Master Task Scheduling with AOV and AOE Networks: From Class Timetables to Project Plans

Life is a series of schedules; to avoid chaos we need to know which task comes first. AOV (Activity on Vertex) and AOE (Activity on Edge) networks are classic graph‑based tools for modeling task dependencies and durations.

AOV Network: Task Dependency

What is an AOV network?

An AOV network represents tasks as vertices and precedence relations as directed edges in a DAG. No cycles are allowed.

Vertex : a task (e.g., “Math class”).

Edge : a dependency (e.g., “Math must precede Physics lab”).

Acyclic : prevents contradictory loops.

One‑sentence summary: AOV is a “who‑goes‑first” dependency graph.

Practical use: Course scheduling

Example constraints:

Math must be taught before Physics lab and Programming.

Physics lab must finish before Engineering practice.

Programming must finish before Software testing.

Converted to an AOV graph:

<code>Math → Physics Lab<br/>Math → Programming<br/>Physics Lab → Engineering Practice<br/>Programming → Software Testing<br/></code>

Topological sorting solves the schedule:

Find tasks with no predecessors (in‑degree 0).

Place them, remove their outgoing edges.

Repeat until all tasks are ordered.

Resulting order:

<code>Math → (Physics Lab, Programming) → (Engineering Practice, Software Testing)<br/></code>

AOE Network: Project Progress Management

What is an AOE network?

In an AOE network vertices represent events (start or finish), and edges represent tasks with time weights.

One‑sentence summary: AOE is a “task + time” progress graph.

Practical use: Construction project

Tasks with durations (days):

Foundation construction: 5 days (event 1→2)

Rebar tying: 3 days (event 2→3)

Wall building: 4 days (event 2→4)

Roof construction: 2 days (event 3→5)

Finishing: 3 days (event 4→5)

Converted to an AOE graph:

<code>Foundation → Rebar (5d)<br/>Foundation → Wall (5d)<br/>Rebar → Roof (3d)<br/>Wall → Finishing (4d)<br/>Roof → Project Completion (2d)<br/>Finishing → Project Completion (3d)<br/></code>

Critical path analysis

Project managers need the earliest completion time and the tasks that cannot be delayed. The critical path is found by computing earliest start (ES) and latest start (LS) times:

<code>ES(j) = max(ES(i) + w(i,j))</code>

<code>LS(i) = min(LS(j) - w(i,j))</code>

The critical path (all zero‑slack tasks) is:

<code>Foundation → Wall → Finishing → Project Completion<br/></code>

Tasks on this path must not be delayed, otherwise the whole project slips.

AOV vs. AOE

Focus : AOV – task order; AOE – task time.

Vertex represents : AOV – task; AOE – event.

Edge represents : AOV – dependency; AOE – task with duration.

Key question : AOV – ensure correct dependencies; AOE – compute duration and critical path.

Main algorithm : AOV – topological sort; AOE – critical‑path method.

Typical scenarios : AOV – course scheduling, production dispatch; AOE – project planning, construction management.

Both AOV and AOE provide simple yet powerful ways to eliminate scheduling chaos, whether you are arranging classes or managing a construction timeline.