17-1

Project Management

Operations Management

William J. Stevenson

8th edition

17-2

Project Management

CHAPTER

17

Project
Management

McGraw-Hill/Irwin

Operations Management, Eighth Edition, by William J. Stevenson

Copyright 2005 by The McGraw-Hill Companies, Inc. All rights

17-3

Project Management

INTRODUCTION

Network: A network is a graphical representation

of a project, depicting the flow as well as the
sequence of well-defined activities and events.

Activity is the actual work to be performed.

An Event marks the beginning or end of an

activity

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Project Management

Network Models Developed during 1950s

PERT (Program Evaluation & Review

Technique)

CPM (Critical Path

Method)

The purpose of this chapter is to present these

models with specific examples.

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Project Management

PHILOSOPHICAL FOUNDATIONS OF
PERT & CPM

Planning and Control are two of the most important

functions of management.
Planning involves the formulation of objectives and
goals that are subsequently translated into specific plans
and projects.
Control: The function of control is to institute a
mechanism that shows actual performance (in terms of
time, cost, or some other measure of effectiveness) is
deviating from the plan. If such a deviation is
unacceptable to the manager, he will take corrective
action to bring performance in conformity with the
plans.

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Project Management

EVOLUTION OF PERT AND CPM

PERT and CPM models are based upon, and have
evolved from GANTT-TYPE bar charts and
milestone charts.
GANTT CHART is a popular tool for planning
and scheduling simple projects. It shows the start
and finish times of overall project.

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Project Management

GENERAL PURPOSE AND FRAMEWORK

OF PERT & CPM
PERT & CPM models are useful for the purpose of
planning, analyzing, scheduling and controlling the
progress and completion of large and complex
projects.
In PERT & CPM the working procedure consists of
five steps:

1.
2.
3.

Analyze and break down the project in terms of

specific activities and/or events;
Determine the interdependence and sequence of
activities and produce a network;
Assign estimates of time, cost or both to all the
activities of the network;

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Project Management

GENERAL PURPOSE AND FRAMEWORK

OF PERT & CPM
4.

5.

Identify the longest or critical path through the

network; and
Monitor, evaluate and control the progress of the
project by re-planning, rescheduling and
reassignment of resources.

The central task in these models is to identify the

longest path through the network. It is also called the
critical path of the project. If for some reason the
project need to be completed in less time than the
critical path time, additional resources must be devoted
to expedite one or more activities of the critical path.

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Project Management

Paths other than critical path (i.e. non-critical or

SLACK PATHS) offer flexibility in scheduling and
transferring resources, because they take less time to
complete than the critical path.
PERT & CPM models are similar in basic structure,
rationale, and mode of analysis. However, in general,
two distinctions are made between PERT & CPM.
1.The way in which activity times are estimated;
2.The cost estimates for completing various activities

17-10 Project Management

Project Management

What are the tools?

Work breakdown structure
Gantt charts
GANTT CHART is a popular tool for
planning and scheduling simple projects. It
shows the start and finish times of overall
project. Figure 1
Milestone Charts
Network Diagram

17-11 Project Management

Fig. 1 Gantt Chart

Gantt Chart
Locate new
facilities
Interview staff
Hire and train staff
Select and order
furniture
Remodel and install
phones
Move in/startup

MAR

APR

MAY

JUN

JUL

AUG

SEP

OCT

NOV

DEC

Milestone Chart PERT NETWORK

diagram

17-12 Project Management

Milestone Chart (Fig 2) is an improvement on the

bar chart because it identifies significant milestones
or events and shows dependencies within tasks.

Milestone chart does not show inter-relationships

and inter-dependencies of events among tasks. This
deficiency is eliminated by the PERT NETWORK as
shown in Figure 3.

17-13 Project Management

MILESTONE CHART

Task 1

Task 2

Task 3

Task 4

Figure 2

17-14 Project Management

PERT NETWORK

Task 1

Task 2

Task 3

Task 4

Figure 3

17-15 Project Management

Difference between PERT & CPM

PERT

CPM

Activity time estimates are probabilistic three Activity times are deterministic. A single
different time estimates, based on the
time estimate is made for each activity.
concept of probability of completing the
activity are made for each activity.
Activity costs are not explicitly provided.

CPM model gives explicit estimates of

activity costs. Two sets of estimates are
provided. One set gives normal time and
normal cost required to complete each
activity under normal condition.
Second set gives crash time and crash cost
required to complete each activity under
condition to reduce project completion time
by expending more money.

17-16 Project Management

PERT

PERT was developed in 1958 by the special

projects office of US Navy. The development was
a result of the research conducted for the purpose
of coordinating and expediting the work of several
thousand contractors involved in the POLARIS
MISSILE PROGRAM. It is claimed that, with the
aid of PERT, the completion of the Polaris Missile
Program was expedited by almost two years.

17-17 Project Management

PERT & CPM

PERT WAS EVOLVED THROUGH THE JOINT
EFFORTS OF Lockheed Aircraft, Booz, Allen &
Hamilton in the effort to speed up the Polaris Missile
Project.
CPM was developed by J.E. Kelly of the Remington
Rand Corporation and M.R. Walker of DuPont in
1957 in connection with building and maintenance
projects in chemical plants. As its name implies it
identifies the critical (largest) path through the
network and use it to exercise control on the
progress of the project.

17-18 Project Management

PERT AND CPM

By using PERT OR CPM, managers are able to
obtain:
1.A graphical display of project activities;
2.An estimate of how long the project will take;
3.An indication of which activities are the most
critical to timely project completion;
4.An indication of how long any activity can be
delayed without lengthening the project

17-19 Project Management

PERT
As PERT is a probalistic model; and factor of
uncertainties involved in the time required to complete
work activities.
PERTs originator decided three different time estimates:

(a) Most Optimistic Time (this is short test time, assuming

most favorable conditions)
(m) Most likely time (most realistic time required to
complete an activity);
(b) Most pessimistic time (this is the longest time,
assuming most unfavorable conditions)
Three times can be represented by Beta Distribution and
can be skewed to the right as shown in Figure 4.

17-20 Project Management

Figure 4, Beta Distribution

Figure 17.2: Activity Time

Distribution for the Activity B of
the R.C. Coleman Project

17-21 Project Management

Probabilistic Estimates, Another

notations

17-22 Project Management

Figure 17.8

to
Activity
start

Optimistic
time

tm

te

Most likely
time (mode)

tp
Pessimistic
time

17-23 Project Management

Discussion of the Beta Distribution for Activity

Completion time and Probabilistic Time Estimates

a most optimistic time estimate. Assume that a =

10 weeks, then the probability of completing the
activity within 10 or less weeks is 1/100.
m most likely time. Assume m = 16 weeks
means that most of the time this activity will take
16 weeks to complete.
b most pessimistic time estimate. Assume b =
40 weeks; then the probability that the activity will
take more than 40 weeks is 1/100.

17-24 Project Management

The Network Diagram

Network (precedence) diagram

Activity-on-arrow (AOA)

Activity-on-node (AON)

Activities

Events

17-25 Project Management

NETWORK CONVENTIONS

17-26 Project Management

Network Conventions
a
c

c
b

c
Dummy
activity

17-27 Project Management

The Network Diagram (contd)

Path

Critical path

The longest path; determines expected project

duration

Critical activities

Sequence of activities that leads from the starting

node to the finishing node

Activities on the critical path

Slack

Allowable slippage for path; the difference the

length of path and the length of critical path

Expected Time for Completing the

Activity, te

17-28 Project Management

17-29 Project Management

Variance
2
(t

t
)
p o
36

= variance
to = optimistic time
tp = pessimistic time

Table 17.1: Activities for the

Coleman Automated-Warehouse
Project

17-30 Project Management

Activities
A Determine equipment needs
B Obtain vendor proposals

Immediate Predecessors

C Select vendor

A, B

D Order system

E Design new warehouse layout

F Layout warehouse

G Design computer interface

H Interface computer
I Install system

D, F, G
D, F

J Train system operators

K Test system

I, J

17-31 Project Management

Figure 17.1: PERT/CPM Network for

the R.C. Coleman Project

17-32 Project
Management
Table
17.2:

Optimistic, Most Probable, and

Pessimistic Activity Time Estimates in Weeks for
the R.C. Coleman Project

Activity

Optimistic a

Most Probable m

Pessimistic b

11

12

10

10

17-33 Project Management

Table 17.3: Expected Times and Variances

for the R.C. Coleman Activities

Variance 2

Activity

Expected Time t (in Weeks)

.11

1.78

.11

1.78

.44

.11

1.78

.11

1.78

.11

.11

17-34 Project Management

Table 17.4: Activity Schedule in Weeks for the R.C.

Coleman Project

Activity

Earliest
Start

Earliest
Finish

Latest
Start

Latest
Finish

Slack
(LS-ES)

Critical
Path

13

15

12

12

12

15

12

15

11

11

15

15

18

15

18

15

19

16

20

18

20

18

20

20

22

20

22

17-35 Project Management

Activity

Earliest Start
Time
A [0, 3]
3

Earliest Finish
Time

Expected Activity
Time

35

Figure
17.8: R.C. Coleman Project with
17-36 Project Management
Latest Start and Latest Finish Times in
Parentheses

17-37 Project Management

Project Network Activity on Arrow

Figure 17.4

AOA
Locate
facilities

Order
furniture

4
Furniture
setup

2
Remodel

Interview

Hire and
train

Move
in

17-38 Project Management

Project Network Activity on Node

Figure 17.4
Order
furniture

AON

Locate
facilities

Furniture
setup

Move
in

Remodel

S
Interview

Hire and
train

17-39 Project Management

Time Estimates

Deterministic

Time estimates that are fairly certain

Probabilistic

Estimates of times that allow for variation

17-40 Project Management

Example 1
Figure 17.5

8 weeks
ate s
c
Lo ilitie
fac

er e
d
r
O
tu r
i
n
fu r
Rem
od e
l
11 weeks

re
n i tu
Fur p
u
set

Deterministic
time estimates

6 weeks

3 weeks

In
te
rv
i
4 weeks ew

in
a
r
t

nd
a
re
Hi
9 weeks

Move
in
1 week

17-41 Project Management

Example 1 Solution
Critical Path

P a th

L e n g th

S la c k

(w e e k s )

1 -2 -3 -4 -5 -6
1 -2 -5 -6
1 -3 -5 -6

18
20
14

2
0
6

17-42 Project Management

Computing Algorithm

Network activities
ES: early start
EF: early finish
LS: late start
LF: late finish

Used to determine
Expected project duration
Slack time
Critical path

17-43 Project Management

Example 5
Optimistic
time

2-4-6
b
3-5-7
e

6
32- g

4-6-8
h

Pessimistic
time

5
32- c

4
3
1a
3-4-5
d

Most likely
time

5-7-9
f
-6
4
3- i

17-44 Project Management

Path Probabilities
Z =

Specified time Path mean

Path standard deviation

Z indicates how many standard deviations of the path distribution

the specified time is beyond the expected path duration.

17-45 Project Management

Example 6
17
Weeks
1.00
a-b-c
Weeks

10.0

d-e-f
16.0

Weeks

1.00
g-h-i
13.5

Weeks

17-46 Project Management

Time-cost Trade-offs: Crashing

Crash shortening activity duration

Procedure for crashing

Crash the project one period at a time

Only an activity on the critical path

Crash the least expensive activity

Multiple critical paths: find the sum of

crashing the least expensive activity on each
critical path

17-47 Project Management

Time-Cost Trade-Offs: Crashing

Figure 17.11
Total
cost
Expected indirect costs

Shorten

CRASH

Cumulative
cost of
crashing

Shorten
Optimum

17-48 Project Management

Example 7

10

2
f
9

4
d

17-49 Project Management

Advantages of PERT

Forces managers to organize

Provides graphic display of activities

Identifies

Critical activities

Slack activities
1

17-50 Project Management

Limitations of PERT

Important activities may be omitted

Precedence relationships may not be correct

Estimates may include

a fudge factor

May focus solely

on critical path

4
2

142 weeks
3

17-51 Project Management

Technology for Managing Projects

Computer aided design (CAD)
Groupware (Lotus Notes)
Project management software

CA Super Project
Harvard Total Manager
MS Project
Sure Track Project Manager
Time Line

17-52 Project Management

Advantages of PM Software

Imposes a methodology

Provides logical planning structure

Enhances team communication

Flag constraint violations

Automatic report formats

Multiple levels of reports

Enables what-if scenarios

Generates various chart types

17-53 Project Management

Project Risk Management

Risk: occurrence of events that have

undesirable consequences

Delays

Increased costs

Inability to meet specifications

Project termination

17-54 Project Management

Risk Management

Identify potential risks

Analyze and assess risks

Work to minimize occurrence of risk

Establish contingency plans

EARLIEST START AND EARLIEST

FINISH

17-55 Project Management

Consider the network in Fig (T), using the starting

time of 0, compute an earliest start and earliest
finish time for each activity in the network.
Letting
ES = Earliest Start Time for a particular activity;
EF = Earliest Finish Time for a particular activity;
te = Expected activity time for the activity.

17-56 Project Management

The following expression can be used to find the

earliest finish time for a given activity:
EF = ES + te

e.g.; for activity A, ES=0, & te =3; the earliest

finish time for activity is EF=0+3=3.
We write the earliest start and earliest finish times
directly on the network in brackets next to the
letter of the activity. Using activity A as an
example, we have:

17-57 Project Management

EARLIEST START TIME RULE

The earliest start time for an activity leaving a
particular node is equal to the largest value of the
earliest finish times for all activities entering the
node.
Applying this rule to the portion of the network
involving node 1,2,3 and 4, we obtain the
following (on tran:
Note that after activity C, earliest start time is 5,
which is equal to the largest earliest finish times of
dummy and activity B.

57

17-58 Project Management

EARLIEST START TIME RULE

Proceeding in a FORWARD PASS through the
network, we can establish first an earliest start and
then an earliest finish time for each activity. (see
fig. 3)
Note that finish time for activity K, the last
activity, is 22 week. Thus the earliest completion
time for the entire project is 22 weeks.

58

17-59 Project Management

EARLIEST START TIME RULE

We now continue the algorith for finding the

critical path by making a BACKWARD PASS
calculation.

Starting at the completion point (node 10) and using

a latest finish time of 22 for activity K, we trace
back through the network computing a latest start
and latest finish time for each activity.
LS = Latest starting time for a particular activity;
LF = Latest finishing time for a particular activity
59

17-60 Project Management

EARLIEST START TIME RULE

The following expression can be used to find the

latest start time for a given activity:
LS = LF-t
Given and note that LF = EF for last activity
Therefore LS = 22-2 = 20 Weeks

60

17-61 Project Management

LATEST FINISH TIME RULE

The latest finish time for an activity entering a
particular node is equal to the smallest value of the
latest starting times for all activities leaving the
node.
The calculation for LS and LF are given (in
transparency)
After finding start and finish activity times as
summarized in the bottom figure of
(transparency), we can find the amount of slack or
free time associated with each of the activities.

61

17-62 Project Management

LATEST FINISH TIME RULE

Slack is defined as the length of time and activity can
be delayed w/o affecting the completion date for the
project.
The amount of slack for each activity is computed as
follows:

Slack = LS-ES = LF-EF

(A1)

e.g. slack associated with activity A is LS-ES=2-0=2

weeks. This means that activity A can be delayed upto 2
weeks (start anywhere between weeks 0 and 2) and
entire project can still be completed in 22 weeks.
62

17-63 Project Management

LATEST FINISH TIME RULE

Using equation A the slack association with
activity C is LS-ES=5-5=0. thus activity C has no
slack time and must be held to the 5 weeks start
time schedule.
Since this activity cannot be delayed without
affecting the entire project, it is a critical activity
and is on the critical path.
In general the critical path activities are the
activities with ZERO SLACK.

63

17-64 Project Management

LATEST FINISH TIME RULE

64

17-65 Project Management

LATEST FINISH TIME RULE

*The final assumption of PERT is that the duration

of the project completion time T follows a normal
or Bell-Shaped, distribution. Thus T follows the
distribution shown
*The first assumption was
*The activities are independent in terms of their
variance (i.e. the completion times of the activities
are assumed to be independent)

65

17-66 Project Management

LATEST FINISH TIME RULE

= 1.66
weeks

T
22
Expected Completion Time (22
Weeks)

Fig. PERT Normal Distribution of the Project Completion Time

Variation.
66

17-67 Project Management

LATEST FINISH TIME RULE

With the assumption we can compute the probability
of meeting a specified project completion date.
Suppose we want to finish the Colemans Project in
25 weeks.
While we expect it to finish in 22 weeks.

Q: what is the probability that we will meet 25 weeks

deadline?
Using normal distribution from Fig. 17.9, we are asking
for the probability that T< 25 weeks.
This is shown graphically by shaded area in Fig. 17.10
67

Fig. 17.9 LATEST FINISH TIME

RULE

17-68 Project Management

=
1.66

(Prob.
T<25)

0.5

0.4649

22
Time (Weeks)

25

68

17-69 Project Management

LATEST FINISH TIME RULE

69

Q. What is the time by which

management can be 95% confident of
completing the project?

17-70 Project Management

0.45
0.5
0.05

1.64

te = 22

Tim
e

The question asks for point C in the figure.

Then the area under the curve from te to C is
45% (i.e. 95% of the area is to the left of C)
70

17-71 Project Management

QUESTION 1
Given the information provided in the
accompanying network diagram, determine each of
the following:
a.The length of each path;
b.The critical path;
c.The expected length of the project;
d.Amount of slack time for each path;
e.Compute activity slack times for the precedence
diagram

EARLIEST START AND EARLIEST

FINISH

17-72 Project Management

Consider the network in Fig (T), using the starting

time of O, compute an earliest start and earliest
finish time for each activity in the network.
Letting
ES = Earliest Start Time for a particular activity;
EF = Earliest Finish Time for a particular activity;
te = Expected activity time for the activity.

72

17-73 Project Management

The following expression can be used to find the

earliest finish time for a given activity:
EF = ES + t

e.g.; for activity A, ES=0, & t=3; the earliest finish

time for activity is EF=0+3=3.
We write the earliest start and earliest finish times
directly on the network in brackets next to the
letter of the activity. Using activity A as an
example, we have:

73

17-74 Project Management

Activity

Earliest Start
Time
A [0, 3]
3

Earliest Finish
Time

Expected Activity
Time

74

17-75 Project Management

EARLIEST START TIME RULE

The earliest start time for an activity leaving a
particular node is equal to the largest value of the
earliest finish times for all activities entering the
node.
Applying this rule to the portion of the network
involving node 1,2,3 and 4, we obtain the
following (on transparency):
Note that after activity C, earliest start time is 5,
which is equal to the largest earliest finish times of
dummy and activity B.

75

17-76 Project Management

EARLIEST START TIME RULE

Proceeding in a FORWARD PASS through the
network, we can establish first an earliest start and
then an earliest finish time for each activity. (see
fig. 3)
Note that finish time for activity K, the last
activity, is 22 week. Thus the earliest completion
time for the entire project is 22 weeks.

76

17-77 Project Management

EARLIEST START TIME RULE

We now continue the algorith for finding the

critical path by making a BACKWARD PASS
calculation.

Starting at the completion point (node 10) and using

a latest finish time of 22 for activity K, we trace
back through the network computing a latest start
and latest finish time for each activity.
LS = Latest starting time for a particular activity;
LF = Latest finishing time for a particular activity
77

17-78 Project Management

EARLIEST START TIME RULE

The following expression can be used to find the

latest start time for a given activity:
LS = LF-t
Given and note that LF = EF for last activity
Therefore LS = 22-2 = 20 Weeks

78

17-79 Project Management

LATEST FINISH TIME RULE

The latest finish time for an activity entering a
particular node is equal to the smallest value of the
latest starting times for all activities leaving the
node.
The calculation for LS and LF are given (in
transparency)
After finding start and finish activity times as
summarized in the bottom figure of
(transparency), we can find the amount of slack or
free time associated with each of the activities.

79

17-80 Project Management

LATEST FINISH TIME RULE

Slack is defined as the length of time and activity can
be delayed w/o affecting the completion date for the
project.
The amount of slack for each activity is computed as
follows:

Slack = LS-ES = LF-EF

(A1)

e.g. slack associated with activity A is LS-ES=2-0=2

weeks. This means that activity A can be delayed upto 2
weeks (start anywhere between weeks 0 and 2) and
entire project can still be completed in 22 weeks.
80

17-81 Project Management

LATEST FINISH TIME RULE

Using equation A the slack association with
activity C is LS-ES=5-5=0. thus activity C has no
slack time and must be held to the 5 weeks start
time schedule.
Since this activity cannot be delayed without
affecting the entire project, it is a critical activity
and is on the critical path.
In general the critical path activities are the
activities with ZERO SLACK.

81

17-82 Project Management

LATEST FINISH TIME RULE

82

17-83 Project Management

LATEST FINISH TIME RULE

*The final assumption of PERT is that the duration

of the project completion time T follows a normal
or Bell-Shaped, distribution. Thus T follows the
distribution shown
*The first assumption was
*The activities are independent in terms of their
variance (i.e. the completion times of the activities
are assumed to be independent)

83

17-84 Project Management

LATEST FINISH TIME RULE

= 1.66
weeks

T
22
Expected Completion Time (22
Weeks)

Fig. PERT Normal Distribution of the Project Completion Time

Variation.
84

17-85 Project Management

LATEST FINISH TIME RULE

With the assumption we can compute the probability
of meeting a specified project completion date.
Suppose we want to finish the Colemans Project in
25 weeks.
While we expect it to finish in 22 weeks.

Q: what is the probability that we will meet 25 weeks

deadline?
Using normal distribution from Fig. 17.9, we are asking
for the probability that T< 25 weeks.
This is shown graphically by shaded area in Fig. 17.10
85

17-86 Project Management

LATEST FINISH TIME RULE

=
1.66

(Prob.
T<25)

0.5

0.4649

22
Time (Weeks)

25

86

17-87 Project Management

LATEST FINISH TIME RULE

87

Q. What is the time by which

management can be 95% confident of
completing the project?

17-88 Project Management

0.45
0.5
0.05

1.64

te = 22

Tim
e

The question asks for point C in the figure.

Then the area under the curve from te to C is
45% (i.e. 95% of the area is to the left of C)
88

17-89 Project Management

89

17-90 Project Management

90

17-91 Project Management

91

17-92 Project Management

ADVANTAGES OF USING PERT AND

POTENTIAL SOURCE OF ERROR

PERT and similar project scheduling techniques can provide

important services for the project manager. Among the most
useful features are these:
Use of these techniques forces the manager to organize and
quantify available information and to recognize where
additional information is needed.
The techniques provide a graphic display of the project and
its major activities.
The identify (a) activities that should be closely watched
because of the potential for delaying the project and (b) other
activities that have slack time and so can be delayed without
affecting project completion time. This raises the possibility
of reallocating resources to shorten the project.
92

17-93 Project Management

No analytical technique is without potential errors. Among the more

important sources of errors are:
When developing the project network, managers may unwittingly
omit one or more important activities.
Precedence relationships may not all the be correct as shown.
Time estimates may include a fudge factor; mangers may feel
uncomfortable about making time estimates because they appear to
commit themselves to completion within a certain time period.
There may be a tendency to focus solely on activities that are on the
critical path. As the project progresses, other paths may become
critical. Further, major risk events may not be on the critical path.

93

17-94 Project Management

17-95 Project Management

QUESTION 2
The network diagram for a project is shown in the
accompanying figure, with three time estimates for
each activity. Activity times are in months. Do the
following:
a.Compute the expected time for each activity and the
expected duration for each path;
b.Identify the critical path;
c.Compute the variance for each activity and the
variance for each path;
d.Find the probability that the project will be completed
within 17 months of its start;

17-96 Project Management

QUESTION 2
e.

f.
g.

h.

Find the probability that the project will take

longer than 18 months 2%;
Can the paths be considered independent? Why?
Determine the probability that the project will be
completed within 15 months of its start;
What is the probability that the project will not
be completed within 15 months of its start?

17-97 Project Management

CPM NETWORKS & PROJECT

CRASHING

17-98 Project Management

CPM network is deterministic;

CPM approach is useful when both time and cost
estimates are known with certainty;
Two sets of time and cost figures are obtained for
each activity:

Normal time and normal cost;

Crash time and crash cost;

Further, it is usually assumed that the relationship

between time and cost is linear;

CPM NETWORKS & PROJECT

CRASHING

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The idea of project crashing is that, under certain

circumstances, it is necessary and desirable to expedite
project completion even though it will result in higher
costs.
In CPM we know both the minimum project completion
time and the cost-time relationships of all activities; our
objective is to design a program that will yield minimum
project completion time with the least increase in costs
over the normal costs. Let us consider these concepts
with the aid of the network shown in the figure (on
transparency).

17-101 Project Management

Normal and crash times and cost data are shown;

Note that cost/time will always yield a negative
slope;
Normal time is without parenthesis; while the
crash time is within parenthesis. (This is the usual
convension to represent)

17-102 Project Management

THE MECHANICS OF CPM

First four steps are the same as that of PERT network

i.e.;
Step 1: Define the overall project, including the project
objective and target completion date;
Step 2: Break down the project into well-defined
activities i.e. Identify beginning event (source event)
and Terminal Event;
Step 3: Give serial nos. to each event and arrange then
in proper sequence as required by planning and
technological requirements; this establishes the
precedence relationships;

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Step 4: Construct the actual CPM network, inter

connecting all the activities and events.
Next steps are the Analysis Phase of CPM

Step 5: Identify the normal critical path and the

crash critical path;
The network shown in the figure has got three
different paths as follows:

The

normal critical path is 1-3-6 (45 weeks) and

that, under normal conditions, the cost of the entire
project is $24,800.

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Length of Path (Weeks)

Path

Normal

Crash

1-3-6

45

30

1-2-5-6

42

32

1-2-4-5-6

44

34

Crash critical path is 1-2-4-5-6 (34 weeks) and that, under

crash conditions, the cost of the entire project is $32,600.
Our task, then, is to design a program that will complete
the project within 34 weeks with the least increase in cost
above $24,800.
It is logical to start by crashing the least expensive activity
on the normal critical path.

17-105 Project Management

Step 6: On the normal critical path, identify the

least expensive activity to crash;
Crash this activity and note whether the critical path
has changed. If not, crash the next least expensive
activity on the critical path and so on. Until a new
critical path emerges, with its own least expensive
activity to crash.
Burning Time:
Assumption of CPM is that Normal and Crash
estimates are linearly related

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In many cases the

relationship is not linear.
Special computer
programs have been
developed to accept nonlinear time/cost tradeoffs,
but these are beyond the
scope of this text (course)

Crash
Effort

Cost

Normal
Effort

Time

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Finding the Minimum Time Minimum

Cost Network:
One of the principal questions CPM can answer is:
What is the least cost to complete a project in
minimum time?
Continue this process until an irreducible critical
path on which all activities are on their crash times
has been obtained.

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FIRST CRASH
Start our analysis with the normal critical path.
There are only two uncrashed activities on the
normal critical path (i.e.; 1-3 and 3-6);
Activity 1-3 is the least expensive activity
($200/week), and hence crash it by 5 weeks. (see
transparency). This reduces path 1-3-6 to 40
weeks and changes the critical path to (1-2-4-5-6)
44 weeks.

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SECOND CRASH
There are four uncrashed activities on (1-2-4-5-6)
the new critical path i.e.; (1-2, 2-4, 4-5, & 5-6).
Of these activity 1-2 is the least expensive (i.e.;
$200/week); (see transparency). Hence we crash it
by 3 weeks.
The critical path is still 1-2-4-5-6; but it has
decreased to 41 weeks.

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THIRD CRASH
There are three uncrashed activities on the current
critical path (2-4, 4-5 and 5-6).
Of these activities 2-4 and 5-6 are least expensive
$400/week). We can crash any of them; however,
we crash activity 5-6 as it yields a larger reduction
in completion time (3 weeks). As shown in table.
This changes the critical path to 1-3-6 (40 weeks).

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FOURTH CRASH
There is only one uncrashed activity (i.e.; 3-6) the
current critical path. We crash it by 10 weeks (as
shown in the transparency) and the critical path is
again 1-2-4-5 (38 weeks).
Of these activities 2-4 and 5-6 are least expensive
$400/week). We can crash any of them; however,
we crash activity 5-6 as it yields a larger reduction
in completion time (3 weeks). As shown in table.
This changes the critical path to 1-3-6 (40 weeks).

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FIFTH CRASH

Now there are only two uncrashed activities on

the current critical path (3-4 and 4-5). Of these
activities 2-4 is least expensive $400/week) and
hence we crash it by 2 weeks. (as shown in table).
This means that we now have two critical paths
(1-2-5-6 and 1-2-4-5-6) of 36 weeks.

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SIXTH CRASH

If we compare the two critical paths (1-2-5-6 and

1-2-4-5-6), we find that only two uncrashed
activities remain (2-5 and 4-5). Since activity 2-5
is least expensive $200/week), we crash it by 4
weeks. (as shown in transparency). This results in
making path 1-2-4-5-6 the critical path.

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SEVENTH CRASH

There now remains only one uncrashed activity

on the current critical path (activity 4-5). We
crash it, and note that the critical path is still 1-24-5-6, but it is irreducible (i.e.; all activities on this
path are at their crash times). Therefore, we have
completed step 2.

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Step 3: Examine the non-critical paths and uncrash

activities on such paths (beginning with the most
expensive activity) to the point after which further
uncrashing will create a longer critical path.
It has been observed uptill now:
1)The minimum completion time of 34 weeks has
been achieved;
2)The non-critical path 1-3-6 is 30 weeks long and
hence can be uncrashed by no more than 4 weeks;
and

17-116 Project Management

3)

The other non-critical path 1-2-5-6 is 32 weeks long

and hence can be uncrashed by no more than 2
weeks;

4)
i.

ii.

Of the non-critical path 1-3-6, the activity 3-6 is the

most expensive ($240/week), hence we uncrash it by
4 weeks;
Of the three activities on the non-critical path 1-2-56, activities 1-2 and 5-6 cannot be uncrashed, as they
are included in the crash critical path 1-2-4-5-6.
However, we can uncrash activity 2-5 ($200/week),
and hence we uncrash it by 2 weeks.

17-117 Project Management

We have now executed step 3, and our analysis of

CPM Network of Fig. is complete.
We have obtained:

Minimum completion time with the least increase in

costs above the normal project cost. The final time
status of the activities and their cost consequences
are summarized in table (transparency e.g.)

17-118 Project Management

Activity

Normal

Crash

Cost ($)

1-2

1600

1-3

10

3000

2-4

2600

2-5

18 weeks total

4900

3-6

24 weeks total

8640

4-5

12

6000

5-6

4500

It should be noted that the minimum project completion time

of 34 weeks has been obtained at a cost of 31,240, as
compared to the crash project cost of 32,600.

17-119 Project Management

It should be emphasized that the CPM model is

well suited to accommodate the reality of
budgetary constraints. (e.g.; we can answer the
questions):
What is the minimum project completion time if
our budget for the network in figure is $27,000?
To answer the question we start at the end of step
3 and, of the crashed activities at the stage, we
uncrash the most expensive activity first.

17-120 Project Management

Then keeping an eye on the critical path, we keep

on uncrashing activities until the project cost is
reduced to the level of the budget constraints. For
H.W. find the new minimum project completion
time, assuming a budget of $27,000.
We can also answer the question:
What is the minimum cost to complete the project
in 39 days?