PROJECT SCHEDULING
A step-by-step guide to Problem Solving, Network Construction, and the
Critical Path Method (CPM) Algorithm.
Problem Solving Core
1
Network Construction
(AOA vs AON)
2
Simple Path Analysis
3
CPM Algorithm
(Forward/Backward
Pass)
Network Construction
Activity-on-Node (AON): The standard method.
Activities are rectangles. Arrows show dependency.
No dummy activities needed.
Activity-on-Arrow (AOA): Older method. Activities
are arrows. Nodes are events.
The Dummy Activity: Essential in AOA when two
activities share the same start and end nodes, or to
preserve logic during merges. It has 0 duration.
Sample Problem Data for Simple Path Analysis
Activity
Precedence
Duration (Weeks)
a
—
5
b
—
4
c
a
6
d
b
2
e
b
5
f
c, d
8
Simple Path Analysis
Methodology
Path Sequence
Calculation
Total Duration
a → c → f → (End)
5+6+8
19 weeks
b → d → f → (End)
4+2+8
14 weeks
b → e → (End)
4+5
9 weeks
For small networks, you don't need complex formulas. You can
simply list every possible path from Start to End and sum their
durations.
Note: This method becomes inefficient for large, complex
projects, which is why we use the Algorithm (Part 3).
In this hypothetical adjustment, a-c-f is the longest.
The CPM Algorithm
The systematic approach to determining project duration and float for every
single activity.
The Four Key Values for CPM Algorithm
ES (Early Start)
EF (Early Finish)
LS (Late Start)
LF (Late Finish)
The earliest possible
The earliest an activity
The latest an activity
The latest an activity
time an activity can
can finish. Calculated
can start without
can finish without
begin, assuming all
as ES + Duration.
delaying the entire
delaying the project.
predecessors are
finished.
project completion.
Sample Problem Data for CPM Algorithm
Activity
Precedence
Duration (Weeks)
a
—
4
b
—
6
c
a, b
5
d
b
7
e
c, d
2
f
e
1
Step 1: The Forward Pass
We move from Start to End to find ES and EF.
The Merge Rule
If an activity has multiple predecessors (a merge), its
ES is the MAXIMUM EF of all predecessors.
Why? Because you can't start until ALL previous steps are done.
Step 2: The Backward Pass
We move from End to Start to find LF and LS.
Start Condition
Set the LF of the last activity equal to the Project
The Burst Rule
Duration calculated in the Forward Pass.
If an activity is a predecessor to multiple tasks (a
burst), its LF is the MINIMUM LS of all successors.
Why? Because you must finish in time for the earliest requirement
of the next steps.
LF (Final Node) = 19
Step 3: Slack & Critical Path
Calculating Slack
Activities with Zero Slack have no flexibility. Any
delay in these tasks delays the whole project.