Project network
A project network, commonly referred to as a project schedule network diagram, is a visual tool in project management that depicts the sequential arrangement of project activities, their interdependencies, durations, and logical relationships to aid in planning, scheduling, and execution.[1] It represents tasks as nodes or arrows connected by lines indicating dependencies, such as finish-to-start or start-to-start relationships, enabling project managers to identify the critical path—the longest sequence of dependent activities that determines the minimum project duration.[2] This diagram is a key output of the activity sequencing process in standard project management frameworks like the PMBOK Guide.[1] The origins of project networks trace back to the late 1950s, with the development of two foundational techniques: the Critical Path Method (CPM) and the Program Evaluation and Review Technique (PERT). CPM was pioneered in 1957 by James E. Kelley of Remington Rand and Morgan R. Walker of DuPont to optimize the scheduling of plant maintenance shutdowns, focusing on deterministic time estimates and cost trade-offs.[2] PERT emerged in 1959 from a U.S. Navy team led by Willard Fazar, including Donald G. Malcolm, John H. Roseboom, and Charles E. Clark, for managing the Polaris missile program's complex uncertainties using probabilistic time estimates based on optimistic, most likely, and pessimistic durations.[2] By the early 1960s, the U.S. Department of Defense required PERT/CPM for major contracts, accelerating their adoption across industries like construction, defense, and research and development.[2] Project networks typically employ one of two diagramming methods: Activity on Arrow (AOA), where activities are shown as arrows between event nodes, or the more prevalent Activity on Node (AON), where activities are boxes connected by dependency arrows.[1] These diagrams support critical functions such as calculating early and late start/finish times for activities, resource leveling, and risk assessment by highlighting float or slack in non-critical paths.[1] Modern software tools like Microsoft Project or Primavera automate their creation, incorporating extensions like the Precedence Diagramming Method (PDM) for handling complex dependencies including leads, lags, and four types of relationships (finish-to-start, start-to-start, finish-to-finish, start-to-finish).[2] Overall, project networks remain essential for ensuring timely delivery, cost efficiency, and stakeholder alignment in diverse project environments.[2]Overview
Definition and Purpose
A project network is a graphical representation of the logical relationships among project activities, illustrating their sequence, interdependencies, and durations to model the workflow of a project.[3] It is typically structured as a directed acyclic graph (DAG), where directed edges denote the precedence of one activity over another without cycles to ensure a logical progression from start to finish.[4] This visualization aids project managers in understanding the overall structure of the project without delving into execution details. The primary purpose of a project network is to facilitate the identification of task sequences and dependencies, enabling accurate estimation of the total project duration and efficient resource allocation.[5] By highlighting the critical path—the longest sequence of dependent activities that determines the minimum project completion time—it helps prioritize efforts to avoid delays and ensure timely delivery.[6] Additionally, it supports risk assessment by revealing potential bottlenecks where delays in one activity could impact the entire timeline.[3] Key components of a project network include nodes, which represent activities or milestones, and arrows, which indicate the directional dependencies between them.[7] Each activity is assigned a duration to quantify the time required, while lags or leads may be incorporated as attributes on arrows to account for mandatory waits or overlaps between tasks.[8] These elements collectively provide a schematic framework for planning and control.[9] Common representations include the activity-on-node (AON) method, where nodes depict activities and arrows show relationships, and the activity-on-arrow (AOA) method, which reverses this convention.[10] For instance, in a simple construction project, a network might feature nodes for "site preparation" (duration: 5 days), "foundation work" (duration: 10 days), and "framing" (duration: 7 days), with arrows connecting them to enforce that foundation work follows site preparation and precedes framing, thereby illustrating the sequential dependencies.[11]Historical Development
The development of project networks originated in the late 1950s amid complex industrial and military projects requiring advanced scheduling techniques. The Critical Path Method (CPM) emerged in 1957, created by James E. Kelley of Remington Rand and Morgan R. Walker of DuPont to optimize the scheduling of plant maintenance shutdowns, addressing inefficiencies in traditional bar chart methods by modeling dependencies and identifying the longest sequence of tasks.[12][13] Concurrently, the Program Evaluation and Review Technique (PERT) was introduced in 1958 by the U.S. Navy's Special Projects Office for the Polaris missile program, incorporating probabilistic time estimates to manage uncertainty in a high-stakes defense initiative involving thousands of interdependent activities.[14][15] These foundational methods quickly gained traction, with early implementations leveraging mainframe computers in the 1960s to perform network analysis calculations that were impractical manually, such as forward and backward passes to compute critical paths.[16] By the 1970s, project networks transitioned from purely manual diagramming to computer-aided tools, enabling iterative updates and simulations on emerging systems like minicomputers, which facilitated broader application in engineering and research and development.[17] The evolution continued into the 1980s with formal standardization; the Project Management Institute (PMI) incorporated CPM and PERT as core network-based techniques in the first edition of the PMBOK Guide, published in 1996, establishing them as essential for systematic project planning across industries.[18] By the 1990s, adoption surged in construction, with widespread adoption among major U.S. contractors for large-scale projects, and in information technology, where network methods supported structured scheduling in software development and infrastructure builds amid rapid technological expansion.[19]Network Representations
Activity-on-Node (AON)
The Activity-on-Node (AON) representation, also known as the Precedence Diagramming Method (PDM), is a fundamental approach in project network diagramming where individual project activities are depicted as nodes, typically rectangular boxes, and the logical dependencies between them are shown as directed arrows connecting the nodes.[20] This method was pioneered by John W. Fondahl in 1961 as a non-computerized extension of the Critical Path Method (CPM), allowing for more flexible modeling of activity sequences without relying solely on arrow-based representations.[21] In AON diagrams, each node encapsulates key attributes of an activity, including its unique identifier, estimated duration, required resources, and potential start and end times, providing a centralized view of all relevant details for that task.[3] Arrows in an AON network illustrate precedence relationships, most commonly finish-to-start (FS), where the predecessor activity must finish before the successor can begin, but also supporting start-to-start (SS), finish-to-finish (FF), and start-to-finish (SF) types, often with incorporated lags (delays) or leads (advances) to refine timing.[20] This structure enables the representation of parallel activities and complex interdependencies, such as multiple predecessors or successors for a single node, without the need for auxiliary elements in most cases.[3] AON offers several advantages, particularly its intuitive visualization that makes it accessible for beginners by focusing directly on activities rather than abstract events, facilitating easier identification of workflows and bottlenecks.[20] It accommodates diverse dependency types and lags more naturally than earlier methods, reducing diagramming complexity, and aligns seamlessly with modern project management software tools like Microsoft Project, which natively support AON for automated scheduling and resource allocation. As of 2025, AON continues to dominate due to its compatibility with agile and hybrid project frameworks.[3][22] To convert an Activity-on-Arrow (AOA) network to AON, the process involves redrawing the diagram by creating a node for each activity (previously represented by arrows in AOA), then connecting these nodes with arrows to mirror the original precedence logic; the need for dummy activities in AOA is typically eliminated in AON by allowing multiple incoming and outgoing dependencies on nodes, avoiding the use of auxiliary elements.[21] For illustration, consider a simplified AON network for a software development project with six activities: Node A (Requirements Gathering, duration 5 days) leads to Node B (System Design, 10 days) and parallel Node C (UI/UX Design, 7 days) via FS arrows; Node B and Node C both precede Node D (Coding, 15 days) with SS relationships (allowing coding to start once design begins, with a 2-day lag on C); Node D then connects via FS to Node E (Testing, 8 days); finally, Node E leads to Node F (Deployment, 3 days) via FF to ensure deployment completes only after testing finishes. This setup highlights sequential core development alongside parallel design tasks, with arrows denoting dependencies to visualize the overall flow.[3]Activity-on-Arrow (AOA)
The Activity-on-Arrow (AOA) representation is a traditional method for depicting project networks, where nodes, typically shown as circles, represent events or milestones such as the start or completion of tasks, and directed arrows connecting these nodes illustrate the activities along with their durations.[23][24] This structure enforces a strict sequence, permitting only finish-to-start dependencies between activities without support for lags or other relationship types like start-to-start.[3] Historically, AOA served as the primary format in the original Program Evaluation and Review Technique (PERT) developed by the U.S. Navy in 1958 and in early implementations of the Critical Path Method (CPM) introduced by DuPont in the late 1950s.[24][25] In these foundational approaches, the event-node and activity-arrow convention facilitated the calculation of project timelines by tracing paths through the network.[25] A key limitation of AOA arises in handling non-sequential or converging dependencies, often requiring the introduction of dummy arrows—fictitious activities with zero duration depicted as dashed lines—to preserve logical precedence without duplicating real activities. This can lead to increased diagram complexity, particularly for projects involving extensive parallel tasks, making AOA less adaptable to intricate modern workflows compared to alternatives like Activity-on-Node.[26]Example: AOA Network for a Manufacturing Project
Consider a simplified manufacturing project to produce a mechanical assembly, involving design, procurement, tooling, fabrication, and machining. The network uses event nodes numbered sequentially (1 through 6) and arrows for activities with durations in days:- Event 1 (start) connects via arrow A: Prepare design (4 days) to event 2.
- From event 2, arrow B: Order materials (5 days) leads to event 3.
- Also from event 2, arrow C: Build mold (3 days) leads to event 4.
- From event 3, arrow D: Pour metal (2 days) leads to event 5.
- A dummy arrow (0 days, dashed line) from event 4 to event 5 ensures event 5 occurs only after both C and D are complete, maintaining the dependency for machining.
- Finally, from event 5, arrow E: Machine parts (6 days) leads to event 6 (project completion).
Construction Process
Steps for Building a Network Diagram
Building a project network diagram involves a systematic process to visually represent the sequence and interdependencies of project activities, ensuring a logical flow from initiation to completion. This procedure aligns with principles in the current Project Management Body of Knowledge (PMBOK Guide, 8th Edition, 2025), though detailed processes are described in earlier editions like the 6th; it applies to both activity-on-node (AON) and activity-on-arrow (AOA) representations, with precedence diagramming method (PDM) commonly used for manual creation in AON formats to accommodate various dependency types.[27][28] The first step is to identify all project activities using the work breakdown structure (WBS), which decomposes the project scope into manageable, verifiable work packages that form the basis for individual activities. This ensures every element of the project is accounted for, drawing from detailed planning outputs like scope baseline and historical information.[29][30] Next, determine the dependencies between activities and sequence them logically, categorizing relationships as mandatory (inherent in the work, such as one task requiring the completion of another), discretionary (based on best practices or preferences), external (involving outside factors like vendor deliveries), or internal (within the project team). These dependencies are mapped using PDM, which supports four relationship types—finish-to-start (FS), start-to-start (SS), finish-to-finish (FF), and start-to-finish (SF)—to accurately reflect how activities interconnect without assuming strict linearity.[28][31] The third step involves estimating durations for each activity, typically through expert judgment, analogous estimating from similar past projects, or parametric models based on historical data and resource availability. These estimates provide the temporal scale needed for the diagram, often expressed in workdays or weeks, and are refined iteratively as more information becomes available.[3][32] Finally, draw the diagram by plotting activities and their connections, resolving any loops (circular dependencies that could invalidate sequencing) or redundancies (unnecessary parallel paths) through review and adjustment, then validate for completeness by confirming all activities, dependencies, and durations align with project objectives. Manual tools like PDM facilitate this by allowing flexible arrow connections between nodes, ensuring the diagram is error-free and ready for further use. These steps align with predictive planning in the PMBOK 8th Edition (2025), which integrates hybrid methods and AI tools for enhanced accuracy.[1][30][27]Notation and Symbols
In project network diagrams, nodes typically represent activities or events, often depicted as rectangles or circles to denote tasks with associated durations and descriptions. Arrows connecting these nodes illustrate logical dependencies between activities. In AOA diagrams, solid arrows represent activities, with dashed arrows used for dummy activities (zero-duration placeholders) to clarify complex sequencing without implying actual work. In the more common AON diagrams using PDM, arrows are typically solid lines for dependencies, with lags (delays) or leads (overlaps) notated numerically on the arrows (e.g., FS+2 for a 2-day lag). These conventions ensure visual clarity in representing project workflows.[27][33] Activity labels within nodes commonly include unique identifiers (e.g., "Task A"), estimated durations (e.g., "5 days"), and resources, while post-analysis annotations add timing metrics such as early start (ES), early finish (EF), late start (LS), and late finish (LF) to support critical path calculations. These labels are integrated after initial diagram construction to reflect schedule computations without altering the core dependency structure.[33] Dependency types are notated using standardized abbreviations to specify how activities interrelate, including finish-to-start (FS), where a successor activity begins only after the predecessor finishes; start-to-start (SS), where the successor starts upon or after the predecessor's start; finish-to-finish (FF), requiring the successor to finish after or with the predecessor; and start-to-finish (SF), a less common type where the successor finishes upon the predecessor's start. Lags, representing mandatory delays, are denoted with a plus sign (e.g., FS+2 for a 2-day lag after finish-to-start), while leads (advances) use a minus sign (e.g., SS-1 for overlapping starts).[33][34] These notations align with the standards outlined in the Project Management Body of Knowledge (PMBOK Guide, 8th Edition, 2025) by the Project Management Institute (PMI), which endorses the precedence diagramming method (PDM) for professional project diagrams to promote consistency across predictive and adaptive approaches. In activity-on-node (AON) representations, nodes focus on activities with arrows for dependencies, whereas activity-on-arrow (AOA) uses arrows for activities and nodes for events, often requiring dummies for clarity.[27][35]Analysis Methods
Critical Path Method (CPM)
The Critical Path Method (CPM) is a deterministic scheduling technique used in project management to analyze project networks by assuming fixed activity durations, enabling the identification of the critical path—the longest sequence of dependent activities that determines the minimum project completion time.[36] Developed in the late 1950s for industrial projects, CPM represents activities as nodes in a network diagram connected by precedence relationships, allowing managers to calculate the earliest and latest possible start and finish times for each activity to highlight those with zero slack, where any delay would extend the overall project duration.[37] Unlike probabilistic methods such as PERT, CPM focuses on precise time estimates without incorporating uncertainty.[38] The analysis begins with a forward pass to compute the earliest start (ES) and earliest finish (EF) times for each activity, starting from the project beginning where ES is zero. For an activity, the ES is the maximum EF of its immediate predecessors, ensuring no activity starts before all dependencies are complete; the EF is then calculated as ES plus the activity's fixed duration.[36] Mathematically, this is expressed as: \text{ES}_j = \max(\text{EF}_i) \quad \text{for all predecessors } i \text{ of } j \text{EF}_j = \text{ES}_j + D_j where D_j is the duration of activity j.[37] This pass yields the earliest possible project completion time at the final activity's EF. Following the forward pass, a backward pass determines the latest allowable start (LS) and finish (LF) times, beginning from the project end where LF equals the EF from the forward pass. The LF for an activity is the minimum LS of its immediate successors, and the LS is LF minus the duration, providing a latest schedule that still meets the project deadline.[38] The formulas are: \text{LF}_i = \min(\text{LS}_j) \quad \text{for all successors } j \text{ of } i \text{LS}_i = \text{LF}_i - D_i These calculations reveal the total slack for each activity, defined as the difference between LS and ES (or equivalently, LF minus EF), representing the amount of time an activity can be delayed without impacting the project finish.[36] Thus, \text{Total Slack} = \text{LS} - \text{ES} = \text{LF} - \text{EF} The critical path consists of all activities with zero slack, forming the sequence where delays directly extend the project duration.[37] To illustrate, consider a simple project network with four activities: A (duration 3 days, start activity), B (5 days, successor to A), C (2 days, successor to A), and D (4 days, successor to both B and C). The network has two paths: A-B-D (total 12 days) and A-C-D (total 9 days). Step 1: Forward Pass- ES(A) = 0, EF(A) = 0 + 3 = 3
- ES(B) = EF(A) = 3, EF(B) = 3 + 5 = 8
- ES(C) = EF(A) = 3, EF(C) = 3 + 2 = 5
- ES(D) = max(EF(B), EF(C)) = max(8, 5) = 8, EF(D) = 8 + 4 = 12 (project duration)
- LS(D) = 12 - 4 = 8, LF(D) = 12
- For B (predecessor to D): LF(B) = LS(D) = 8, LS(B) = 8 - 5 = 3
- For C (predecessor to D): LF(C) = LS(D) = 8, LS(C) = 8 - 2 = 6
- For A (predecessor to B and C): LF(A) = min(LS(B), LS(C)) = min(3, 6) = 3, LS(A) = 3 - 3 = 0
The values are summarized in the table below:
| Activity | Duration | ES | EF | LS | LF | Slack (LS - ES) |
|---|---|---|---|---|---|---|
| A | 3 | 0 | 3 | 0 | 3 | 0 |
| B | 5 | 3 | 8 | 3 | 8 | 0 |
| C | 2 | 3 | 5 | 6 | 8 | 3 |
| D | 4 | 8 | 12 | 8 | 12 | 0 |