Chapter 13
Project Management
Characteristics of a “project”:
 A project
is unique (not routine),
 A project is composed of interrelated
sub-projects/activities,
 It is associated woth a large
investment.
What is Project Management
 To
schedule and control the progress
and cost of a project.
PERT/CPM:

Input:
–
–
–

Activities in a project;
Precedence relationships among tasks;
Expected performance times of tasks.
Output:
–
–
–
–
–
The earliest finish time of the project;
The critical path of the project;
The required starting time and finish time of each
task;
Probabilities of finishing project on a certain date;
...
PERT/CPM is supposed to
answer questions such as:
How long does the project take?
 What are the bottle-neck tasks of the
project?
 What is the time for a task ready to start?
 What is the probability that the project is
finished by some date?
 How additional resources are allocated
among the tasks?

PERT Network:
•
•
•
•
•
It is a directed network.
Each activity is represented by a node.
An arc from task X to task Y if task Y
follows task X.
A ‘start’ node and a ‘finish’ node are
added to show project start and project
finish.
Every node must have at least one outgoing arc except the ‘finish’ node.
Example of Foundry Inc., p.523
Activity
A
B
C
D
E
F
G
H
Immediate Predecessors
A
B
C
C
D, E
F, G
PERT Network for Foundry Inc. Example
Example of a Hospital Project:
Activity
Immediate Predecessor(s)
A
—
B
—
C
A
D
B
E
B
F
A
G
C
H
D
I
A
J
E, G, H
K
F, I, G
PERT Network for Hospital Project
Performance Time t of an Activity

t is calculated as follows:
t 
a  4m  b
6
where
a=optimistic time,
b=pessimistic time,
m=most likely time.

Note: t is also called the expected
performance time of an activity.
Variance of Activity Time t
•
If a, m, and b are given for the optimistic,
most likely, and pessimistic estimations of
activity k, variance k2 is calculated by
the formula
k
2
ba


 6 
2
Variance, a Measure of Variation
Variance is a measure of variation of
possible values around the expected
value.
 The larger the variance, the more spreadout the random values.
 The square root of variance is called
standard deviation.

Example, Foundry Inc., p.525
Activity
a
m
b
A
1
2
3
B
2
3
4
C
1
2
3
D
2
4
6
E
1
4
7
F
1
2
9
G
3
4
11
H
1
2
3
t
variance
Critical Path
•
•
•
It is the “longest” path in the PERT
network from the start to the end.
It determines the duration of the
project.
It is the bottle-neck of the project.
Time and Timings of an Activity:
•
•
•
•
•
•
t=estimated performance time;
ES=Earliest starting time;
LS=Latest starting time;
EF=Earliest finish time;
LF=Latest finish time;
s=Slack time of a task.
Uses of Time and Timings
Earliest times (ES and EF) and latest
times (LS and LF) show the timings of an
activity’s “in/out” of project.
 ES and LS of an activity tell the time
when the preparations for that activity
must be done.
 For calculating the critical path.

Computing Earliest Times
Step 1. Mark “start” node: ES=EF=0.
Step 2. Repeatedly do this until finishing all
nodes:
For a node whose immediate predecessors are all
marked, mark it as below:
•
•
ES = Latest EF of its immediate predecessors,
EF = ES + t
•
Note: EF=ES at the Finish node.
Computing Latest Times:
Step 1. Mark “Finish” node:
LF = LS = EF of “Finish” node.
Step 2. Repeatedly do this until finishing all
nodes:
For a node whose immediate children’s are all
marked with LF and LS, mark it as below:
•
•
LF = Earliest LS of its immediate children,
LS = LF – t
•
Note: LS=LF at Start node.
Computing Slack Times
•
For each activity:
–
slack = LS – ES = LF – EF
Foundry Inc. Example
•
Calculate ES, EF, LS, LF, and slack for each
activity of the Foundry Inc. example on its
PERT network, given the data about the
project as in the next slide.
Example, Foundry Inc.
Activity
a
m
b
t
variance
A
1
2
3
B
2
3
4
C
1
2
3
D
2
4
6
E
1
4
7
F
1
2
9
G
3
4
11
H
1
2
3
2
3
2
4
4
3
5
2
0.111
0.111
0.111
0.444
1
1.777
1.777
0.111
A
2
C
F
2
3
ES=
EF=
ES=
EF=
ES=
EF=
LS=
LF=
LS=
LF=
LS=
LF=
slack=
slack=
slack=
Start
E
4
H
ES=EF=
ES=
EF=
ES=
EF=
ES=EF=
LS=LF=
LS=
LF=
LS=
LF=
LS=LF=
slack=
B
3
D
4
slack=
G
5
ES=
EF=
ES=
EF=
ES=
EF=
LS=
LF=
LS=
LF=
LS=
LF=
slack=
slack=
Network for Foundry Inc.
slack=
2
Finish
Example of Hospital Project:
 Calculate
ES, EF, LF, LS and slack of
each activity in this project on its
PERT network, given the data about
the project as in the next slide.
Example: A Hospital Project
Activity
Immediate
Predecessor(s)
Performance time t
(weeks)
A
—
12
B
—
9
C
A
10
D
B
10
E
B
24
F
A
10
G
C
35
H
D
40
I
A
15
J
E, G, H
4
K
F, I, G
6
A
12
ES=
EF=
LS=
F
10
ES=
EF=
LS=
LF=
slack=
I
LF=
slack=
K
15
ES=
EF=
LS=
LF=
slack=
6
ES=
EF=
LS=
LF=
slack=
G
35
Start
C
10
ES=
EF=
ES=EF=
ES=
EF=
LS=
LF=
LS=LF=
LS=
LF=
slack=
D
10
H
40
9
ES=
EF=
ES=
EF=
ES=
EF=
LS=
LF=
LS=
LF=
LS=
LF=
Finish
ES=EF=
LS=LF=
slack=
B
slack=
slack=
slack=
E
24
ES=
EF=
LS=
LF=
slack=
J
4
ES=
EF=
LS=
LF=
slack=
A Hospital Project
Slack and the Critical Path
•
•
The slack of any activity on the
critical path is zero.
If an activity’s slack time is zero,
then it is must be on the critical
path.
Critical Path, Examples
•
What is the critical path in the Foundry Inc.
example?
•
What is the critical path in the Hospital
project example?
Calculate the Critical Path:
Step 1. Mark earliest times (ES, EF) on all
nodes, forward;
Step 2. Mark latest times (LF, LS) on all
nodes, backward;
Step 3. Calculate slack of each activity;
Step 4. Identify the critical path that
contain the activities with zero slack.
C
A
ES=
EF=
LS=
LF=
2
ES=
EF=
LS=
LF=
4
slack=
slack=
D
Start
ES=EF=
LS=LF=
3
Finish
ES=
EF=
ES=EF=
LS=
LF=
LS=LF=
slack=
B
7
E
2
ES=
EF=
ES=
EF=
LS=
LF=
LS=
LF=
slack=
Calculate the critical path
slack=
Example: Draw diagram and find critical path
Activity
A
B
C
D
E
F
G
H
I
Predecessor
B
A
C
A,D
E,G
G
t
5
3
6
4
8
12
7
6
5
Example: Draw diagram and find critical path
Activity
A
B
C
D
E
F
G
H
Predecessor
A
B
A,B
C
D,E,F
E,F
t
3
4
6
5
8
2
4
5
Solved Problem 13-1&2, p.547-548
Calculate the Critical Path
Activity
A
B
C
D
E
F
G
a
1
2
4
8
2
4
1
m b
2 3
3 4
5 6
9 10
5 8
5 6
2 3
Immediate predecessor
A
B
C, D
B
E
Steps for Solving 13-1&2
1.
2.
3.
4.
Calculate activity performance time t for
each activity;
Draw the PERT network;
Calculate ES, EF, LS, LF and slack of
each activity on PERT network;
Identify the critical path.
Probabilities in PERT
•
•
Since the performance time t of an
activity is from estimations, its
actual performance time may
deviate from t;
And the actual project completion
time may vary, therefore.
Probabilistic Information for
Management
 The
expected project finish time and
the variance of project finish time;
 Probability the project is finished by
a certain date.
Project Completion Time and
its Variance
•
The expected project completion time T:
T = earliest completion time of the project.
•
The variance of T, T2 :
T2 = (variances of activities on the critical path)
Example, Foundry Inc.
Activity
a
m
b
t
variance
A
1
2
3
B
2
3
4
C
1
2
3
D
2
4
6
E
1
4
7
F
1
2
9
G
3
4
11
H
1
2
3
2
3
2
4
4
3
5
2
0.111
0.111
0.111
0.444
1
1.777
1.777
0.111
Critical path: A-C-E-G-H
Project completion time, T =
Variance of T, T2 =
Solved Problem 13-1&2, p.547-548
Project completion time and variance
Activity
a
m
b
t
variance
A
1
2
3
2
0.111
B
2
3
4
3
0.111
C
4
5
6
5
0.111
D
8
9
10
9
0.111
E
2
5
8
5
1
F
4
5
6
5
0.111
G
1
2
3
2
0.111
Critical path: B-D-E-G
Project completion time, T =
Variance of T, T2 =
Probability Analysis
•
To find probability of completing project
within a particular time x:
1. Find the critical path, expected project
completion time T and its variance T2 .
2.
Calculate
Z 
xT
T
2
3. Find probability from a normal distribution
table (as on page 698).
The Idea of the Approach
•
•
The table on p.698 gives the probability
P(z<=Z) where z is a random variable with
standard normal distribution, i.e. zN(0,1);
Z is a specific value.
P(project finishes within x days)

x T

 P z 
2

T



  P(z  Z )


Notes (1)
•
•
P(project is finished within x days)
= P(z<=Z)
P(project is not finished within x days)
= 1P(project finishes within x days)
= 1P(z<=Z)
Notes (2)
•
•
•
If x<T, then Z is a negative number.
But the table on p.698 is only for positive Z
values.
For example, Z= 1.5, per to the symmetry
feature of the normal curve,
P(z<=1.5) = P(z>=1.5) = 1P(z<=1.5)
Example of Foundry Inc. p.530-531
•
•
•
Project completion time T=15 weeks.
Variance of project time, T2=3.111.
We want to find the probability that project is
finished within 16 weeks. Here, x=16, and
Z 
xT
T
•
2

16  15
3 . 111

1
 0 . 57
1 . 76
So, P(project is finished within 16 weeks)
= P(z<=Z) = P(z<=0.57) = 0.71566.
Examples of probability analysis
•
If a project’s expected completing time is
T=246 days with its variance T2=25, then
what is the probability that the project
–
–
–
–
is actually completed within 246 days?
is actually completed within 240 days?
is actually completed within 256 days?
is not completed by the 256th day?
A Comprehensive Example

Given the data of a project as in the next
slide, answer the following questions:
–
–
–
–
–
What is PERT network like for this project?
What is the critical path?
Activity E will be subcontracted out. What is
earliest time it can be started? What is time it
must start so that it will not delay the project?
What is probability that the project can be
finished within 10 weeks?
What is the probability that the project is not
yet finished after 12 weeks?
Data of One-more-example:
Activity
A
B
C
D
E
Predecessor
A
A
B,D
a
1
5.5
3.5
2
0
m
2
7
4
3
2
b
3
8.5
4.5
4
4
Example (cont.)
Activity
A
B
C
D
E
Predecessor
A
A
B,D
t
2
7
4
3
2
v
0.111
0.250
0.028
0.111
0.444
C
A
ES=
EF=
LS=
LF=
2
ES=
EF=
LS=
LF=
4
slack=
slack=
D
Start
ES=EF=
LS=LF=
3
Finish
ES=
EF=
ES=EF=
LS=
LF=
LS=LF=
slack=
B
7
E
2
ES=
EF=
ES=
EF=
LS=
LF=
LS=
LF=
slack=
Calculate the critical path
slack=
Solving on QM
Critical path, ES, LS, EF, LF, and slack can
be calculated by QM for Windows. We
need to enter activities’ times and
immediate predecessors.
 But QM does not provide the network.
