CPM and PERT Network Scheduling

Critical Path Method

The Critical Path Method (CPM) is a deterministic network-scheduling technique that calculates activity timing and identifies the longest-duration path through the network. That path determines the minimum project duration under the stated logic and durations.

Quantity	Symbol	Calculation
Earliest start	$ES$	Maximum $EF$ among immediate predecessors
Earliest finish	$EF$	$ES+d$
Latest finish	$LF$	Minimum $LS$ among immediate successors
Latest start	$LS$	$LF-d$
Total float	$TF$	$LS-ES=LF-EF$
Free float	$FF$	$\min(ES_{\text{successors}})-EF$

CPM timing quantities

Here $d$ is activity duration. With the usual finish constraint, a critical activity has zero total float. A delay to a critical activity delays project completion unless time is recovered elsewhere or the network logic changes.

CPM procedure

1. List activities, durations, and precedence relationships.

2. Draw a valid activity-on-node or activity-on-arrow network.

3. Make a forward pass from left to right to calculate $ES$ and $EF$.

4. Make a backward pass from right to left to calculate $LF$ and $LS$.

5. Calculate total and free float; trace the continuous zero-float path(s).

6. Check calendar, resource, and constraint assumptions before baselining.

Note

Example — CPM network and critical path

The activities are:

Activity	Duration (days)	Immediate predecessor(s)
A	3	–
B	4	–
C	2	A
D	5	B
E	3	A, B
F	4	C, D, E

The forward pass gives $ES_E=\max(3,4)=4$ and $ES_F=\max(5,9,7)=9$.

Activity	$d$	$ES$	$EF$	$LS$	$TF$
A	3	0	3	3	3
B	4	0	4	0	0
C	2	3	5	7	4
D	5	4	9	4	0
E	3	4	7	6	2
F	4	9	13	9	0

The zero-float chain is

$$\boxed{B\rightarrow D\rightarrow F},\qquad T_{\text{project}}=4+5+4=13\text{ days}.$$

Activity-on-node network for the CPM example

Program Evaluation and Review Technique

The Program Evaluation and Review Technique (PERT) is a probabilistic network technique that represents uncertain activity duration by optimistic, most likely, and pessimistic estimates.

Estimate	Symbol	Interpretation
Optimistic	$t_o$	Plausible minimum duration under favorable conditions
Most likely	$t_m$	Modal duration under normal conditions
Pessimistic	$t_p$	Plausible maximum duration under unfavorable conditions

PERT three-point estimates

Caution

Key Formula — PERT activity moments

$$t_e=\frac{t_o+4t_m+t_p}{6},\qquad \sigma=\frac{t_p-t_o}{6},\qquad \sigma^2=\left(\frac{t_p-t_o}{6}\right)^2.$$

For one selected critical path, the classical approximation assumes independent activity durations:

$$\mu_{\mathrm{path}}=\sum t_e, \qquad \sigma^2_{\mathrm{path}}=\sum\sigma_i^2.$$

Feature	CPM	PERT
Duration model	One deterministic estimate	Three estimates summarized by an expected value and variance
Typical context	Repetitive or well-understood work such as construction	Novel, research, or uncertain work
Main emphasis	Schedule logic and time–cost trade-off	Duration uncertainty and completion probability
Limitation	Precision may conceal uncertain inputs	Approximation assumes a distribution and may ignore path switching and correlation

CPM and PERT compared

Note

Example — PERT expected times and critical paths

Consider the following estimates:

Activity	$t_o$	$t_m$	$t_p$	Predecessor(s)
A	2	4	6	–
B	1	3	11	–
C	3	5	7	A
D	2	6	10	B
E	4	5	12	A, B

Expected durations are

$$t_{e,A}=4, \quad t_{e,B}=4, \quad t_{e,C}=5, \quad t_{e,D}=6, \quad t_{e,E}=6\text{ days}.$$

Thus $A\!\to\!C=9$ days, while $B\!\to\!D=10$ days and both predecessor branches feeding $E$ are tight at $4+6=10$ days. The expected completion is $10$ days, with critical chains $A\!\to\!E$, $B\!\to\!E$, and $B\!\to\!D$ under these expected durations.

For the selected path $B\!\to\!D$,

$$\sigma_B^2=\left(\frac{11-1}{6}\right)^2=2.78, \qquad \sigma_D^2=\left(\frac{10-2}{6}\right)^2=1.78,$$$$\sigma_{B-D}^2=4.56, \qquad \sigma_{B-D}=\sqrt{4.56}=2.14\text{ days}.$$

Caution: Because several critical paths coexist, $2.14$ days is the standard deviation of the selected $B$–$D$ path, not an exact standard deviation for the maximum of all project paths. Correlation and path switching require simulation or a fuller network-risk analysis.

[!TIP]

Exam Focus — Network calculations

Show the network logic, forward pass, backward pass, float, and continuous critical path. A longest numerical sum that violates precedence is not a valid critical path.