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INTRODUCTION
 CPM (Critical Path Method) was the discovery of M.R.Walker of E.I.Du Pont de Nemours & Co. and
J.E.Kelly of Remington Rand, circa 1957.
 The methods are essentially network-oriented techniques using the same principle. PERT and CPM are
basically time-oriented methods in the sense that they both lead to determination of a time schedule
for the project.
 The significant difference between two approaches is that the time estimates for the different activities
in CPM were assumed to be deterministic while in PERT these are described probabilistically.
 These techniques are referred as project scheduling techniques.
In CPM activities are shown as a network of precedence relationships using activity-on node network
construction – Single estimate of activity time – Deterministic activity times

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Benefits of CPM
1.Useful at many stages of project management
2.Mathematically simple
3.Give critical path and slack time
4.Provide project documentation
5.Useful in monitoring costs

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Applications of CPM
These methods have been applied to a wide variety of problems in industries and have
found acceptance even in government organizations. These include
1.Construction of a dam or a canal system in a region
2.Construction of a building or highway
3.Maintenance or overhaul of airplanes or oil refinery
4.Space flight
5.Cost control of a project using PERT / COST
6.Designing a prototype of a machine
7. Development of supersonic planes

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Limitations of CPM
1.Clearly defined, independent and stable activities
2.Specified precedence relationships
3.Over emphasis on critical paths

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Basic Steps in CPM Project scheduling
1. Planning
2.Scheduling
3.Allocation of resources
4.Controlling

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Network Diagram Representation
In a network representation of a project certain definitions are used
 Activity Any individual operation which utilizes resources and has an end and a
beginning is called activity. An arrow is commonly used to represent an activity with
its head indicating the direction of progress in the project. These are classified into
four categories
 Predecessor activity – Activities that must be completed immediately prior to the
start of another activity are called predecessor activities.
 Successor activity – Activities that cannot be started until one or more of other
activities are completed but immediately succeed them are called successor activities.
 Concurrent activity – Activities which can be accomplished concurrently are known as
concurrent activities. It may be noted that an activity can be a predecessor or a
successor to an event or it may be concurrent with one or more of other activities.

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Dummy activity – An activity which
does not consume any kind of resource
but merely depicts the technological
dependence is called a dummy activity.

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Event: An event represents a point in time signifying the completion of some activities and the
beginning of new ones. This is usually represented by a circle in a network which is also called a node
or connector. The events are classified in to three categories
1. Merge event – When more than one activity comes and joins an event such an event is known as
merge event
2. Burst event – When more than one activity leaves an event such an event is known as burst event.
3. Merge and Burst event – An activity may be merge and burst event at the same time as with
respect to some activities it can be a merge event and with respect to some other activities it may be
a burst event.

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Rules for Drawing Network Diagram
Rule 1 Each activity is represented by one and only one arrow in the network
 Rule 2 No two activities can be identified by the same end events
Rule 3 In order to ensure the correct precedence relationship in the arrow diagram, following questions
must be checked whenever any activity is added to the network
 What activity must be completed immediately before this activity can start?
 What activities must follow this activity?
 What activities must occur simultaneously with this activity? In case of large network, it is essential
that certain good habits be practiced to draw an easy to follow network

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 Try to avoid arrows which cross each other
 Use straight arrows
 Do not attempt to represent duration of activity by its arrow
length
 Use arrows from left to right. Avoid mixing two directions,
vertical and standing arrows may be used if necessary.
 Use dummies freely in rough draft but final network should
not have any redundant dummies.
 The network has only one entry point called start event and
one point of emergence called the end event

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Critical Path in Network Analysis
Basic Scheduling Computations
The notations used are
(i, j) = Activity with tail event i and head event j
Ei = Earliest occurrence time of event i
Lj = Latest allowable occurrence time of event j
Dij = Estimated completion time of activity (i, j)
(Es)ij = Earliest starting time of activity (i, j)
(Ef)ij = Earliest finishing time of activity (i, j)
(Ls)ij = Latest starting time of activity (i, j)
(Lf)ij = Latest finishing time of activity (i, j)

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Activity(i, j)
Normal Time
(Dij)
(1,2) 2
(1,3) 2
(1,4) 1
(2,6) 4
(3,7) 5
(3,5) 8
(4,5) 3
(5,9) 5
(6,8) 1
(7,8) 4
(8,9) 3

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STEP 1: NETWORK DIAGRAM

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STEP 2: Finding Earliest start time and latest finish time

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STEP 3: The earliest time and the latest time are obtained below
Activity(i, j)
Normal
Time
(Dij)
Earliest Time Latest Time
Float Time (Li -
Dij ) - Ei
Start
(Ei)
Finish
(Ei + Dij )
Start
(Li - Dij )
Finish
(Li)
(1,2) 2 0 2 5 7 5
(1,3) 2 0 2 0 2 0
(1,4) 1 0 1 6 7 6
(2,6) 4 2 6 7 11 5
(3,7) 5 2 7 3 8 1
(3,5) 8 2 10 2 10 0
(4,5) 3 1 4 7 10 6
(5,9) 5 10 15 10 15 0
(6,8) 1 6 7 11 12 5
(7,8) 4 7 11 8 12 1
(8,9) 3 11 14 12 15 1

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The cri cal path is 1 → 3 → 5 → 9
THANKS

Cpm network analysis and problems

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