PERT / CPM Network

About PERT charts

A PERT chart (Program Evaluation and Review Technique) — and its near-twin CPM (Critical Path Method) — is the foundational project-scheduling diagram of modern project management. Born in 1959 (PERT from the US Navy Polaris program, CPM from DuPont), the two converged into the Activity-on-Node / Precedence Diagramming Method taught by PMI's PMBOK Guide. You write the tasks, their durations, and which task depends on which; the schedule — Early Start, Early Finish, Late Start, Late Finish, slack, and the critical path — is computed.

That last point is what makes Schematex's pert engine different from every drag-and-drop "PERT chart maker". Most tools ship a shape library and make you fill the six-field box and compute the critical path with a calculator. Schematex runs the forward pass and backward pass itself and highlights the critical path in red automatically. Distinct from flowchart (no scheduling), timeline (no critical-path math), and bpmn (an organisational process, not a one-off project schedule).

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1. Your first diagram

Every document starts with the pert keyword, an optional header, then one task line per activity:

pert
unit: days

task A "Market research" duration: 5
task B "Design mockups"  duration: 8 after: A
task C "Backend API"     duration: 15 after: A
task D "Frontend build"  duration: 10 after: B, C

Each task carries an <id>, a quoted <label>, a duration:, and an optional after: list of predecessors. The engine adds an invisible Start and Finish, runs the schedule, and draws the six-field activity box for every task. You never type ES/EF/LS/LF — they are computed.

The header accepts:

title: "…" — a heading drawn above the diagram.
unit: days | weeks | hours | abstract — the time unit (default days; purely a label, the engine is calendar-agnostic).
direction: LR | TB — left-to-right (default) or top-to-bottom.
layout: network | timescaled — the layered network (default) or a time-proportional view (§6).
critical-tolerance: <n> — slack ≤ this counts as critical (default 0; set 0.001 for three-point projects).
show-sentinels: true — draw the synthetic Start / Finish nodes (hidden by default).

2. The six-field activity box

Every task renders as the canonical 3×2 PERT/CPM rectangle:

┌──────────┬────────────┬──────────┐
│  ES      │  Duration  │   EF     │
├──────────┴────────────┴──────────┤
│         Task Name (ID)           │
├──────────┬────────────┬──────────┤
│  LS      │  Slack     │   LF     │
└──────────┴────────────┴──────────┘

ES / EF — Early Start / Early Finish, from the forward pass.
LS / LF — Late Start / Late Finish, from the backward pass.
Slack (total float) = LS − ES = LF − EF. Zero slack means the activity is on the critical path.

Critical activities get a red border and a bold 0 slack; the project's critical chain reads as an unbroken red line. Every field is also mirrored onto data-* attributes (data-es, data-slack, data-critical, …) so you can query the SVG without re-running the math.

On the two-colour palette. A PERT chart is deliberately drawn in just two colours. Red for the critical path is a genuine industry convention — MS Project, Oracle Primavera P6, and PMBOK figures all use it, because "critical vs. not" is the one distinction the diagram exists to surface. Everything else stays a neutral house-blue. Adding more colours would imply categories that PERT's semantics don't have; if you need to group by team or phase, use swimlanes (lane:) instead of colour.

3. Dependencies (FS / SS / FF / SF + lag/lead)

after: takes a comma-separated list of predecessor references. The default relationship is Finish-to-Start (FS) — a task starts once its predecessor finishes:

task D "Frontend build" duration: 10 after: B, C

Modern scheduling needs the other three Precedence-Diagramming relationships and lag (delay) or lead (negative lag):

task B "Stakeholder interviews" duration: 6 after: A SS+1   # start 1d after A starts
task I "Documentation"          duration: 4 after: D+3      # FS with a 3-day lag
task J "Translation"            duration: 3 after: I SS-1   # start 1d before I starts (lead)
task K "Sign-off"               duration: 1 after: I FF     # finishes when I finishes
task L "Press release"          duration: 2 after: G SF+1   # start-to-finish (rare)

Form	Meaning
`after: A`	Finish-to-Start, no lag (the common case)
`after: A+2` or `after: A+2d`	FS with a 2-unit lag
`after: A SS` / `A FF` / `A SF`	Start-to-Start / Finish-to-Finish / Start-to-Finish
`after: A SS+2d` / `A FF-1d`	any type with lag (`+`) or lead (`-`)

A lag unit suffix (d / w / h) must match the diagram's unit: or be omitted; mixed units are rejected. FS with zero lag is unlabelled; SS/FF/SF always show their type on the edge.

4. Three-point (PERT) estimation

Write a duration as O/M/P (optimistic / most-likely / pessimistic) and the engine computes the beta-distribution expected duration te = (O + 4M + P) / 6 and the variance σ² = ((P − O)/6)²:

pert
critical-tolerance: 0.01
task A "Spec"   duration: 2/3/5    # te = 3.17, σ² = 0.25
task B "Build"  duration: 5/8/14   after: A   # te = 8.50, σ² = 2.25
task C "Test"   duration: 3/4/6    after: B
task D "Deploy" duration: 1/2/3    after: C

The Duration field shows te; a small σ=… annotation appears under the name; and the project-level standard deviation (√ of the summed critical-path variances) is reported in the footer. Use critical-tolerance: 0.01 so floating-point te values don't displace the visible critical path.

5. Milestones

A milestone is a zero-duration checkpoint, drawn as a diamond. Use the milestone flag or duration: 0:

task P "Cutover weekend" milestone after: O
task Q "Go-live"         duration: 0 after: P

6. Time-scaled layout

layout: timescaled switches from the layered network to a network-Gantt hybrid: each activity's x-position is proportional to its ES and its width is proportional to its duration, with a unit time axis along the bottom. Activities are packed into lanes so nothing overlaps.

pert
layout: timescaled
unit: days

task A "Inventory"     duration: 5
task B "Interviews"    duration: 6  after: A SS+1
task C "Design"        duration: 10 after: A, B
task D "Build"         duration: 15 after: C
task E "Pilot"         duration: 7  after: D

This is a bridge to a Gantt view, not a replacement: there's no calendar, working-time mask, or resource swimlane.

Activity-on-arrow (`layout: aoa`)

layout: aoa renders the older activity-on-arrow (ADM) notation you'll find in textbooks and on Investopedia: numbered event circles, activities as labelled arrows, and dotted dummy activities auto-inserted wherever an activity has two or more predecessors. You write the same activity-on-node DSL — Schematex builds the event graph and numbers the events for you.

pert
layout: aoa
unit: days

task A "create schedule" duration: 10
task B "buy hardware"    duration: 5
task C "programming"     duration: 20 after: A
task D "installation"    duration: 5  after: B
task E "conversion"      duration: 15 after: D
task F "test code"       duration: 20 after: C, E
task G "write manual"    duration: 15 after: E

AOA is a legacy notation (PMBOK 7 dropped it; AON is the modern standard) and it can only express finish-to-start logic — SS/FF/SF and lag/lead are flattened to FS with a warning. Use it for teaching, exam prep, or matching an existing textbook figure; use the default network (AON) layout for real scheduling.

7. Swimlanes, tags, classes, and comments

Add lane: "…" to any task and the network re-groups into horizontal swimlanes — by responsible team, phase, or owner — while still computing the schedule exactly as before:

task T1 "Support Account Deletion" duration: 3  lane: "Customer Account"
task T2 "Design a New Theme"        duration: 8  lane: "Shopping Site"
task T3 "Apply New Theme"           duration: 15 after: T2 lane: "Shopping Site"

Lanes appear in first-declared order, each as a banded row with a label gutter on the left. Swimlanes are a presentation of the same AON network, not a different layout mode — they activate automatically when any task declares a lane.

task X "External vendor work" duration: 10 after: A tags: vendor, external class: secondary
# this is a comment

tags: emit data-tag="…" on the node group and class: adds a CSS class for downstream theming. # and // start a comment to end of line.

8. Grammar (EBNF)

document    = "pert" NEWLINE header* task+
header      = "title:" quoted
            | "unit:" ("days" | "weeks" | "hours" | "abstract")
            | "direction:" ("LR" | "TB")
            | "layout:" ("network" | "timescaled" | "aoa")
            | "critical-tolerance:" number
            | "show-sentinels:" boolean

task        = "task" IDENT quoted "duration:" duration ("after:" reflist)? "milestone"? attrs?
            | "task" IDENT quoted "milestone" ("after:" reflist)? attrs?
duration    = number | number "/" number "/" number          ; deterministic or O/M/P
reflist     = ref ("," ref)*
ref         = IDENT (WS deptype)? laglead?
            | IDENT "+" number unit?                          ; attached FS lag sugar
deptype     = "FS" | "SS" | "FF" | "SF"
laglead     = ("+" | "-") number unit?
unit        = "d" | "w" | "h"
attrs       = ("tags:" idlist)? ("class:" IDENT)? ("lane:" quoted)?

9. Standard compliance

Schematex pert implements the Activity-on-Node / Precedence Diagramming Method per PMI PMBOK 7 and Moder, Phillips & Davis (1983), with the six-field box convention from Kerzner and Oracle Primavera P6. The forward/backward pass, total slack, critical path, all four PDM dependency types with lag/lead, and three-point estimation (te + variance) are computed exactly. The older Activity-on-Arrow notation is available as an opt-in legacy view (layout: aoa) for teaching and textbook parity. Out of scope for v0.1: resource leveling / RCPSP, Monte Carlo schedule-risk simulation, calendar-aware durations, and MS Project / Primavera import-export. See docs/reference/32-PERT-STANDARD.md for the full specification.

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