Project Management
Introduces Pert / CPM as a tool for
planning, scheduling, and controlling
projects
Applied Management Science for Decision Making, 2e © 2014 Pearson Learning Solutions
Philip A. Vaccaro , PhD
Project Management Overview
Pert / CPM
History
Conventions
Project
Scheduling
Budgeting
Crashing
Phases
Probabilistic
PERT
Pert / CPM
Building
Blocks
Task
Times
ES, EF,
LS, LF,
S
Project Applications
Product
Development
and
Rollout
Product
Promotion
Campaign
Broadway
Shows
Defense
Contracts
Software
Conversion
Corporate
Restructure
Bridges
and
Highways
Applied Management Science for Decision Making, 2e © 2014 Pearson Learning Solutions
Virtually
every
company
engages
in one
or more
projects
on a
continual
basis
Consequences
of Failure
Lost Revenue, Profit,
and
Contract Penalties
Resource Waste
and
Reputation Damage
Failed Project
Repercussions
Loss
of
Clientele
Cost
and
Time Overruns
Managers
personally
suffer
the
loss
of
career
advancement,
if not
outright
termination
Project Task Prerequisites
Clear Start &
Finish Points
Alternate
Execution
Sequences
Several
Possible Time
Estimates
Running
Parallel Over
Time
Potential to
be Shortened
THESE PREREQUISITES
STRENGTHEN MANAGEMENT
ACCOUNTABILITY
AND PROVIDE
FLEXIBILITY IN THE
FACE OF FUNDING CHANGES,
TIME CHANGES, DELAYS,
STAFFING, & TECHNICAL
PROBLEMS
Project Task Prerequisites
Tasks with
Clear Start
and Finish
Points
• A new home
foundation starts when
the excavation crew
arrives on site and
ends when the poured
foundation has dried
and the frames are
removed
The
“Meeting
of the
Trades”
refers to the
visitation
schedule
that the
carpenters,
electricians,
plumbers,
and finishers
agree to, in
order to keep
site congestion
to a minimum
Project Task Prerequisites
Tasks with
Alternate
Execution
Sequences
• Tasks that can be
reordered might result
in shorter overall
execution times and
less cost
Project Task Prerequisites
Tasks with
Several
Possible Time
Estimates
• We identify the best case,
worst case, and most
likely time estimates for
each project task
• This allows us to better
adopt to changes in
funding, deadlines, and
unforeseen technical
issues.
Project Task Prerequisites
Assistance
may take
the
form of
extra funds,
personnel,
and
equipment.
Tasks that
Run Parallel
to Each
Other
• Tasks should be
scheduled to run at
the same time with
one or two others
• If one task is in
danger of falling
behind, others stand
ready to assist
This will,
in turn,
save the
entire
project
from
falling
behind
schedule !
Critical Path Method History
CPM was developed in 1957
by J.E. Kelly of Remington
Rand and M.R. Walker of
Dupont Chemical
First used to reduce
chemical refinery
construction from 7 to 4
years
Requires only one time
estimate for each project
task
Program Evaluation & Review Technique
Developed in 1958 by the U.S.
Navy, Lockheed Missile
Systems Division, and Booz,
Allen and Hamilton
Consultants
First used to plan and control
the Polaris attack submarine
program, reducing sub build
time from 7 to 4 years
In 1960, PERT and CPM were
combined, hence the term
PERT / CPM !
PERT
requires
3
time
estimates
for each
project
task
U.S. Navy Special Projects Office
Grace Murray Hopper
1906 - 1992
Famous
Staff
Member
PhD, Yale
University
1934
1st
Woman
Admiral
Developed
COBOL
Lecturer
programming
and the
compiler
Consultant
Developed
International
Standards
for Computer
Languages
Engineer
Operations
Researcher
Professor
Vassar
College
1931-1941
47
Honorary
Degrees
Grace Murray Hopper
While she was working on the Mark II computer at Harvard
University, her associates discovered a moth stuck on a
relay, thereby impeding operation.
Whereupon she remarked that they were “debugging” the
system.
The remains of the moth can be found at the Smithsonian
Museum of American History in Washington, D.C.
ANECDOTE
PERT / CPM
The Two Conventions
Activity-on-Arc
TASKS ARE SHOWN AS
ARROWS ( ARCS )
Activity-on-Node
NODES REPRESENT
TASK
START AND FINISH
TASKS ARE SHOWN AS
SQUARES ( NODES )
Activity-on-Node
ARROWS REPRESENT
TASK PREDECESSOR
RELATIONSHIPS
Activity-on-Node Convention
The network is
cleaner and
uncluttered
It is natural to view
nodes as tasks
It is easier to use
than the AOA
convention
The U.S.
Government
converted to the
AON convention in
2001
Firms desiring
government
contracts must use
AON convention
ADVANTAGES
Activity-on-Node Building Blocks
Nodes represent the project tasks
Small nodes represent the project start
and finish
Arcs / arrows indicate the predecessor
relationships among the tasks
start
1st
Task
2nd
Task
3rd
Task
Here,
the 2nd task
cannot begin
until the
1st task
has been
completed.
The 3rd task
cannot begin
until the
2nd task
has been
completed.
end
GENERAL FOUNDRY INC.
Task A Build
Internal
Component
Task B Modify
Roof and Floor
Task D Pour
Concrete
Task C
Construct
Collection Stack
Task E
Build Burner
Task G Install
Pollution Device
Task F Install
Control
System
Task H Inspect
and Test
General Foundry Inc.
A
2
START
Time
can be
expressed
in
days, weeks,
or months
C
F
2
3
E
TIME in WEEKS
H
4
B
3
2
D
4
G
5
FINISH
Task Interpretation
EARLIEST TIME
TASK “A” CAN
START IS AT THE
END OF WEEK “0”
THAT IS, THE START
OF WEEK “1”
ES = 0
TASK “A”
EXPECTED
DURATION
TIME IS 2
WEEKS
A
2
TASK “A”
EF = 2
EARLIEST TIME
TASK “A” CAN
FINISH IS AT THE
END OF WEEK “2”
THAT IS, THE START
OF WEEK “3”
Expected Task or Activity Time
A WEIGHTED AVERAGE TIME FORMULA
𝑡𝑒 =
optimistic
time
estimate
[ 1𝑎 + 4𝑚 + 1𝑏]
6
17%
Sum of the Weights
most likely
time
estimate
67%
pessimistic
time
estimate
17%
Weights
Expected
times
are usually
used
for each
task in
the project
Expected Task or Activity Time
EXAMPLE
GIVEN: a = 1 week , b = 3 weeks , m = 2 weeks
OPTIMISTIC
TIME
𝑡𝑒 =
𝑡𝑒 =
PESSIMISTIC
TIME
MOST LIKELY
TIME
(1𝑎 + 4𝑚 + 1𝑏)
6
[1 1 +4 2 +1 3 ]
6
=
12
6
=
= 2 weeks
The BETA Distribution
Skewed
Distribution
The probability
distribution
commonly used
to describe the
inherent variability
in task time
estimates
TASK TIME IS NOT
ASSUMED TO BE
NORMALLY
DISTRIBUTED
a
Optimistic
time
m
te
b
Most likely
time
(mode)
Expected
time
Pessimistic
time
The BETA DISTRIBUTION
CHARACTERISTICS & COMMENTS

Symmetrical, right, or left-skewed based
on the nature of a particular task

Unimodal with a high concentration of
probability surrounding the most likely
time estimate (m)

No strong empirical reason for using the
BETA distribution
 Attractive however, because the mean (μ)
and the variance (𝝈𝟐 ) can be easily obtained
from the three time estimates “a”, “m”, and “b”
Even
if a task
actually
had a
normally
distributed
time, we would
still use the
Beta
Distribution !
The Critical Path ( CP )
A
B
C
start
end
D
The chain of tasks from project
start to end that consumes the
longest amount of time.
Any delay in one or more of
those tasks will delay the entire
project !
The critical path is the project’s
expected or mean completion
time
E
Critical Path Characteristics
Several critical paths may exist
within the project network at
any given time.
These critical paths may change
or disappear entirely at any time
as the project progresses.
Management must monitor all
critical paths closely.
General Foundry Inc
A
2
START
C
2
3
E
TIME in WEEKS
H
4
B
3
F
2
D
4
G
5
FINISH
General Foundry Inc.
1st Critical Path Candidate
A
2
START
C
2
3
E
A-C-F-H
Nine (9) Weeks
H
4
B
3
F
2
D
4
G
5
FINISH
General Foundry Inc.
2nd Critical Path Candidate
A
2
START
C
2
3
E
A-C-E-G-H
Fifteen (15) weeks
B
3
F
H
4
2
D
4
G
5
FINISH
General Foundry Inc.
3rd Critical Path Candidate
A
2
START
C
2
3
E
B-D-G-H
Fourteen (14) weeks
B
3
F
H
4
2
D
4
G
5
FINISH
The Critical Path
A-C-F-H
( 9 weeks )
A-C-E-G-H
( 15 weeks )
B-D-G-H
( 14 weeks )
The
expected,
mean,
or
average
project
completion
time
is
15 weeks
General Foundry Inc.
The Critical Path
A
2
START
C
F
2
3
E
Fifteen (15) Weeks
H
4
B
3
2
D
4
G
5
FINISH
Expected, Mean, or Average
Project Completion Time
50% CHANCE OF
COMPLETION AFTER
μ (15 weeks)
50% CHANCE OF
COMPLETION BEFORE
μ (15 weeks)
µ = 15
THE
CRITICAL
PATH
EQUALS
MEAN
PROJECT
COMPLETION
TIME
EARLY START TIME ( ES )
 The earliest time that
each task can begin.
 Computed from left
to right, that is, from
the network’s beginning node to the network’s finish node.
The technique is called
FORWARD PASS
EARLY START TIME FORMULA
IF THERE
ARE
SEVERAL
CANDIDATES
FOR
THE
FOLLOWER
TASK
ES ,
THE
LONGEST
ES
IS
SELECTED
THE
VERY
FIRST
TASK
IN A
PROJECT
HAS
AN EARLY
START
TIME
OF ZERO
Predecessor
Task
Early Start
Predecessor
Task
Expected Time
Follower
Task
Early Start
As we
progress
through
the project,
follower
tasks
become
predecessor
tasks
themselves !
𝑷𝒓𝒆𝒅𝒆𝒄𝒆𝒔𝒔𝒐𝒓 𝑬𝑺 + 𝑷𝒓𝒆𝒅𝒆𝒄𝒆𝒔𝒔𝒐𝒓 𝒕𝒆 = 𝑭𝒐𝒍𝒍𝒐𝒘𝒆𝒓 𝑬𝑺
General Foundry Inc.
EARLY START TIMES
ES = 0
ES = 2
A
2
ES = 4
C
F
2
3
ES = 4
E
TIME in WEEKS
START
ES = 13
H
4
2
ES = 0
ES = 3
ES = 8
B
D
G
3
4
5
FINISH
EARLY START TIME FORMULA
IF THERE
ARE
SEVERAL
CANDIDATES
FOR
THE
FOLLOWER
TASK
ES ,
THE
LONGEST
ES
IS
SELECTED
THE
VERY
FIRST
TASK
IN A
PROJECT
HAS
AN EARLY
START
TIME
OF ZERO
Predecessor
Task A
Predecessor
Task A
Early Start (0)
Expected Time (2)
Follower
Task C
Early Start (2)
As we
progress
through
the project,
follower
tasks
become
predecessor
tasks
themselves !
𝑷𝒓𝒆𝒅𝒆𝒄𝒆𝒔𝒔𝒐𝒓 𝑬𝑺 + 𝑷𝒓𝒆𝒅𝒆𝒄𝒆𝒔𝒔𝒐𝒓 𝒕𝒆 = 𝑭𝒐𝒍𝒍𝒐𝒘𝒆𝒓 𝑬𝑺
General Foundry Inc.
EARLY START TIMES
ES = 0
ES = 2
A
C
2
ES = 4
F
2
3
ES = 13
ES = 4
E
START
H
4
ES = 0
ES = 3
ES = 8
D
G
B
3
2
4
5
FINISH
EARLY START TIME FORMULA
IF THERE
ARE
SEVERAL
CANDIDATES
FOR
THE
FOLLOWER
TASK
ES ,
THE
LONGEST
ES
IS
SELECTED
THE
VERY
FIRST
TASK
IN A
PROJECT
HAS
AN EARLY
START
TIME
OF ZERO
Predecessor
Task B
Predecessor
Task B
Early Start (0)
Expected Time (3)
Follower
Task D
Early Start (3)
As we
progress
through
the project,
follower
tasks
become
predecessor
tasks
themselves !
𝑷𝒓𝒆𝒅𝒆𝒄𝒆𝒔𝒔𝒐𝒓 𝑬𝑺 + 𝑷𝒓𝒆𝒅𝒆𝒄𝒆𝒔𝒔𝒐𝒓 𝒕𝒆 = 𝑭𝒐𝒍𝒍𝒐𝒘𝒆𝒓 𝑬𝑺
General Foundry Inc.
EARLY START TIMES
ES = 0
ES = 2
A
C
2
ES = 4
F
2
3
ES = 4
ES = 13
E
START
H
4
ES = 0
ES = 3
ES = 8
D
G
B
3
2
4
5
FINISH
ES Candidate Selection
ES=4
E
4
COMING IN FROM TASK “E”
EARLY START TIME FOR TASK “G”
WOULD BE “8” ( 4 + 4 = 8 )
ES=8
G
ES=3
D
4
5
COMING IN FROM TASK “D”
EARLY START TIME FOR TASK “G”
WOULD BE “7” ( 3 + 4 = 7 )
THE
HIGHER
EARLY START
CONTROLS
General Foundry Inc.
EARLY START TIMES
ES = 0
ES = 2
A
C
2
ES = 0
3
ES = 4
ES = 13
E
H
4
2
ES = 3
ES = 8
D
G
B
3
F
2
TIME in WEEKS
START
ES = 4
4
5
FINISH
ES Candidate Selection
ES=4
F
3
COMING IN FROM TASK “F”
EARLY START TIME FOR TASK “H”
WOULD BE “7” ( 4 + 3 = 7 )
ES=13
H
ES=8
G
5
2
COMING IN FROM TASK “G”
EARLY START TIME FOR TASK “H”
WOULD BE “13” ( 8 + 5 = 13 )
THE
HIGHER
EARLY START
CONTROLS
EARLY FINISH TIME ( EF )
• The earliest time that
each task can finish.
• Computed from left to
right, that is, from the
network’s beginning
node to the network’s
finish node.
This technique is also called
FORWARD PASS
EARLY FINISH TIME FORMULA
Task
Early
Start
Time
Task
Expected
Time
(te)
Task
Early
Finish
Time
NEEDLESS
TO SAY,
EARLY
FINISH
TIMES
CANNOT
BE
COMPUTED
UNTIL
EARLY
START
TIMES
ARE
IDENTIFIED
General Foundry Inc.
EARLY FINISH TIMES
ES=0
EF=2
ES=2
EF=4
ES=4
EF=7
A
C
F
2
2
3
ES=4
EF=8
ES=13
EF=15
E
H
TIME in WEEKS
START
4
2
ES=0
EF=3
ES=3
EF=7
ES= 8
EF=13
B
D
G
3
4
5
FINISH
EARLY FINISH TIME
SELECTED CALCULATIONS
TASK
EARLY
START
TIME
ES = 0
A
+
TASK
EXPECTED
TIME
te = 2
A
=
TASK
EARLY
FINISH
TIME
EF = 2
A
NEEDLESS
TO SAY,
EARLY
FINISH
TIMES
CANNOT
BE
COMPUTED
UNTIL
EARLY
START
TIMES
ARE
IDENTIFIED
General Foundry Inc.
EARLY FINISH TIMES
ES=0
EF=2
ES=2
EF=4
ES=4
EF=7
A
C
F
2
2
3
TIME in WEEKS
START
ES=4
EF=8
ES=13
EF=15
E
H
4
2
ES=0
EF=3
ES=3
EF=7
ES= 8
EF=13
B
D
G
3
4
5
FINISH
EARLY FINISH TIME
SELECTED CALCULATIONS
TASK
EARLY
START
TIME
ES = 2
C
+
TASK
EXPECTED
TIME
te = 2
C
=
TASK
EARLY
FINISH
TIME
EF = 4
C
NEEDLESS
TO SAY,
EARLY
FINISH
TIMES
CANNOT
BE
COMPUTED
UNTIL
EARLY
START
TIMES
ARE
IDENTIFIED
General Foundry Inc.
EARLY FINISH TIMES
EF=2
EF=4
EF=7
A
C
F
2
2
3
TIME in WEEKS
START
EF=8
EF=15
E
H
4
2
EF=3
EF=7
EF=13
B
D
G
3
4
5
FINISH
LATE FINISH TIME ( LF )
The latest time that
each task can finish
without jeopardizing
the project’s expected
completion time.
Computed from right
to left, that is, from the
network’s finish node
to the network’s start
node.
This technique is called
BACKWARD PASS
LATE FINISH TIME FORMULA
FOLLOWER
TASK
LATE
FINISH
TIME
(LF)
-
FOLLOWER
TASK
EXPECTED
TIME
(te)
PREDECESSOR
=
TASK
LATE
FINISH
TIME
(LF)
IF
THERE
ARE
SEVERAL
CANDIDATES
FOR THE
PREDECESSOR
TASK LF,
SELECT
THE
SHORTEST LF
General Foundry Inc.
LATE FINISH TIMES
LF = 2
LF = 4
A
C
2
LF = 13
F
2
3
LF = 8
E
TIME in WEEKS
START
H
4
LF = 4
LF = 8
B
D
3
LF = 15
4
2
LF = 13
G
5
FINISH
LATE FINISH TIME
SELECTED CALCULATIONS
FOLLOWER
TASK
LATE
FINISH
TIME
(LF = 15)
H
-
FOLLOWER
TASK
EXPECTED
TIME
(te = 2)
H
PREDECESSOR
=
TASK
LATE
FINISH
TIME
(LF = 13)
F
IF THERE
ARE
SEVERAL
CANDIDATES
FOR THE
PREDECESSOR
TASK LF,
SELECT
THE
SHORTEST
LF
General Foundry Inc.
LATE FINISH TIMES
LF = 2
A
2
LF = 13
LF = 4
C
F
2
3
LF = 8
E
TIME in WEEKS
START
H
4
LF = 4
LF = 8
B
D
3
LF = 15
4
2
LF = 13
G
5
FINISH
LF Candidate Selection
LF=4
LF=13
C
2
THE SMALLER
LATE FINISH
TIME
CONTROLS
F
COMING IN FROM TASK “F”.
THE LATE FINISH TIME FOR
TASK “C” IS “10” (13-3=10)
3
LF=8
E
4
COMING IN FROM TASK “E”,
THE LATE FINISH TIME FOR
TASK “C” IS “4” (8-4=4)
LATE START TIME ( LS )
 The latest possible
time that each task
can start without
jeopardizing the
project’s expected
completion time.
 Computed from right
to left, that is, from
the network’s finish
node to the network’s
start node.
This technique is also called
BACKWARD PASS
LATE START TIME FORMULA
TASK
LATE
FINISH
TIME
(LF)
-
TASK
EXPECTED
TIME
(te)
=
TASK
LATE
START
TIME
(LS)
NEEDLESS
TO SAY,
TASK
LATE
START
TIMES
CANNOT BE
COMPUTED
UNTIL
TASK
LATE
FINISH
TIMES
ARE
IDENTIFIED
General Foundry Inc.
LATE START TIMES
LS=0
LF=2
LS=2
LF=4
A
C
2
F
2
3
TIME in WEEKS
START
LS=10
LF=13
LS=4
LF=8
LS=13
LF=15
E
H
4
2
LS=1
LF=4
LS=4
LF=8
LS= 8
LF=13
B
D
G
3
4
5
FINISH
LATE START TIME
SELECTED CALCULATIONS
TASK
LATE
FINISH
TIME
(LF = 15)
H
-
TASK
EXPECTED
TIME
(te = 2)
H
=
TASK
LATE
START
TIME
(LS = 13)
H NEEDLESS
TO SAY,
TASK
LATE
START
TIMES
CANNOT BE
COMPUTED
UNTIL
TASK
LATE
FINISH
TIMES
ARE
IDENTIFIED
General Foundry Inc.
LATE START TIMES
LS=0
LF=2
LS=2
LF=4
LS=10
LF=13
A
C
F
2
START
2
3
TIME in WEEKS
LS=4
LF=8
LS=13
LF=15
E
H
4
2
LS=1
LF=4
LS=4
LF=8
LS= 8
LF=13
B
D
G
3
4
5
FINISH
LATE START TIME
SELECTED CALCULATIONS
TASK
LATE
FINISH
TIME
(LF = 4)
B
-
TASK
EXPECTED
TIME
(te = 3)
B
=
TASK
LATE
START
TIME
(LS = 1)
B
NEEDLESS
TO SAY,
TASK
LATE
START
TIMES
CANNOT
BE COMPUTED
UNTIL
TASK
LATE
FINISH
TIMES
ARE
IDENTIFIED
General Foundry Inc.
LATE START TIMES
LS=0
LS=2
A
C
2
LS=10
F
2
3
LS=4
LS=13
E
H
TIME in WEEKS
START
4
LS=1
LS=4
B
3
2
LS=8
D
4
G
5
FINISH
SLACK TIME ( S )
ALSO
KNOWN
AS
PRIMARY

SLACK
The time
each
task
may be
postponed
without
jeopardizing
the project’s
expected
completion
time.
The
chain
of
zero
slack
tasks
in the
network
will also
identify
the
critical
path
SLACK TIME FORMULAE
THERE
ARE
TWO
VERSIONS
S = Task LS – Task ES
S = Task LF – Task EF
EITHER
ONE
PRODUCES
SAME
VALUES
BUT DO
NOT
“MIX
AND
MATCH”
General Foundry Inc
ES = 0 EF = 2
LS = 0 LF = 2
S=0
ES=2 EF=4
LS=2 LF=4
S=0
A
2
C
F
2
3
ES=4 EF=8
LS=4 LF=8
S=0
START
ES=13 EF=15
LS=13 LF=15
S=0
E
H
4
ES=0 EF=3
LS=1 LF=4
S=1
2
ES=3 EF=7
LS=4 LF=8
S=1
B
3
All
slack
time
calculations
ES= 4 EF= 7
LS=10 LF=13
S=6
D
4
G
5
ES=8 EF=13
LS=8 LF=13
S=0
FINISH
General Foundry Inc.
S=0
S=0
S=6
C
F
A
2
2
3
S=0
S=0
E
TIME in WEEKS
START
H
4
S=1
S=1
B
3
Primary
Slack
Times
for all
Tasks
2
S=0
D
4
G
5
FINISH
General Foundry Inc.
S=0
S=0
S=6
C
F
A
2
2
3
S=0
S=1
H
4
S=1
B
3
S=0
E
A-C-E-G-H
START
Critical
path
via
zero
slack
times
2
S=0
D
4
G
5
FINISH
Probabilistic PERT
Generates probabilities for
completing a project
both before and after its
expected completion date.
1. Critical path time ( CP or μ )
2. CP tasks’ optimistic times ( a )
REQUIRES 4
STATISTICS
3. CP tasks’ pessimistic times ( b )
4. CP tasks’ time variances ( 𝝈𝟐 )
Task Time Variance Formula
FOR THE BETA DISTRIBUTION
MUCH
SIMPLER
FORMULA
THAN THE
ONE FOR
THE
NORMAL
PROBABILITY
DISTRIBUTION
THE
PRIMARY
REASON
WHY
WE
ASSUME
THE
BETA
DISTRIBUTION
FOR
TASKS
2
𝝈
𝟐
b-a
=
6
where:
a = optimistic time
b = pessimistic time
6 = constant ( k )
Assume
the
critical
path
is
36.33
days
• CP = µ
Assume
the
tasks
along
the
critical
path
are:
•C,D,E,F,H,K
Assume
the
critical path
task time
variances
(in days) are:
• C = .11
• D = .11
• E = .44
• F= 1.78
• H = 1.00
• K = 1.78
An all
new
example
and
variances
are
fabricated
Requirements
What are the
chances of
finishing the
project in 30
days or less?
In other words,
P ( t =< 30 ) = ?
What are the
chances of
finishing the
project in 40
days or less?
In other words,
P ( t =< 40 ) = ?
Solution
Project
Variance = ∑ CP Task Variances
2
(σ )
5.22 days =
Project
Std Dev
(σ)
.11
.11
.44
1.78
1.00
1.78
= √5.22 = 2.28 days
σ = 2.28 days
.99728
The no. of standard deviates
between the mean ( μ ) and the
value of interest ( X )
.00272
-2.78 z
X = 30 days
Z = X – μ = 30.00 – 36.33 = - 2.78
σ
2.28
μ = 36.33 days
Project completion time is
normally distributed.
Therefore, a normal curve can
be drawn with a μ and σ.
Z
.08
2.7
.99728
The percentage of the normal
curve covered to a point that is
“2.78” standard deviates to the
left of the mean = 99.728%
Therefore, the
probability of
finishing the
project in 30
days or less is:
1 - .99728 = .00272
P( t =< 30 ) ≈ 0%
Conversely, the
probability of
finishing the
project in more
than 30 days is:
.99728
P( t > 30 ) ≈ 100%
σ = 2.28 days
Z = X – μ = 40.00 – 36.33 = +1.61
σ
2.28
.94630
+ 1.61 Z
μ = 36.33 days
.0537
X = 40 days
Project completion time is
normally distributed.
Therefore, a normal curve can
be drawn with a μ and σ.
The no. of standard deviates
between the mean (μ) and the
value of interest (X)
Z
.01
1.6
.94630
The percentage of the normal
curve covered to a point that is
“1.61” standard deviates to the
right of the mean = 94.630%
Therefore, the
probability of
finishing the
project in 40
days or less is:
.94630
P( t=<40 ) ≈ 95%
Conversely, the
probability of
finishing the
project in more
than 40 days is:
.0537
P( t>40 ) ≈ 5%
PERT / CPM with QM for WINDOWS
We scroll to
PROJECT MANAGEMENT
( PERT / CPM )
Applied Management Science for Decision Making, 2e © 2014 Pearson Learning Solutions
We have only one time
estimate for each task
in this project
We select the
SINGLE TIME ESTIMATE
program
There are 8 tasks
in the project
The tasks are labeled
A, B, C, D, etc.
“Precedence List”
is another term for
Activity-on-Node
Convention
The Data Input
Table
 “Prec” is an abbreviation for “Predecessor
Task”.
Here, the program provides for listing as
many as 7 predecessor tasks for each
task.
Project Estimated
Completion Time
( Critical Path )
Zero Slack Time Tasks
Are
Highlighted In Red
The 2nd Solution Is “CHARTS”
Four Different Charts
Can Be Brought Up
By Clicking Their Titles
Early Start
Early Finish
Late Start
Late Finish
Critical Path
Tasks
Here, the program displays
a precedence relationship
diagram based on what we
entered in the
“predecessor” columns
The Critical Path
Tasks Are Shown
In Red
Template
and
Sample Data
Template
and
Sample Data
Project Management