) EF at a merge node in the forward pass.
• Confusing total float with free float.
• Assuming a single critical path when parallel critical paths exist.
• Summing variances of non-critical activities for PERT probability.

Quick revision checklist

Before attempting cpm and pert problems, confirm you can:

1. Forward pass: max EF; backward pass: min LS
2. Crash cost slope for time-cost trade-off on critical activities
3.

PERT probability: z = \frac{(D - T_{E})}{\sigma_{project}}

Revise the solved examples in Construction Management — PS Ghai and one previous-year GATE or university paper for this unit.

Worked examples

Try the problem first — open the solution when you are ready to check.

PERT expected time and variance

Problem

An activity has optimistic time t_o = 4 days, most-likely t_m = 6 days and pessimistic t_p = 14 days. Find the expected duration and the variance.

Solution

Expected time t_e = (t_o + 4t_m + t_p)/6 = (4 + 4 × 6 + 14)/6 = (4 + 24 + 14)/6 = 42/6 = 7 days. Standard deviation σ = (t_p − t_o)/6 = (14 − 4)/6 = 1.67 days, so variance σ² = 1.67² = 2.78 days². These values feed into the project variance along the critical path for probability calculations.

Conceptual check — CPM and PERT

Problem

In a Construction Management semester or GATE paper you are asked: "State the main assumption, the governing relation, and one practical consequence of cpm and pert." What should a complete answer include?

Exams & GATE

PS Ghai — network drawing and critical path identification.

📖 Standard books (India)

Construction Management — PS Ghai
Read: Syllabus unit
CPM, PERT, and project planning

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