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5.3 Critical Path Method and Network Analysis

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NCEES PE Civil Construction Study Guide Exam

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1 Chapter 1: Soil Mechanics

1.1 Lateral Earth Pressure 1.2 Soil Consolidation and Settlement 1.3 Effective and Total Stresses 1.4 Bearing Capacity 1.5 Slope Stability

2 Chapter 2: Site Layout and Development

2.1 Layout and Control 2.2 Horizontal and Vertical Curves 2.3 Site Investigations and Utilities

3 Chapter 3: Material Properties

3.1 Soils, Rock, and Aggregates 3.2 Concrete Material Properties 3.3 Steel Material Properties 3.4 Wood Material Properties

4 Chapter 4: Estimating Quantities and Costs

4.1 Quantity Take-Off Methods 4.2 Cost Estimating 4.3 Engineering Economics 4.4 Work Measurement and Productivity (Earned Value)

5 Chapter 5: Project Planning and Scheduling

5.1 Construction Sequencing 5.2 Activity Time Analysis 5.3 Critical Path Method and Network Analysis 5.4 Resource Scheduling and Leveling 5.5 Time–Cost Trade-Off and Crashing

6 Chapter 6: Material Production, Execution, and Quality Control

6.1 Specification Conformance and Hold-Points 6.2 Field Testing: Concrete, Soils, and Aggregates 6.3 Compaction: Principles, Calculations, and Equipment 6.4 Concrete Placement, Consolidation, and Finishing 6.5 Bolting and Welding: Structural Steel Connections 6.6 Acceptance Sampling and Statistical Quality Control

7 Chapter 7: Structural Mechanics

7.1 Loads (Dead, Live, Construction, Wind, and Preloading) 7.2 Statics and Stresses (Bending, Shear, Deflection) 7.3 Beams, Columns, and Slabs 7.4 Foundation and Retaining Wall Mechanics

8 Chapter 8: Hydraulics and Hydrology

8.1 Stormwater Collection and Drainage (Open-Channel Flow) 8.2 Runoff Analysis 8.3 Detention and Retention Ponds

9 Chapter 9: Operations and Methods

9.1 Lifting and Rigging 9.2 Crane Capacity and Stability 9.3 Dewatering and Pumping 9.4 Equipment Selection and Production Rates 9.5 Deep Foundation Installation 9.6 Excavation and Embankment (Cut and Fill)

10 Chapter 10: Support of Construction Loads

10.1 Formwork Design 10.2 Falsework and Scaffolding 10.3 Shoring and Reshoring 10.4 Bracing and Anchorage 10.5 Support of Excavation (Trenching, Cofferdams) 10.6 Erection and Equipment Loads on Permanent Structures

11 Chapter 11: Health and Safety

11.1 OSHA Construction Regulations and Safety Management 11.2 Work Zone and Public Safety (Traffic Control)

5.3 Critical Path Method and Network Analysis

Key Takeaways Explain This Topic AI

The Precedence Diagramming Method (PDM) uses Activity-on-Node (AON) representation to build the schedule network.

The Forward Pass calculates the Early Start (ES) and Early Finish (EF) by adding duration and selecting the largest EF among predecessors.

The Backward Pass calculates the Late Finish (LF) and Late Start (LS) by subtracting duration and selecting the smallest LS among successors.

Total Float (TF) is the amount of time an activity can be delayed without delaying the project completion date. TF = LS - ES or LF - EF.

The Critical Path is the longest path through the network, consisting of activities with zero Total Float. Any delay on the critical path delays the entire project.

Last updated: July 2026

Critical Path Method (CPM) and Network Analysis

Why this topic matters for the PE Construction exam: CPM analysis is arguably the most important scheduling topic on the exam. You will be expected to calculate early and late dates, determine float, identify the critical path, and understand how changes in activity durations affect the overall project timeline.

The Critical Path Method (CPM) is a mathematical algorithm used to schedule a set of project activities. It is the industry standard for planning and executing construction projects. The foundation of CPM is the Precedence Diagramming Method (PDM) , which represents activities as boxes (nodes) linked by arrows showing logical dependencies. This is also known as Activity-on-Node (AON).

Anatomy of a Node

To perform a network analysis, each activity node must track six key pieces of data:

  • Activity Name / ID
  • Duration (D)

Early Start (ES): The earliest possible time the activity can begin.

Early Finish (EF): The earliest possible time the activity can end ( E S + D ES + D ES + D ).

Late Start (LS): The latest time the activity can begin without delaying the project.

Late Finish (LF): The latest time the activity can end without delaying the project ( L S + D LS + D L S + D ).

A common node layout is a 3x2 grid:

  • ES Duration EF
  • LS Float LF

(Note: There are different conventions for Node layouts, but the mathematical relationships remain identical).

The Forward Pass: Calculating Early Dates

The Forward Pass determines the shortest possible project duration by calculating the ES and EF for every activity, moving from the start of the project to the end.

Rules for the Forward Pass:

The first activity's Early Start (ES) is usually 0 (or day 1, depending on convention). For the PE exam, standard convention usually uses end-of-day 0, so start at 0.

Calculate Early Finish: E F = E S + Duration EF = ES + \text{Duration} EF = ES + Duration

The Early Start (ES) of a successor activity is equal to the Early Finish (EF) of its predecessor.

Merge Rule: If an activity has multiple predecessors , its ES is the maximum (largest) EF of those predecessors. It must wait for all prior work to finish.

The Backward Pass: Calculating Late Dates

The Backward Pass calculates the Late Start (LS) and Late Finish (LF) for every activity, moving backward from the end of the project to the start.

Rules for the Backward Pass:

Set the Late Finish (LF) of the final activity equal to its Early Finish (EF). This establishes the project completion date.

Calculate Late Start: L S = L F − Duration LS = LF - \text{Duration} L S = L F − Duration

The Late Finish (LF) of a predecessor activity is equal to the Late Start (LS) of its successor.

Burst Rule: If an activity has multiple successors , its LF is the minimum (smallest) LS of those successors. It must finish in time so as not to delay the most critical following task.

Float (Slack) Calculations

Float, also known as slack, represents scheduling flexibility.

Total Float (TF)

The amount of time an activity can be delayed without delaying the overall project completion date . Formula: T F = L S − E S TF = LS - ES TF = L S − ES (or T F = L F − E F TF = LF - EF TF = L F − EF )

Free Float (FF)

The amount of time an activity can be delayed without delaying the Early Start of any immediate successor activity . Formula: F F = ( Minimum ES of all successors ) − E F FF = (\text{Minimum ES of all successors}) - EF FF = ( Minimum ES of all successors ) − EF

Tip

Exam Strategy: If you only need to find the critical path, you only need to calculate the Forward Pass. The path with the longest total duration is the critical path. However, calculating the Backward Pass is necessary to find the exact Total Float of non-critical activities.

Identifying the Critical Path

The Critical Path is defined as the sequence of activities that dictates the overall project duration. It is the longest continuous path through the network.

Activities on the critical path have a Total Float of zero ( T F = 0 TF = 0 TF = 0 ) .

Any delay to a critical path activity will result in a day-for-day delay to the overall project finish date.

A project can have multiple critical paths if several paths share the exact same longest duration.

Worked Example: Network Analysis

Consider the following simplified project:

  • Activity Predecessors Duration (days)
  • A None 3
  • B A 4
  • C A 6
  • D B, C 2
  • Step 1: Forward Pass (Early Dates)

Activity A: Starts at day 0. E S = 0 ES = 0 ES = 0 . E F = 0 + 3 = 3 EF = 0 + 3 = 3 EF = 0 + 3 = 3 .

Activity B: Follows A. E S = 3 ES = 3 ES = 3 . E F = 3 + 4 = 7 EF = 3 + 4 = 7 EF = 3 + 4 = 7 .

Activity C: Follows A. E S = 3 ES = 3 ES = 3 . E F = 3 + 6 = 9 EF = 3 + 6 = 9 EF = 3 + 6 = 9 .

Activity D: Follows B and C. Merge Rule: Max(EF_B, EF_C) = Max(7, 9) = 9. E S = 9 ES = 9 ES = 9 . E F = 9 + 2 = 11 EF = 9 + 2 = 11 EF = 9 + 2 = 11 .

Project Duration: 11 days.

Step 2: Backward Pass (Late Dates)

Activity D: L F = 11 LF = 11 L F = 11 . L S = 11 − 2 = 9 LS = 11 - 2 = 9 L S = 11 − 2 = 9 .

Activity C: Precedes D. L F = 9 LF = 9 L F = 9 . L S = 9 − 6 = 3 LS = 9 - 6 = 3 L S = 9 − 6 = 3 .

Activity B: Precedes D. L F = 9 LF = 9 L F = 9 . L S = 9 − 4 = 5 LS = 9 - 4 = 5 L S = 9 − 4 = 5 .

Activity A: Precedes B and C. Burst Rule: Min(LS_B, LS_C) = Min(5, 3) = 3. L F = 3 LF = 3 L F = 3 . L S = 3 − 3 = 0 LS = 3 - 3 = 0 L S = 3 − 3 = 0 .

  • Step 3: Calculate Total Float and Critical Path
  • A: T F = 0 − 0 = 0 TF = 0 - 0 = 0 TF = 0 − 0 = 0
  • B: T F = 5 − 3 = 2 TF = 5 - 3 = 2 TF = 5 − 3 = 2
  • C: T F = 3 − 3 = 0 TF = 3 - 3 = 0 TF = 3 − 3 = 0
  • D: T F = 9 − 9 = 0 TF = 9 - 9 = 0 TF = 9 − 9 = 0

Conclusion: The Critical Path is A → \rightarrow → C → \rightarrow → D (11 days). Activity B has 2 days of Total Float, meaning it can be delayed by up to 2 days without impacting the 11-day project duration.