CPM (Critical Path Method) and PERT (Program Evaluation and Review Technique) are the two foundational project-scheduling techniques used to plan, monitor, and control construction and engineering projects — from building the network diagram itself, through forward/backward pass and float calculations, time-cost trade-off (crashing), PERT's probabilistic treatment of uncertain durations, to Gantt charts and resource scheduling. Every formula, worked step-by-step example, diagram, and exam-pattern table is included.

Learning Objectives

After studying this chapter you will be able to:

• Construct AOA and AON project networks, correctly identifying where dummy activities are required
• Perform the forward pass (earliest times) and backward pass (latest times) through a network
• Compute Total Float, Free Float, and Independent Float, and identify the critical path
• Apply CPM time-cost trade-off (crashing) using cost slope to find the optimal project duration
• Apply PERT's three-time-estimate model to compute expected duration, variance, and completion probability
• Interpret Gantt charts and recognise their limitations relative to network techniques
• Distinguish resource levelling from resource-constrained allocation and apply basic heuristic scheduling rules

Exam Relevance

Concept Map

Where This Fits

Prerequisite: Highway Engineering (large-scale road and pavement construction projects rely on the network scheduling techniques developed here to plan and monitor progress). CPM-PERT completes the civil engineering study material series on Mock Test Mitra — from here, head back to Study Material to revisit any subject or start practising topic-wise mock tests.

1.1 Network Terminology

1.2 AOA vs AON Networks

AON networks have largely superseded AOA in modern practice specifically because they eliminate the need for dummy activities altogether and natively support the four dependency types (Finish-to-Start, Start-to-Start, Finish-to-Finish, Start-to-Finish) with lags — but AOA remains the standard teaching format for GATE/ESE since the classical PERT technique was originally developed on arrow diagrams.

1.3 Rules for AOA Network Construction

A valid AOA network follows strict conventions: each activity gets exactly one arrow; whenever two activities would otherwise share the same start and end node (making them indistinguishable), a dummy activity (drawn as a dashed arrow with zero duration and zero resource consumption) is inserted to break the ambiguity; the whole network must reduce to a single start event and single end event; no circular dependencies (loops) are permitted; and every event is uniquely numbered.

2.1 Event Times

2.2 Activity Times

2.3 Forward Pass (Earliest Times)

The forward pass sweeps through the network from the start event (time zero) toward the end, computing the earliest time each event can occur. Whenever a node has multiple incoming activities (a merge point), its earliest event time must be the maximum of all the incoming paths' earliest finish times — because an event genuinely cannot occur until every activity leading into it has completed, so the slowest incoming path governs. The earliest time of the final node directly gives the overall project duration.

2.4 Backward Pass (Latest Times)

The backward pass sweeps in the opposite direction, from the end event (set equal to the project deadline) back toward the start, computing the latest time each event can occur without delaying the project. Where a node has multiple outgoing activities (a burst point), its latest event time must be the minimum of all the outgoing paths' latest start times — since the event must be ready early enough to satisfy the most demanding (earliest-deadline) successor. A completed backward pass should always recover \(L_1=E_1=0\) at the start node — this is the standard consistency check that the calculation was done correctly.

3.1 Types of Float

3.2 Relationships Between Float Types

The three float measures always satisfy \(IF\le FF\le TF\), reflecting progressively looser assumptions about the surrounding activities: Independent Float assumes the worst case for both neighbours (predecessors finish as late as possible, successors must start as early as possible), Free Float assumes predecessors finish at their earliest, and Total Float makes no allowance for neighbours at all — it's the delay possible if the activity had the entire path's slack to itself. Free Float can always be consumed without affecting any other activity's schedule, whereas Total Float is shared along a path — using it on one activity reduces the float remaining for others sharing that same slack.

3.3 Critical Path

The critical path is the longest path through the network from start to finish, and every activity on it has zero total float (and consequently zero free float and zero independent float too) — meaning any delay on a critical activity delays the entire project. The critical path can be identified either by marking every activity with \(TF=0\) and tracing the connected chain from start to end, or by exhaustively listing every path through the network and comparing total durations. A network is not guaranteed to have only one critical path — multiple critical paths can coexist, all sharing the same (longest) duration, which is a critical consideration when crashing (since shortening only one critical path leaves the others still governing project duration).

4.1 Normal vs Crash

4.2 Cost Slope

The cost slope expresses how much direct cost rises for every unit of time saved by crashing a particular activity — the additional resources (overtime, extra crews, expedited materials) needed to compress that activity's duration. Since the goal is to reduce project duration as cheaply as possible, the activity with the lowest cost slope among the critical activities should always be crashed first.

4.3 Crashing Procedure

Crashing proceeds iteratively: identify all currently critical activities (\(TF=0\)), select the one with the lowest cost slope, crash it by one time unit, then recheck the critical path — because crashing one activity can reduce its path's duration enough that a previously non-critical path becomes newly critical (or ties for critical) alongside it. When multiple critical paths exist simultaneously, activities on every critical path must be crashed together in that step (potentially by crashing a single shared activity, if one exists on all paths, or several activities in parallel otherwise) — crashing only one path leaves project duration unchanged since the other critical path(s) still govern.

4.4 Indirect Costs and the Optimal Duration

Direct costs (labour, materials, equipment for the activities themselves) rise as duration is compressed, while indirect costs (site overheads, supervision, equipment rental) fall as duration shortens, since less time on site means less overhead accumulated. The total project cost is the sum of both, and it traces out a U-shaped curve as duration varies — the optimal duration is the point of minimum total cost, found where the marginal cost of crashing one more day just equals the marginal indirect-cost saving from finishing one day sooner. Beyond this point, further crashing costs more than it saves.

5.1 PERT vs CPM

CPM assumes each activity's duration is known with confidence, since construction tasks are typically well-understood and repeatable; PERT was developed for exactly the opposite situation — genuinely novel R&D work (originally, missile development) where a single duration estimate would be unreliable, so PERT instead models each activity's duration as a random variable.

5.2 Three Time Estimates

5.3 Expected Time and Variance

PERT assumes each activity's duration follows a Beta distribution (bounded between \(t_o\) and \(t_p\), and unimodal at \(t_m\)) — chosen because it can flexibly represent skewed uncertainty (e.g., more room for things to go wrong than right) unlike a symmetric normal distribution. The expected duration weights the most-likely estimate four times as heavily as either extreme, since the mode is considered the single best guess, while the optimistic and pessimistic bounds each get equal but lesser weight. The variance is derived directly from the spread between the two extreme estimates — a wider gap between optimistic and pessimistic times signals greater uncertainty (larger variance) in that activity's actual duration.

5.4 Project Completion Probability

Because the Central Limit Theorem justifies treating the sum of many independent random activity durations as approximately normally distributed, the expected project duration is simply the sum of the individual expected activity times along the critical path, and — critically — the project's variance is the sum of the individual variances (never the sum of standard deviations, which is a very common mistake) along the same critical path. The probability of finishing by a target date is then found by converting the gap between the target and the expected duration into a standard normal Z-score, and reading the corresponding probability from the standard normal cumulative distribution.

6.1 Description

A Gantt chart (bar chart) is a horizontal bar graph representing the project schedule, developed by Henry Gantt around 1910–1915 — making it far older than CPM or PERT, yet it remains the most widely used tool for communicating schedules to non-technical stakeholders because of its intuitive visual format.

6.2 Features

6.3 Advantages and Limitations

A Gantt chart's core strength is also the source of its core weakness: its simple, visual bar-per-activity format is easy to read at a glance and communicates well with clients and field staff, but that same simplicity means it does not clearly show the logical dependencies between activities, nor does it explicitly identify the critical path — two things a network diagram shows immediately. For large, complex projects (roughly beyond 50 activities), a Gantt chart also becomes visually unwieldy and difficult to update when the underlying logic changes. In practice, the two tools are complementary rather than competing: the CPM/PERT network is used to plan the logic and identify the critical path, while the Gantt chart derived from it is used to communicate and track the schedule day-to-day.

7.1 Resource Levelling vs Resource Allocation

These two techniques represent opposite trade-offs: levelling holds the schedule (project duration) fixed and adjusts resource deployment within that constraint, whereas limited resource allocation holds the resource ceiling fixed and allows the schedule to stretch if the available workforce or equipment can't handle the theoretical peak demand.

7.2 Resource Levelling Procedure

Levelling begins from the earliest-start schedule, plotting a resource histogram (resource units required versus time) to visualise demand peaks and troughs. The key principle is that only non-critical activities may be shifted — shifting them later (within their available float) smooths the resource profile without delaying the project, whereas shifting a critical activity has no available float to absorb the shift and directly extends the project duration. Levelling quality is often measured by minimising the sum of squared resource usage across periods, since squaring penalises sharp peaks much more heavily than a simple sum would, favouring a genuinely smoother profile over one with occasional extreme spikes.

7.3 Heuristic Rules for Scheduling

When resources are limited and multiple activities compete for the same resource simultaneously, priority must be assigned by some rule — common heuristics include scheduling the activity with the least total float first (protecting the most schedule-critical work), the shortest-duration activity first (freeing up the resource quickly for others), or the earliest latest-finish-time first. Resource-constrained project scheduling is known to be an NP-hard combinatorial problem in general, meaning no single algorithm guarantees the mathematically optimal schedule efficiently for large projects — these heuristic priority rules instead give practical, near-optimal solutions used throughout the industry.

9.1 Purpose & Types of Estimate

An estimate is the anticipated cost of a work computed before execution, from drawings and rates. Its accuracy grows as the design matures:

9.2 Methods of Taking Out Quantities

For a building, the quantity of each item (earthwork, foundation concrete, masonry) is obtained by two standard methods:

• Long-wall & short-wall (separate / individual wall) method: walls running in one direction are "long walls", those perpendicular are "short walls". Long-wall length is measured out-to-out = centre-line length + one breadth; short-wall length is measured in-to-in = centre-line length − one breadth. Quantity = length × breadth × depth for each.
• Centre-line method: the total centre-line length of all walls is multiplied once by the breadth and depth of the item. It is quicker but requires a deduction of one breadth for each junction (T- or cross-junction) of the centre lines to avoid double counting.

9.3 Earthwork Volumes

Earthwork for roads, canals and embankments is computed from cross-sectional areas taken at a regular interval \(d\) using the mid-section, trapezoidal (average end area), or prismoidal (Simpson's) rule — the prismoidal rule is the most accurate but needs an odd number of sections. A bar bending schedule (BBS) lists the shape, size, number and cut length of every reinforcement bar, giving the steel weight (via \(\text{weight} = d^2/162\ \text{kg/m}\) for a bar of diameter \(d\) mm).

9.4 Analysis of Rates

Rate analysis builds up the unit rate of an item by adding the cost of materials, labour and machinery for a unit quantity, plus water charges, contractor's profit (~10%) and overheads. It is the bridge between measured quantities and money, and is what makes a detailed estimate a priced tender.

9.5 Tendering, Contracts & Quality Control

Once an estimate is priced, work is awarded and executed through a management framework:

• Tendering process: invitation (open / limited / single / global), issue of tender documents, submission of the earnest-money deposit (EMD), opening and comparative statement, and award to the lowest responsive bidder (L1).
• Contract types: item-rate (schedule/unit-price), lump-sum, percentage-rate, cost-plus, and labour contracts — each allocating cost and risk differently between owner and contractor.
• Quality control (QC): the operational techniques — sampling, testing (cube tests, slump, field density) and statistical acceptance criteria — used to verify that materials and workmanship meet specification, distinct from quality assurance (QA), which is the planned system of procedures that gives confidence the QC will be met. IS 456 acceptance of concrete uses running-mean and individual-result criteria on characteristic strength.
• Productivity & operation cost: output per unit of labour/plant time; monitored to control the direct cost component and to feed realistic rate analysis.

10.1 Cost, Price & Value

Cost is the money spent to produce a property; value is its present worth in the market. Common value terms:

10.2 Depreciation & Obsolescence

Depreciation is the gradual loss in value of a property due to age, wear, and decay. Obsolescence is loss of value due to the asset going out of date (better designs, changed fashion, new regulations) even while physically sound. Common methods to compute depreciation:

• Straight-line method: equal depreciation each year; annual depreciation \(D = (C - S)/n\).
• Constant-percentage (declining-balance) method: a fixed percentage of the reducing book value is written off each year, so book value \(= C(1-r)^t\).
• Sinking-fund method: depreciation set aside is invested to accumulate the replacement cost over the life.
• Quantity-survey method: depreciation assessed by a detailed survey of the actual physical deterioration of each part.

10.3 Sinking Fund & Years' Purchase

A sinking fund is an amount set aside annually (and invested at compound interest) so that a required sum — the cost of replacement, less scrap value — is accumulated by the end of the asset's life. Years' purchase (YP) is the capital sum whose interest equals one unit of annual net income; for a perpetual income \(YP = 1/i\). The capitalised value of a property = net annual income × YP.

A. Key Formulae at a Glance

B. Exam-Angle Summary

C. Mnemonics & Memory Aids

Practice Questions

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