This document provides an overview of program evaluation and review technique (PERT) and critical path method (CPM) for project scheduling. It defines key concepts like activities, events, critical path, floats and provides examples of how to draw network diagrams and calculate event and activity times. The examples demonstrate how to identify the critical path, calculate earliest and latest start/finish times, and use PERT to determine the probability of completing a project by a certain deadline while accounting for uncertainty in activity durations.

1. Department of Management
19th
Batch
University of Dhaka
Management Science (MGT-301)
Pert & CPM
Table of Contents
1 PERT & CPM........................................................................................................................................................2
1.1 PERT............................................................................................................................................................2
1.2 CPM.............................................................................................................................................................2
1.3 Differences between PERT & CPM.............................................................................................................2
1.4 CRITICAL PATH........................................................................................................................................2
1.5 Rules for drawing the network diagrams. ....................................................................................................3
1.6 Network Representation:..............................................................................................................................3
1.7 Float (Slack).................................................................................................................................................3
1.8 Examples......................................................................................................................................................5
Courtesy of- Kamrul, Zonayed, and Fahad || Department of Management || University of Dhaka

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1 PERT & CPM
1.1 PERT
Program Evaluation & Review Technique – It is generally used for those projects where time required to
complete various activities are not known as a priori. It is probabilistic model & is primarily concerned for
evaluation of time. It is event oriented.
1.2 CPM
Critical Path Analysis – It is a commonly used for those projects which are repetitive in nature & where one has
prior experience of handling similar projects. It is a deterministic model & places emphasis on time & cost for
activities of a project.
 A project can be defined as a set of large number of activities or jobs (with each activity consuming time &
resources) that are performed in a certain sequence determined.
 A network is a graphical representation of a project, depicting the flow as well as the sequence of well-
defined activities & events.
 An activity (Also known as task & job) is any portion of a project which consumes time or resources and
has definable beginning & ending.
 Event (Also known as node & connector) is the beginning & ending points of an activity or a group of
activities.
1.3 Differences between PERT & CPM
PERT CPM
It is a technique for planning scheduling & controlling
of projects whose activities are subject to uncertainty in
the performance time. Hence it is a probabilistic model.
It is a technique for planning scheduling & controlling
of projects whose activities not subjected to any
uncertainty and the performance times are fixed. Hence
it is a deterministic model
It is an Event oriented system. It is an Activity oriented system.
Basically does not differentiate critical and non-critical
activities.
Differentiates clearly the critical activities from the
other activities.
Used in projects where resources (men, materials,
money) are always available when required.
Used in projects where overall costs is of primarily
important. Therefore better utilized resources.
Suitable for Research and Development projects where
times cannot be predicted.
Suitable for civil constructions, installation, ship
building etc.
1.4 CRITICAL PATH
Meaning: The longest path in a project network which determine the duration of the project is known as critical
path.
Determination of Critical Path
Step 1. List all the possible sequences from start to finish
Step 2. For each sequence determine the total time required from start to finish.
Step 3. Identify the longest path (Critical Path)

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1.5 Rules for drawing the network diagrams.
 In a network diagram, arrows represent the activities and circles represent the events.
 The tail of an arrow represents the start of an activity and the head represent the completion of the activity.
 The event numbered 1 denotes the start of the project and is called initial event.
 Event carrying the highest number in the network denotes the completion of the project and is called
terminal event.
 Each defined activity is represented by one and only arrow in the network.
 Determine which operation must be completed immediately before other can start.
 Determine which other operation must follow the other given operation.
 The network should be developed on the basis of logical, analytical and technical dependencies between
various activities of the project.
1.6 Network Representation:
Each activity of the project is represented by arrow pointing in direction of progress of project. The events of the
network establish the precedence relationship among different activities.
Three rules are available for constructing the network.
Rule 1. Each activity is represented by one & only one, arrow.
Rule 2. Each activity must be identified by two distinct events & No two or more activities can have the same tail
and head events.
Rule 3. To maintain correct precedence relationship, the following questions must be answered as each activity is
added to the network:
 What activities must be immediately precede the current activity?
 What activities must follow the current activity?
 What activities must occur concurrently with the current activity?
There are two types of systems –
i. AOA system (Activity on Arrow system)
ii. AON system (Activity on Node system )
1.7 Float (Slack)
Float (Slack) refers to the amount of time by which a particular event or an activity can be delayed without
affecting the time schedule of the network. Float (Slack) is defined as the difference between latest allowable and
the earliest expected time.
Event Float/Slack = LS – ES
Where LS = Latest start time
ES = Early start time.
1. Earliest start: Earliest start time is the earliest possible time by which the activity can be started. Denoted
as ‘ES’
2. Early finish time: Early finish time is the earliest possible time by which the activity can be completed.
Denoted as ‘EF’
3. Latest start time: Latest start time is the latest possible time by which the activity can be started. Denoted
as ‘LS’
4. Late finish time: Late finish time is the latest possible time by which the activity can be completed.
Denoted as ‘LS’
5. Total float (TF) / Total slack (TS): Total float of the job is the differences between its Late start and
Early start ‘or’ Late finish and Early finish. i.e.
TF (CA) = LS (CA) - ES (CA) or TF (CA) = LF (CA) - EF (CA)
CA = Current activity

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6. Free float (FF) Free float is the amount of time a job can be delayed without affecting the Early start time
of any other job.
 FF (CA) = ES (SA) – EF (CA)
 CA = Current Activity
 SA = Succeeding Activity
7. Independent Float (IF): Independent Float is the amount of time that can be delayed without affecting
either predecessor or successor activities.
 IF = ES (SA) – LF (PA) - Duration of CA
 ES = Early Start
 LF = Late Finish
 SA = Succeeding Activity
 PA = Preceding Activity
 CA = Current Activity

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1.8 Examples
Example-1
Develop the network for a project with following activities and immediate predecessors:
Activity Immediate
predecessors
A -
B -
C B
D A,C
E C
F C
G D,E,F
Solution
Example-2
Scheduling with activity time
 Draw the net work
 What is the critical path
 Find out earliest start and earliest finish
 Find out latest start and latest finish
 Find out slack or free time

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Solution
This information indicates that the total time required to complete activities is 51 weeks.
However, we can see from the network that several of the activities can be conducted
simultaneously (A and B, for example).
i. Network with ES & EF time
ii. Network with LS & LF time
iii. Activity schedule for our example
iv. Critical path activities: A, E, F, G, and I.

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Example-3
PERT For Dealing with Uncertainty
Solution
i. Determination of Expected Value, Variance and Standard Deviation
Let m= most likely time estimate, mode.
a = optimistic time estimate,
b = pessimistic time estimate, and
Expected Value (TE) = (a + 4m + b) /6
Variance (V) = (( b – a) / 6 ) 2
Std Deviation (δ) = SQRT (V)

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ii. The complete network
iii. The complete Network with critical path
iv. Critical Path Analysis (PERT)
v. Assume, Manager promised to complete the project in the fifty days. What are the chances of
meeting that deadline?
Calculate Z, where Z = (D-S) / V
D = 50 S (Scheduled date) = 20+15+8 =43; V = (4+25+4) =33
Z = (50 – 43) / 5.745
= 1.22

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The probability value of Z = 1.22, is 0.888
vi. What deadline are you 95% sure of meeting
Z value associated with 0.95 is 1.645
D = S + 5.745 (1.645)
= 43 + 9.45
= 52.45 days
Thus, there is a 95 percent chance of finishing the project by 52.45 days.
Example-4
Construct the Network for the following Project and determine the following
 Critical Path
 ES,EF,LS,LF
 TF,FF
Solution
i. Project Network

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ii. Determination of ES,EF,LS,LF and TF,FF
Example-5
The following table lists the jobs of a network along with their time estimates.
i. Draw the project network.
ii. What is the probability that the job will be completed in 35 days?
iii. What due date has 90% chance of being met?
Solution
i. Construction of the Network

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ii. Calculation of Expected time for all the activities
Expected Time ( te): ‘te’ can be calculated by the following formula
te = (to + 4tm + tp) / 6
iii. Determination of Critical Path
Expected Duration of the project Te = 32 days
iv. Probability of completing the project within a given date
Z = (TS – TE) / σ
= (35 – 32) / 6
= 0.5
From the Normal distribution Table, we get the probability of completing the project in
35 days is 69.15%
v. The due date for 90% chance of being met.
Probability of completing the project within a given date. The value of Z from the table for a 90%
probability is +1.28
 TS =? (To be calculated),
 TE = 32,
 σ = 6
Z = (TS – TE) / σ
1.28 = (TS– 32) / 6
TS = 39.68 days