C2C Digital Magazine (Spring/Summer 2023)

Credible efficient pathways to project completion with AON diagrams

By Shalin Hai-Jew, Kansas State University

Professionals in project management have a wide range of ICT and other tools to help them manage the work: task schedulers, staffing, calendars, human resources systems, and others. I ran across a basic tool that may be executed in Excel, the so-called Activity-on-Node (AON) diagram (an example of a project network).

This short article explores a simple AON diagram that can be actualized manually for a basic project in a higher education context. While AON diagrams have been built into various computerized project management tools, even a basic manual setup may be informative. The simple math enables a back-of-the-napkin sort of scheduling for a project (although if the estimate is more high-value, more detailed inputs and oversight should be invested).

Infusing Work Efficiencies and Planner and Team Member Awareness with Planning

The basic rationale for an AON diagram is to map out the various chained sequential / semi-sequential tasks in order to better understand task activity dependencies and how much subsequent work depends on prior tasks. The goal of such planning is to minimize time lags on projects and to help project planners and others get a handle on work trajectory and complexity. Those working on a cross-functional team may better synchronize their efforts.

Video: How to Create a Schedule Network Diagram in Excel

[In the video, note the work breakdown structure in which deliverables are recorded, along with the work packages, activities, tasks, prior dependencies, activity sequences, responsible parties (bureaucratic units or staff), and then the “estimate confidence.” Every model involves various assumptions, and spelling out how confident the assumptions are in a systematic way may result in a more accurate model. I also like the table with the column headers of Activity, Predecessor, and Duration, because this also simplifies the AON diagram and captures the most important basic information related to time.]

The mapping out of the various work activities and sequences enables better planning, so that conflicts in the order of the work may be avoided. The idea is that avoiding work re-dos can be beneficial.

A Set of Activities, Required Time, and Task Interrelationships

It may help to loosely define some terms. The work “project” is defined in part as time segments or durations. “Float” or “slack” refers to the amount of time that a particular task activity, a network path, or even a project can be delayed from the early start (ES) without negatively affecting the completion of the project (the “drop-dead date,” the “hard deadline”). A “total float” for a project refers to the difference between the finish data of the last activity of the critical path and the actual project completion date.

A “critical path” refers to a series (ordered sequence) of nodes (activities) and links (precedence relationships, representing preconditions) that has no substitutions for tasks; here, a lag in any part of the critical path will affect the overall amount of time required in the project work. Various parts of a project may be achieved without preemption. Depending on the tasks being modeled, various critical activities will manifest in different ways. The Critical Path Method was developed in the 1950s.

A project may be completed through the achieving of the work tasks along different paths. Some of the paths are more efficient than others. Changes in-world may force switching from one path to another. The ceteris paribus assumption (all things being equal, general conditions remaining the same) at the start of a project may not hold as the project progresses. After all, staff turnover, technology snafus, unplanned costs, and other changes are not unlikely. Clients change their minds. There are dynamic exogenous and endogenous factors.

AON diagrams help the leadership and the team members think through a project and the decisions they are making and the outcomes they are striving to achieve.

Various technologies enable forwards and backwards mapping of tasks, to enable new learning, and new efficiencies that may have been invisible before.

Estimating Time for Each Task

Clearly, the accuracy of the work required, the capabilities of the staff, and the accuracy of estimated time required for each task will affect the overall accuracy of the model. The much-observed “optimism bias” is a weakness for inexperienced AON modelers who assume much more can be done with much less. Some will use timings from prior projects, breakdowns of tasks into unit parts, or other formulas to set a timeline estimate. A formula known as PERT is calculated as follows:

[optimistic estimate + (4* most likely time estimate] + pessimistic estimate]

The equation captures the min-max upper- and lower-bound estimates and weights the most likely (practicable) estimate in the equation. Prior experiences are seen to be informative of the most likely estimate.

About the Links between Task Activities: Four Main Time Relationships based on the Precedence Diagram Method (PDM)

AON diagrams are a sub-type of precedence diagrams, and it uses the “precedence diagram method” (PDM). AON diagrams are also an example or subtype of schedule network diagrams.

Once those time estimates are available (and are labeled with the level of confidence for that estimate), it is possible to capture specific numbers for four main important time relationships between the nodes in a sequence. In the traditional construct, they include the following: start-to-start, finish-to-finish, finish-to-start, and start-to-finish relationships between any two linked nodes (activities). In the visual below, the nodes or activities are the rectangle listed as “prior” and the next as “subsequent.”

Figure 1. AON Time Relationships Types (of Precedence Diagram Method)

A fully mapped project would be a complex and labeled node-link diagram of all activities. Behind all the estimates should be deep knowledge of the work and the “space” around the work. Beyond its design, a model is only as good as the accuracy of its inputs. Once those elements are clear, it may be easier to condense work and project the quickest time a project may complete or the longest time a project may take (if all assumed lagging phenomena come to pass).

The more complex a project—think collaborations across organizations, think engineering or science angles, think supply chains, think regulatory oversights—the more relevant it would be to harness AON diagrams or project management technologies that enable “precedence diagramming” through “precedence networks.” This basic approach can be complexified, and it may be used with dedicated tools (vs. a general spreadsheet tool.

An AON Template

One simple setup for AON templates involves defining basic unit elements of a task activity in a project work chain. The activities should vary by type and should be calculable with the same scale throughout (such as hours, days, weeks, months, or some combination of the prior). Based on common practice in particular fields, there may be conventions used for the depiction of particular work.

The elements are as follows:
One general layout look follows: 

Early Start (ES)


Duration (D)


Early Finish (EF)


Task Activity


Late Start (LS)


Float (F) [or “Slack” or “Leeway”]


Late Finish (LF)

The Early Start, Late Start, Early Finish, and Late Finish, are listed as Days, as in Day 1, Day 2, Day 3, etc., from the start day of the project and counting consecutively until the finish of the project. There may be gaps between tasks, however, given the real-world complexity of projects. The Duration cell is comprised of the numbers of days required to complete the particular task activity (independent of the Early Start-Early Finish, Late Start-Late Finish). The “Float” or “Slack” or “Leeway” is sort of the main point of AON diagrams. The activity task Float identifies the amount of time that an activity may be delayed with affecting subsequent work tasks negatively (by delaying other follow-on parts of the project). This is calculated by subtracting the late start from the early start (you save time by starting early) and by taking paths with work activities that have lower time durations.

A forward pass determines the early starts of the activities (based on the estimated duration for each task).  A backward pass is done to figure out the late finishes.  Then, late starts are used to subtract early starts to capture the possible floats for each of the task activities. With the mapping available, the critical paths are identified.  Critical paths are identified when the Early Finish (EF) and the Late Finish (LF) are the same (as highlighted in the pink in the visual).  In the forward analysis, the largest of the early starts are carried forward if multiple arrows point to the forward cell.  In the backward analysis, the smallest of the Late Starts is carried to the prior on a backward pass.  The idea is that all precursor activities have to be given sufficient time to be achieved.  The layout of the AON diagram is based on a general chronological time sequence, from left to right. 

In the video, note the approach for filling out the ES, LS, EF, and LF variables. The video begins with a Table of Dependencies, which is a useful summary construct.

Video: Use forward and backward pass to determine project duration and critical path

A “critical path” is defined as sequences of zero-slack task activities in a project. These paths are identified through various criticality analyses, and these use the four relationships mentioned above. Some six classes of relationships between task activities are part of this calculation: “normal, reverse, neutral, bicritical, increasing normal and decreasing reverse” (Valls & Lino, 2001, p. 17). A project’s duration cannot be shorter than the sum of its critical paths or its single critical path. In terms of areas that project managers may look for slack (ways to save on time), a critical path is not generally one. After all, any delay or lag in the critical path will affect the project’s complete date. Critical paths with 0 slack are often part of the longest paths of the network, aka the critical paths.

A Simple AON Diagram

So what might a manual AON diagram look like, at least in simple form?  The following shows a 48-day project...with a few days of slack possible only.  Rather, most of the depicted sequence is time-linear and on a critical path. 

Figure 2.  A Quick Sample AON Diagram

As to modeling complexity, there are some types of models where paths can enter and exit particular tasks at various stages, whether from the start point or the end point, so there are complexities. Some paths may enter a task sequence from the Finish, which has implications on shortening the finish time of the project (the activity is already completed). Some activities may be split into different paths, with finer points of duration and early and late starts and finishes. Other task activities are not able to be split at all. Some projects can be parallelized, so that different teams are working on different parts, to speed up the work and to enable an earlier completion date, ceteris paribus.

For a project’s Total Float (TF), the cumulative information is studied to figure out the latest that an activity start time can be delayed…or an activity duration increased…without delaying the project’s completion time. This can also show how much accelerating a start date or lowering the duration of a task may advance the project’s projected completion time. This is the cumulative leeway or slack in a project.

All incoming links to an activity node affects its starting point, and all exiting links affect its ending point (and those activities that are dependent on that particular task activity node).

There are variations on AON mapping, with extended mapping, to include higher dimensional modeling. For the various projects, there are different conditions related to human resources, material resources, budgets, leadership, and other factors, that may affect how the project actually functions.

Different Representations of AON Diagrams as Visuals, as Equations

An AON diagram is represented in different ways visually. A common approach is a node-link (or vertex-arc) diagram. Here, nodes or vertices represent activities and links or arcs show precedence relationships (basic setup) (Valls & Lino, 2001, p. 18).

While “AON” references “nodes” (as the location of the activities), AON diagrams may be depicted as various types of diagrams: linked bar charts, flowcharts, time-scale networks, time-scale timelines, structured trees, and others.

Equation-wise, the network-based schedule diagram is represented as a set: G = (V, A), where G is the AON diagram, and it equals the set of all the vertices and arcs (all the nodes and links). This is for a basic AON diagram.


Video: Calculating Critical Path with Forward and Backward Pass – Key Concepts in Project Management



This work offers a simple sense of a manual Activity-on-Node (AON) diagram. This approach reads as very intuitive to me, and it also reads as practically valuable.

In all projects, some parts are negotiable, and other parts not. Project objectives are often non-negotiable. Branding is non-negotiable. Laws related to intellectual property, privacy protections, publishing, accessibility, and such, are non-negotiable. Working technologies are non-negotiable. Hard deadlines are non-negotiable (or highly expensive to break).

Soft deadlines are negotiable. Work assignments may be somewhat negotiable. Some developmental technologies are negotiable. And so on.

In the real-world, the project mapping can become quite intensive. For those who want to learn more, there are many resources on social media, the academic research literature, commercial software documentation, and others.

AONs are a good way to enable a team to collaborate around their shared understandings of the work.


Valls, V., & Lino, P. (2001). Criticality analysis in activity-on-node networks with minimal time lags. Annals of Operations Research, 102, 17 – 37.


About the Author

Shalin Hai-Jew works as an instructional designer and researcher at Kansas State University. Her email is shalin@ksu.edu.

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