UNIT IV
PROGRAM EVALUATION AND REVIEW TECHNIQUE (PERT)
AND CRITICAL PATH METHOD (CPM)
Introduction
Many managerial problems in areas such as transportation systems design, information
systems design and project scheduling have been successfully solved with the aid of network
models and network analysis techniques.
The techniques to be introduced here have been applied successfully to a wide range of
significant management problems. The Polaris missile project was planned with the aid of PERT
(program evaluation and review technique), which we shall study in this unit. CPM (critical path
method) was successfully used by one of the authors to plan and coordinate the activities of two
governments as they cooperated in an earthquake relief project in Turkey.
Other uses of network models include planning the flow of traffic to minimize congestion
in cities, determining the shortest pickup and delivery routes for package-handling companies
and even finding the best layout for the water system in a new residential subdivision.
Course Objectives



To introduce students to the basic principles behind network models
To give students an overview of two of the most important network models:
Program Evaluation and Review Technique (PERT) and Critical Path Method (CPM)
To demonstrate, using illustrative examples, how PERT can be applied to a business problem
Suggested Timeframe
6 hours
The Framework of PERT and CPM
Program Evaluation and Review Technique (PERT) and the Critical Path Method (CPM)
are two popular quantitative techniques that help managers plan, schedule, monitor, and control
large, complex and a wide variety of projects, such as:
1. Research and development of new products and processes
2. Construction of plants, buildings and highways
3. Maintenance of large and complex equipment
4. Design and installation of new systems
There are six steps common to PERT and CPM, and these are:






Define the project and all of its significant activities and tasks.
Develop the relationships among the activities. Decide which activities must precede and
follow others.
Draw the network connecting all of the activities.
Assign time and/or cost estimates to each activity.
Compute the longest time path through the network. This is called the critical path.
Use the network to help plan, schedule, monitor, and control the project.
Finding the critical path is a major part of controlling a project. The activities on the
critical path represent tasks that will jeopardize the entire project if they are delayed.
Managers derive flexibility by identifying noncritical activities and replanning,
rescheduling and reallocating resources such as manpower and funds.
Main Difference Between PERT and CPM
PERT and CPM essentially follow the same procedure when applied; however, PERT is
a probabilistic technique, whereas CPM is a deterministic one. PERT incorporates statistical
concepts in the calculation of activity time. It was developed with an objective of being able to
handle uncertainties in activity completion times. On the other hand, CPM was developed
primarily for scheduling and controlling industrial projects where job or activity times were
considered known. CPM offered the option of reducing activity times by adding more workers
and/or resources, usually at an increased cost. It simply assumes normal times in the
determination of the critical path.
Program Evaluation And Review Technique (PERT)
PERT was developed in the 1950s by the Navy Special Projects Office in cooperation
with Booz, Allen and Hamilton, a management-consulting firm. It was specifically directed at
planning and controlling the Polaris missile program, a massive project which had 250 prime
contractors and over 9000 subcontractors. Imagine the problems faced by the project director in
attempting to keep track of hundreds of thousands of individual tasks on this project.
The introduction of PERT into the Polaris project helped management answer questions
like these:

When will the project be finished?

When is each individual part of the project scheduled to start and be finished?

Of the hundreds of thousands of “parts” of the project, which ones must be
finished on time to avoid being late?

Is it possible to shift resources to critical parts of the project (those that must be
finished on time) from other noncritical parts of the project (parts which can be
delayed) without affecting the overall completion time of the project?

Among all the hundreds of thousands of parts of the project, where should
management concentrate its effort at any one time? (Levin et.al. 1982)
Other Network Techniques
Aside from PERT and CPM, there are numerous other network models useful for
business decision making. Three of these are listed below:

Minimal-Spanning Tree Technique. This technique determines the path through the network
that connects all the points while minimizing total distance.

Maximal-Flow Technique. This technique finds the maximum flow of any quantity or
substance through a network.

Shortest-Route Technique. This technique finds the shortest path through a network.
EXAMPLE:
The Maginhawa Company has manufactured industrial vacuum cleaning systems for a
number of years. Recently its new research team submitted a report suggesting the company
consider manufacturing a cordless vacuum cleaner that could be powered by a rechargeable
battery. The vacuum cleaner, referred to as Maginhawa-Vacuum, could be used for light
industrial cleaning and could contribute to the company’s expansion into the household market.
Management hoped that the new product could be manufactured at a reasonable cost and that its
portability and no cord convenience would make it extremely attractive.
Maginhawa Company’s top management would like to initiate a project to study the
feasibility of proceeding with the Maginhawa-Vacuum idea. The data are shown in Table 1.
Table 1. Activity list for the Maginhawa -Vacuum Project
Activity Description
A
R & D product design
B
Plan market research
C
Routing (manufacturing engineering)
D
Build prototype model
E
Prepare marketing brochure
F
Cost estimates (industrial engineering)
G
Preliminary product testing
H
Market survey
I
Pricing and forecast report
J
Final report
Immediate Predecessors
A
A
A
C
D
B, E
H
F,G,I
The network for the Maginhawa-Vacuum project is shown in Figure 1.
Routing
C
2
5
Product
Design
A
1
Cost
Estimates
F
Prototype
D
Marketing
Brochure
E
4
Testing
G
7
Final
Report
J
8
Completion
Market
Research
Plan
B
Market
Survey
H
3
Pricing
and
Forecast
I
6
Activity Times
The next step in the PERT procedure is the assignment of estimated time required to
complete each activity. Time is usually given in units of days or weeks. PERT uses a
probability distribution based on three time estimates for each activity. These are:

Optimistic Time (a). The activity time if everything goes as well as possible.

Most Probable Time (m). Most realistic time estimate to complete the activity.

Pessimistic Time (b). The activity time if we encounter significant breakdowns and/or delays
PERT assumes that the time estimates follow the beta probability distribution
Probability
a
m
b
Activity Time
As an illustration of the PERT procedure with uncertain activity times, let us consider the
optimistic, most probable and pessimistic time estimates for the Maginhawa-Vacuum activities
as presented in Table 2.
Table 2. Optimistic, most probable and pessimistic time estimates (in weeks) for the MaginhawaVacuum project
Activity
Optimistic
(a)
4
1
2
3
2
1.5
1.5
2.5
1.5
1
A
B
C
D
E
F
G
H
I
J
Most Probable
(m)
5
1.5
3
4
3
2
3
3.5
2
2
Pessimistic
(b)
12
5
4
11
4
2.5
4.5
7.5
2.5
3
To find the expected time (t) for an activity, the beta distribution weights the estimates as
follows:
a + 4m + b
t=
6
Finding the Critical Path
To find the critical path, we need to determine the following quantities for each activity in the
network:

Earliest Start Time (ES). This is the earliest time an activity can begin without violating any
of the immediate predecessor requirements.

Earliest Finish Time (EF). This is the earliest time an activity can end.

Latest Start Time (LS). This is the latest time an activity can begin without delaying the
entire project.

Latest Finish Time (LF). This is the latest time an activity can end without delaying the
entire project.
Computing ES and EF
Begin at the network’s origin.
For the first event, the starting time is always set to zero (0).
EF = ES + T
ES time rule:
Before any activity can be started, all of its predecessor activities must be
completed, that is, search for the longest path to an activity in determining ES.
Computing LS and LF
Begin at the network’s end. Make a backward pass through the network.
LS = LF – T
The latest time for an activity equals the smallest LS for all activities leaving that event.
Computing Slack Times
After constructing the network and determining the critical path, slack times may be calculated
in one of two ways:
Slack = LS – ES
or
Slack = LF – EF
Variability in the Project Completion Date
To compute the dispersion or variance of T:
b - a
2
Variance of activity time =
6
Table 3. Expected times and variances for the Maginhawa-Vacuum activities
Activity
Expected Time (t)
Variance
(in weeks)
A
6
1.78
B
2
0.44
C
3
0.11
D
5
1.78
E
3
0.11
F
2
0.03
G
3
0.25
H
4
0.69
I
2
0.03
J
2
0.11
Total
32
The total expected time to complete the work for the Maginhawa-Vacuum project is 32
weeks. It can be seen also from the network that several of the activities can be conducted
simultaneously (A and B for example). Being able to work on two or more activities
simultaneously will have the effect of making the total project completion time shorter than 32
weeks.
In order to arrive at a project duration estimate, we will have to analyze the network and
determine what is called the critical path. A path is a sequence of connected activities that leads
from the starting node (1) to the completion node (8). The connected activities defined by nodes
1-2-5-7-8 form a path consisting of activities A, C, F, and J.
The longest path determines the expected total time or expected duration of the project. If
activities on the longest path are delayed, the entire project will be delayed. Thus the longest path
activities are the critical activities of the project and the longest path is called the critical path of
the network.
C
2
5
3
A
5
6
1
D
F
2
J
G
3 E
4
7
3
2
8
B
2
I
2
H
3
6
4
Figure 2. Maginhawa-Vacuum Proj.Network with Expected Activity Times
Table 4. Activity schedule (in weeks) for the Maginhawa-Vacuum project
Activity
A
B
C
D
E
F
G
H
I
J
Earliest
Start
0
0
6
6
6
9
11
9
13
15
Latest
Start
0
7
10
7
6
13
12
9
13
15
Earliest
Finish
6
2
9
11
9
11
14
13
15
17
Latest
Finish
6
9
13
12
9
15
15
13
15
17
Slack
(LS-ES)
0
7
4
1
0
4
1
0
0
0
Critical
Path?
Yes
Yes
Yes
Yes
Yes
The variance in the project duration is given by the sum of the variance of the critical
path activities. Thus the variance for the Maginhawa-Vacuum project completion time is given
by
σ 2 = σ 2 A + σ 2 E+ σ 2 H + σ 2 I + σ 2 J
= 1.78 + 0.11 + 0.69 + 0.03 + 0.11
= 2.72
Since we know that the standard deviation is the square root of the variance, we can
compute the standard deviation σ for the Maginhawa-Vacuum project completion time as
follows:
   2  2.72  1.65
A final assumption of PERT is that the distribution of the project completion time T
follows a normal or bell-shaped distribution. With this, we can compute the probability of
meeting a specified project completion date. For example, suppose that management has allotted
20 weeks for the Maginhawa-Vacuum project. While we expect completion in 17 weeks, what is
the probability that we will meet the 20-week deadline?
The z value for the normal distribution at T = 20 is given by
Z = 20 – 17 = 1.82
1.65
  1.65 Weeks
Expected
Completion Time
T
17
Time (Weeks)
Figure 3. PERT Normal Distribution of the Project Completion Time
Variation for the Maginhawa-Vacuum Project
Using z = 1.82 and the tables for the normal distribution, we see that the probability of
the project meeting the 20-week deadline is 0.4656 + 0.5000 = 0.9656. Thus while activity time
variability may cause the project to exceed the 17-week expected duration, there is an excellent
chance (96.56%) that the project will be completed before the 20-week deadline.
YOUR ACTIVITY
I. Electrika is a company that installs wiring and electrical fixtures in residential construction.
Its general manager has been very concerned with the amount of time it takes to complete wiring
jobs. Some of his workers are very unreliable. A list of activities and their optimistic
completion time, the pessimistic completion time, and the most likely completion time is given
in the following table.
Activity
A
B
C
D
E
F
G
H
I
J
K
a
3
2
1
6
2
6
1
3
10
14
2
m
6
4
2
7
4
10
2
6
11
16
8
b
8
4
3
8
6
14
4
9
12
20
10
Immediate
Predecessors
None
None
None
C
B,D
A,E
A,E
F
G
C
H,I
a. Draw the appropriate PERT network.
b. Determine the expected completion time and the variance for each activity.
c. What is the critical path?
II.
Activity
A
B
C
D
E
F
G
H
Assume that the project has to be completed in 16 weeks. Crashing of the project is
necessary. Relevant information is shown.
Immediate
predecessor
A
B,C
D
E
B, C
F, G
Normal time
(wks)
3
6
2
5
4
3
9
3
Crash time
(wks)
1
3
1
3
3
1
4
2
Normal cost
(P)
900
2000
500
1800
1500
3500
8000
1000
Crash cost
(P)
1900
4000
1000
2400
1850
3900
10000
2000
a. Make the minimum cost crashing decisions. What is the added cost of meeting the 16week completion time?
b. Develop a complete activity schedule using the crashed activity times.
REFERENCES
Andersen, D.R., D.J. Sweeney, & T.A. Williams, An Introduction to Management
Science: Quantitative Approaches to
Decision Making
Andersen, D. R., Dennis J. Sweeny, and Thomas A. Williams. 2001. Quantitative
Methods for Business. Eighth edition. Cincinnati, Ohio: South-Western College
Publishing.
Anderson, Michael Q. 1982. Quantitative Management Decision Making, Belmont,
California: Brooks/ Cole Publishing Co.,
Dunn, Robert A. and K. D. Ramsing. 1981. Management Science: A Practical Approach
to Decision Making. New York, New York: MacMillan Publishing Co.,
Engbino, Diosdado. Management Decision Models: An Overview. Given as a reference
material in the Quantitative Methods for Business class in 1987.
Krajewski, Lee C. and H. E. Thompson. 1981. Management Science: Quantitative
Methods in Context, New York: John Wiley & Sons,
Levin, R.I.., C.A. Kirkpatrich & D.S. Rubin. 1982. Quantitative
Management. 5th ed. McGraw-Hill, Inc.
Approaches to
Zamora, Elvira A. 2004. Basic Quantitative Methods for Business Decisions. UP College
of Business Administration.