Network Models, Notation and Dependency Logic

Introduction

A project schedule is more than a list of activities and dates. It is a logic model that explains how work is expected to progress, which activities constrain others, and where changes may affect completion. Network diagrams provide the structural representation of this logic. They are therefore foundational to the Critical Path Method (CPM), schedule analysis, resource planning, progress control, and delay evaluation.

Two principal network conventions are used in project management:

• Activity-on-Node (AON), also called the Precedence Diagramming Method (PDM), represents activities as nodes and logical relationships as arrows.
• Activity-on-Arrow (AOA), also called the arrow diagram method, represents activities as arrows and events or milestones as nodes.

Modern scheduling software predominantly uses AON/PDM because it can represent four relationship types and schedule constraints directly. AOA remains important for understanding the historical development of CPM and for interpreting older schedules, contracts, and engineering planning documents.

Learning Objectives

By the end of this chapter, the reader should be able to:

1. Explain the purpose and structure of project network models.
2. Compare AON/PDM and AOA representations.
3. Interpret finish-to-start, start-to-start, finish-to-finish, and start-to-finish relationships.
4. Apply leads and lags without confusing them with unjustified schedule constraints.
5. Explain the role and limitations of dummy activities in AOA networks.
6. Construct and evaluate dependency logic in a professional project schedule.
7. Identify weak, ambiguous, or inappropriate relationships in a network model.

1. Networks as Models of Project Logic

A network model is a directed representation of activities and their dependencies. It answers the question: What must occur before, after, or concurrently with what? The model does not automatically describe resource availability, cost, quality, or risk; those dimensions must be integrated separately.

Important terms include:

• Activity: A defined piece of work requiring time and, usually, resources.
• Node: A graphical element representing an activity or event, depending on the notation.
• Predecessor: An activity or event that logically precedes another activity.
• Successor: An activity or event that follows another activity.
• Dependency: A logical relationship that determines how two activities are connected.
• Path: A continuous sequence of connected activities through the network.
• Milestone: A zero-duration event used to mark a significant point, such as approval or handover.
• Open end: An activity with no predecessor or successor other than the project start or finish. Unjustified open ends can indicate incomplete logic.

A network is a model, not a guarantee. It reflects the planner's assumptions about how work will be performed. Consequently, network quality depends on the quality of the work breakdown structure, activity definitions, stakeholder input, and assumptions concerning design, procurement, approvals, and resources.

2. Activity-on-Node and Precedence Diagramming Method

In AON/PDM, each activity is represented by a box or node. Arrows show logical relationships between activities. Activity duration is written inside or beside the node.

A simplified structure is:

[Design] ──FS──> [Procure] ──FS──> [Install]

The arrow expresses logic; it does not necessarily represent the passage of time. The duration belongs to the activity node.

AON is dominant in contemporary project management because it:

• represents all four standard relationship types;
• supports leads and lags directly;
• accommodates milestones and complex logic clearly;
• maps naturally to scheduling software; and
• separates activity identity from relationship representation.

AON diagrams should be constructed so that the direction of flow is unambiguous, normally from left to right or from top to bottom. Every activity should have a clear description, preferably beginning with a verb and an object, such as “Approve structural drawings” rather than “Drawings.”

3. Activity-on-Arrow Models

In AOA, an activity is represented by an arrow connecting two events. The event at the arrow's tail is the start event, and the event at its head is the finish event. Events generally have zero duration.

(10) ── Install cable, 4 days ──> (20)

The numbers identify events, not activities. The activity name and duration are written above or beside the arrow.

AOA was central to early CPM and Program Evaluation and Review Technique (PERT) applications. It is still useful when examining legacy schedules and when teaching the distinction between activities and events. However, AOA has limitations:

1. It naturally expresses finish-to-start logic, but not the full range of modern relationship types.
2. It may require dummy activities to represent logic accurately.
3. Large networks can become difficult to read because several arrows may converge on or depart from the same events.
4. Modifying logic can require the renumbering or restructuring of events.

3.1 Dummy Activities

A dummy activity is an artificial, normally zero-duration and zero-resource arrow introduced into an AOA network to preserve logical relationships. It is often drawn as a dashed arrow.

Consider three activities:

• A must precede C.
• B must precede D.
• C must also precede D.

If A and B finish at the same event, the network might incorrectly imply that both precede C. A dummy activity can distinguish the required relationships without assigning time or resources to the dummy.

Dummy activities must not be interpreted as actual work. They exist to represent network logic. Excessive use of dummies can make a network difficult to understand and may signal that an AON representation would be more appropriate.

A dummy should be used only when necessary to avoid incorrect logical implications. It should not be used to hide missing activity definitions, create artificial urgency, or substitute for a well-written relationship explanation.

4. Comparison of AON and AOA

Neither notation automatically produces a correct schedule. A poorly reasoned AON schedule can be less reliable than a carefully designed AOA schedule. The essential issue is whether the dependencies represent real technical, contractual, safety, or managerial conditions.

5. The Four Precedence Relationships

PDM uses four principal relationship types. The first letter identifies the point on the predecessor activity; the second identifies the point on the successor activity.

5.1 Finish-to-Start (FS)

A finish-to-start relationship means that the successor cannot start until the predecessor has finished, subject to any stated lag or lead.

Example: Testing cannot begin until installation is complete.

FS is the most common relationship and is appropriate when work genuinely requires completion of one activity before another can begin. It should not be used automatically merely because activities are listed in a certain order.

With a lag of L time units, the basic condition is:

where S_S is the successor start, F_P is the predecessor finish, and L is the lag. A positive L creates waiting time; a negative L is a lead.

5.2 Start-to-Start (SS)

A start-to-start relationship means that the successor cannot start until the predecessor has started, subject to lag or lead.

Example: Quality inspection may begin two days after production begins.

The general condition is:

SS is useful for overlapping activities that can proceed concurrently. It does not imply that the activities finish together or that the successor can start immediately. The lag expresses the minimum offset between starts.

5.3 Finish-to-Finish (FF)

A finish-to-finish relationship means that the successor cannot finish until the predecessor has finished, subject to lag or lead.

Example: Final documentation cannot be completed until the related testing activity is complete.

The condition is:

FF does not require simultaneous completion. With a positive lag, the successor finishes after the predecessor by at least the lag amount. The successor may begin before the predecessor finishes if other logic permits.

5.4 Start-to-Finish (SF)

A start-to-finish relationship means that the successor cannot finish until the predecessor has started, subject to lag or lead.

Example: An outgoing operations team cannot complete its handover until the replacement team has started its assigned shift.

The condition is:

SF is uncommon and frequently misunderstood. It should not be used simply to express a normal handover or sequential replacement when FS or SS is more transparent. Because an SF relationship can produce unintuitive schedules, its business or technical rationale should be documented.

5.5 Relationship Summary

The symbols have the following meanings: S_P and F_P are the predecessor's start and finish; S_S and F_S are the successor's start and finish; L is the stated lag, measured in the same time unit as the schedule. These inequalities express minimum logical conditions. Other constraints, such as calendars, resource availability, contractual dates, and imposed deadlines, may produce later dates.

6. Leads and Lags

A lag is a planned delay between related points of two activities. A lead is an intentional overlap, often represented as a negative lag.

For example, if painting may begin two days after plastering begins, the relationship is SS + 2 days. If procurement may begin three days before design is fully complete, the relationship might be FS − 3 days, provided that the technical information available at that point is sufficient.

6.1 Worked Example: Relationship Timing

Suppose Activity A starts on day 1 and lasts 6 working days. Under the convention that a six-day activity beginning on day 1 finishes at the end of day 6, its finish time is represented as day 7 in continuous time, or as the start of day 7. Activity B lasts 4 working days.

• Under FS with a 2-day lag, B starts after A's finish plus two days of waiting.
• Under SS with a 1-day lag, B starts one day after A starts.
• Under FF with a 0-day lag, B cannot finish before A finishes; because B is shorter, it may need to start later.
• Under FS with a 2-day lead, B may start two time units before A finishes, if the project permits partial overlap.

The exact calendar dates depend on the scheduling convention, working calendar, holidays, and whether “day 1” denotes a time point or a working-day label. Professional schedules must document these conventions.

6.2 Appropriate Use of Leads and Lags

Leads and lags should model real conditions, such as:

• curing or drying time;
• statutory review periods;
• delivery and inspection intervals;
• phased release of design information;
• testing after partial installation; and
• controlled overlap between engineering and procurement.

They should not be used to conceal missing activities. For example, “Install equipment FS − 10 days after design” may hide the fact that a separate activity, “Release approved-for-construction drawings,” is required. Explicit activities are usually more auditable than large unexplained lags or leads.

Excessive leads can create out-of-sequence work, rework, safety exposure, or false confidence in the schedule. Lags can also be difficult to manage because they do not always identify an accountable owner. Where a waiting period is important, it may be preferable to model it as a named activity such as “Concrete curing.”

7. Dependency Types and Logic Quality

Relationship type describes the timing connection; dependency type explains why the connection exists.

7.1 Mandatory Dependencies

These are inherent in the nature of the work, such as installing a component before testing it. They are sometimes called hard logic or physical dependencies.

7.2 Discretionary Dependencies

These reflect a preferred method rather than an unavoidable technical requirement. For example, a team may prefer to complete all interior painting before installing fittings, although partial overlap is technically possible. Such assumptions should be documented because they may be changed during schedule compression.

7.3 External Dependencies

These involve parties outside the project team, such as permits, utility connections, regulatory approvals, or supplier deliveries.

7.4 Internal Dependencies

These are controlled within the project, such as the relationship between design review and internal cost approval.

A robust network should have a traceable rationale for each important relationship. The planner should ask:

1. What specifically prevents the successor from occurring earlier?
2. Is the relationship technical, contractual, safety-related, resource-related, or merely customary?
3. Is the selected relationship type correct?
4. Can the dependency be represented by a named activity rather than a hidden lag?
5. Does the logic create unintended loops, excessive constraints, or out-of-sequence work?

8. Constructing a Network Model

A disciplined construction process is as follows:

1. Define the project scope and deliverables.
2. Decompose deliverables into measurable activities.
3. Confirm activity descriptions, ownership, and estimated duration.
4. Identify predecessors and successors using expert judgment and technical input.
5. Select the correct relationship type for each dependency.
6. Add realistic lags only where a waiting or overlap condition exists.
7. Check for missing logic, open ends, circular relationships, and duplicate links.
8. Validate the network with the project team and external stakeholders.
9. Establish the baseline only after logic, durations, calendars, and constraints have been reviewed.

The network should distinguish logical dependency from resource sequencing. If two activities cannot occur together only because the same specialist is needed, that is a resource limitation rather than necessarily a technical dependency. It may be managed through resource levelling or a documented discretionary dependency.

9. Illustrative Case: Municipal Water-Treatment Upgrade

Background

A municipality is upgrading a water-treatment plant while maintaining partial operation. The schedule includes design approval, equipment procurement, civil works, installation, testing, and operator training.

Problem

The initial schedule used FS links for nearly every activity. This produced an unnecessarily long completion date and did not represent the fact that procurement could begin after approved equipment specifications, while civil works could begin after foundation drawings were released.

Analysis

The project team identified the following logic:

Approach and Solution

The team replaced unjustified FS relationships with explicit SS and FF logic. Training was linked SS + 2 days to commissioning because operators could begin training when commissioning had produced stable operating procedures. Handover was linked FF to commissioning because handover could not be completed before commissioning was complete, although preparation for handover could occur earlier.

The project also avoided a large hidden lag between installation and commissioning. Instead, any required inspection and energisation activities were represented as named activities. This improved accountability and made the schedule easier to update.

Outcome

The revised network reduced apparent duration without compromising technical controls. It also showed which assumptions were critical: timely specification approval, pump delivery, foundation acceptance, and commissioning performance. The schedule became more defensible because each overlap had a documented operational rationale.

Learning Points

• A long chain of FS links may reflect habit rather than necessity.
• SS relationships can model legitimate concurrent work.
• FF relationships are useful when completion conditions are linked, but they do not imply identical starts.
• Hidden lags should be replaced by explicit activities when the waiting period has an owner or risk.
• Schedule compression must not remove safety, quality, regulatory, or commissioning requirements.

10. Technology, Governance, and Contemporary Practice

Scheduling software can automatically calculate dates, display relationship types, identify open ends, and perform schedule quality checks. However, automation does not validate the underlying assumptions. Artificial intelligence and machine-learning tools may suggest dependencies from historical schedules or project data, but such suggestions require expert review. A statistically common relationship is not necessarily a technically valid relationship for a particular project.

Governance practices should require:

• documented assumptions for unusual links and large lags;
• approval of baseline logic by responsible subject-matter experts;
• change control for modifications to critical dependencies;
• traceability from schedule activities to scope and contractual deliverables; and
• periodic review of whether dependencies remain valid.

In delay analysis, the chosen network logic can materially affect conclusions about responsibility and criticality. Analysts must therefore preserve schedule versions, explain logic changes, and avoid retrospectively altering relationships merely to fit actual outcomes.

11. Limitations and Best Practices

Network models simplify reality. They normally assume that activity durations and relationships are sufficiently defined, yet actual work may be affected by uncertainty, weather, productivity, design changes, resource contention, and learning effects. A network also cannot by itself determine whether a project is feasible with available resources.

Best practices include:

1. Prefer explicit, meaningful activities over unexplained large lags.
2. Use FS by default only when a finish-before-start condition genuinely exists.
3. Use SS and FF to represent legitimate overlap or linked completion conditions.
4. Use SF rarely and document its rationale.
5. Keep relationship types and lags visible and auditable.
6. Avoid excessive constraints that override network logic.
7. Distinguish technical dependencies from resource-driven sequencing.
8. Validate the network with those who will perform the work.
9. Check calendars and time units before interpreting dates.
10. Treat the network as a living model and update it through controlled change management.

Chapter Summary

Network diagrams model the logical structure of project work. AON/PDM represents activities as nodes and relationships as arrows, while AOA represents activities as arrows and events as nodes. AON is the prevailing modern method because it supports FS, SS, FF, and SF relationships and handles leads and lags directly. AOA remains valuable for historical CPM and event-oriented analysis but may require dummy activities.

FS links a successor's start to a predecessor's finish; SS links starts; FF links finishes; and SF links a successor's finish to a predecessor's start. Positive lags create waiting time, while negative lags or leads create permitted overlap. These relationships must reflect real technical, contractual, safety, or managerial conditions. Dummy activities have no work or duration and are used in AOA to preserve logic.

High-quality networks use explicit activities, documented assumptions, appropriate relationship types, and controlled validation. Software and AI can assist analysis, but professional judgment remains essential. A network is only as reliable as the assumptions and dependencies on which it is built.

Key Terms

Activity: A defined unit of project work with a duration.

Activity-on-Arrow (AOA): Network notation in which activities are arrows and events are nodes.

Activity-on-Node (AON): Network notation in which activities are nodes and relationships are arrows.

Dependency: A logical connection between activities.

Dummy activity: A zero-duration, zero-resource activity used mainly in AOA to preserve logic.

Event: A point in time, normally with zero duration, in an AOA network.

Finish-to-finish (FF): A relationship linking the predecessor's finish to the successor's finish.

Finish-to-start (FS): A relationship linking the predecessor's finish to the successor's start.

Lag: A planned positive delay between related activity points.

Lead: An intentional overlap, commonly represented by a negative lag.

Milestone: A significant zero-duration project event.

Network model: A graphical or logical representation of project activities and dependencies.

Open end: An activity lacking a justified predecessor or successor.

Precedence Diagramming Method (PDM): The standard AON method for representing precedence relationships.

Predecessor: An activity that logically precedes another activity.

Relationship type: The classification of a dependency as FS, SS, FF, or SF.

Start-to-finish (SF): A relationship linking the predecessor's start to the successor's finish.

Start-to-start (SS): A relationship linking the predecessor's start to the successor's start.

Successor: An activity that logically follows another activity.

Review Questions

5 Short Answer Questions

1. What is the principal difference between AON and AOA notation?
2. Define a dummy activity and explain why it may be required in an AOA network.
3. What does an SS relationship represent?
4. How does a lead differ from a lag?
5. Why should large unexplained lags be treated cautiously?

5 Long Answer Questions

1. Compare AON/PDM and AOA with respect to representation, flexibility, software use, and limitations.
2. Explain all four PDM relationship types and provide a project example for each.
3. Discuss how dependency type and relationship type are different concepts.
4. Describe a systematic process for constructing and validating a project network.
5. Evaluate the risks of using excessive constraints, leads, and lags in a baseline schedule.

3 Analytical / Application Questions

1. A design activity and procurement activity are currently linked FS, but procurement can begin when approved specifications for the first equipment package are released. Recommend a revised network logic and justify it.
2. Construct a logical sequence for a construction package involving foundation drawings, foundation construction, equipment delivery, installation, testing, and operator training. Identify where SS, FF, or explicit waiting activities may be preferable to FS.
3. An AOA network appears to imply that two activities must both precede a third, although only one actually does. Explain how a dummy activity could correct the logic and discuss whether converting the network to AON would be preferable.

Multiple Choice Questions with Answers

Question 1

In an Activity-on-Node network, what does a node normally represent?

A. An activity

B. A project event only

C. A dummy relationship

D. A calendar exception

Correct Answer: A. An activity

Question 2

Which relationship means that a successor cannot start until its predecessor has finished, subject to lag?

A. SS

B. FF

C. FS

D. SF

Correct Answer: C. FS

Question 3

Which relationship links the start of a predecessor to the start of a successor?

A. FS

B. SS

C. FF

D. SF

Correct Answer: B. SS

Question 4

What is the principal purpose of a dummy activity in an AOA network?

A. To consume project resources

B. To add planned duration

C. To preserve logical distinctions

D. To represent a high-risk activity

Correct Answer: C. To preserve logical distinctions

Question 5

A positive lag in an FS relationship generally represents what?

A. Permitted overlap

B. A waiting period after predecessor finish

C. A removal of dependency

D. A mandatory simultaneous start

Correct Answer: B. A waiting period after predecessor finish

Question 6

Which relationship should be used cautiously because it links a successor finish to a predecessor start?

A. FS

B. SS

C. FF

D. SF

Correct Answer: D. SF

Question 7

If a successor may begin two days after its predecessor begins, which relationship is most direct?

A. SS + 2 days

B. FF + 2 days

C. FS − 2 days

D. SF − 2 days

Correct Answer: A. SS + 2 days

Question 8

Which statement about an FF relationship is correct?

A. Both activities must start simultaneously

B. The successor cannot finish before the predecessor finishes, subject to lag

C. The successor must start after the predecessor finishes

D. It can be used only in AOA networks

Correct Answer: B. The successor cannot finish before the predecessor finishes, subject to lag

Question 9

Why may explicit activities be preferable to large lags?

A. They eliminate all uncertainty

B. They identify waiting work and accountability more clearly

C. They make every activity critical

D. They remove the need for calendars

Correct Answer: B. They identify waiting work and accountability more clearly

Question 10

Which statement best describes a network model?

A. A complete substitute for resource planning

B. A guaranteed forecast of project completion

C. A representation of assumed activity logic and dependencies

D. A cost-accounting ledger

Correct Answer: C. A representation of assumed activity logic and dependencies