This document provides an overview of network techniques for project management, including PERT and CPM models. Key points covered include: developing a project network diagram; determining critical paths and calculating floats; time and cost estimation; scheduling activities based on available resources; and using the network to project costs and monitor project progress. PERT uses probabilistic analysis while CPM is deterministic, focusing on time-cost tradeoffs when crashing activities. The network allows visualization of activity relationships and quantitative analysis of schedule options and resource constraints.

1. CHAPTER 22
NETWORK TECHNIQUES FOR
PROJECT MANAGEMENT

2. OUTLINE
• Development of project network
• Time estimation
• Determination of critical path
• Scheduling when resources are limited
• PERT model
• CPM model
• Network cost system

3. PERT and CPM
 There are two basic network techniques: PERT and CPM. PERT, and
acronym for Program Evaluation Review Technique has a ‘probabilistic’
orientation. CPM, an acronym for Critical Path Method has a
deterministic orientation.
 Widely diverse projects are amenable to analysis by PERT and CPM.
 The common characteristics of the projects which make them amenable
to analysis by PERT or CPM are :
1. The project can be broken down into a well-defined set of jobs or
activities.
2. The activities must be performed in a certain sequence which is
technologically ordered.
3. Within a defined sequence, the activities may be started and stopped
in an independent manner.

4. Development of Project Network
Basic to PERT as well as CPM is the network diagram. The network
diagram, also referred to as the project graph, shows the activities and
events of the project and their logical relationships. A simplified net work
diagram for a dinner project is shown below
1
3
2
4
Receive
guests
Take dinner
Network Diagram for a Dinner Project

5. The network diagram is constructed in terms of activities and events.
An activity is a definite task, job, or function to be performed in a
project. For example, ‘prepare dinner’ is an activity. An activity is
represented by an arrow. The head of the arrow marks the completion of
the activity and the tail of the arrow marks its beginning. (The length
and ‘compass’ direction of the arrow have no significance). An event is
a specific point in time indicating the beginning or end of one or more
activities. It represents a milestone and does not consume time or
resources. For example, event 2 marks completion of the activity ‘send
invitation’.

6. Rules for Network Construction
1. Each activity must have a preceding and a succeeding event.
2. Each event should have a distinct number.
3. There should be no loops in the project network.
4. No more than one activity can have the same preceding and
succeeding events.

7. Time Estimation
Once the logic and detail of the network have been established, time estimates
must be assigned to each activity. Generally, three time values are obtained for
each activity:
1. Optimistic time (to)
2. Most likely time (tm)
3. Pessimistic time (tp)
The optimistic time, to, is the time required if no hurdles or complications
arise. The most likely time, tm, is the time in which the activity is most likely
to be completed. This estimate takes into consideration normal circumstances,
making allowance for some unforeseen delays. The pessimistic time, tp, is the
time required if unusual complications and / or unforeseen difficulties arise.

8. Obtaining Time Estimates
Time estimates should be obtained by the PERT planner from persons who are
responsible for estimation. The following points should be borne in mind while
obtaining time estimates:
1. Time estimates should be obtained by skipping around the network rather than by
following a specific path. If estimates are obtained by following one path, there is
a tendency for the person providing the estimates to add them mentally and
compare them with a previously conceived notion of the time of the total path
2. The estimates of to, tm, and tp , should be defined independently of each other.
3. The time available for completing the project should not influence the estimates
of to, tm, and tp.
4. It should be made known that to, tm, and tp are estimates and not schedule
commitments.
5. The estimates of to, tm, and tp should include allowances for occurrences which
are generally considered as random variables (weather conditions, administrative
delays, etc.) but not for occurrences that are normally not considered as random
variables (flood, wars, etc.)

9. Average Time
Once the three estimates for each activity are obtained, the expected
value of activity duration is calculated. The expected value, te, is
usually obtained by the formula
to + 4 tm + tp
6
Where te = weighted arithmetic average time
to = optimistic time
tm = most likely time
tp = pessimistic time
te =

10. Determination of the Critical Path
Once the network diagram with single time estimates has been
developed, the following computational procedure may be
employed for determining the critical path/s, event slacks, and
activity floats.
1. Calculate the earliest occurrence time (EOT) for each event
2. Calculate the latest occurrence time (LOT) for each event
3. Calculate the slack for each event
4. Obtain the critical and slack paths
5. Compute the activity floats

11. Earliest Occurrence Time (EOT)
The general formula for EOT is :
EOT (i) = Max [ EOT (k) + d (k,i)]
Where EOT(i) = earliest occurrence time for event i
EOT(k) = earliest occurrence time for event k (k
precedes i and there may be several ks)
d (k,i) = duration of activity (k,i)
The maximisation shown is considering all activities (k,i) leading to
event node i.

12. Earliest Starting Time (EST) and Earliest Finishing
Time (EFT)
The formulae for EST and EFT are :
EST (i, j) = EOT (i) (22.3)
EFT (i, j) = EOT (i) + d (i,j) (22.4)
where EST (i, j) = earliest starting time for activity (i,j)
EOT (i) = earliest occurrence time of event (i)
EFT (i, j) = earliest finishing time for activity (i, j)
d (i, j) = duration of activity ( i, j)

13. Latest Occurrence Time (LOT)
The general formula for LOT is :
LOT (i) = Min [LOT (j) - d(i,j)]
Where LOT (i) = latest occurrence time for i
LOT (j) = latest occurrence time for j( j follows i
and there may be several j’s)
d(i, j) = duration of activity (i,j)
The minimisation shown here is done with respect to all activities
(i,j) starting from i.

14. Latest Finishing Time (LFT) and Latest Starting
Time (LST)
Given the LOT for various events we can calculate the latest
finishing time (LFT) and latest starting time (LST) for various
activities using the formulae:
LFT (i, j) = LOT (j) (22.6)
LFT (i, j) = LFT (i, j) – d (i,j) (22.7)
Where LFT (i, j) = latest finishing time for activity (i,j)
LOT (j) = latest occurrence time for event j
LST ( i, j) = latest starting time for activity (i,j)
d ( i, j) = duration of activity ( i,j)

15. Event Slacks
The slack for an event is the difference between its LOT and EOT. The slacks for
various events of our illustrative project are shown below.
Event LOT EOT Slack = LOT-EOT
5
4
3
2
1
28
26
18
13
00
28
26
18
13
00
28
20
12
13
00
0
6
6
0
0
(in weeks)

16. Critical and Slack Paths
 The critical path starts with the beginning event, terminates with the
end event, and is marked by events which have a zero slack. This is
obviously the path on which there is no slack, no cushion. Other
paths are slack paths with some cushion
 The critical path is the longest path from the beginning event to the
end event. Since the end can be reached only when this longest path
is traversed, the minimum time required for completing the project is
the duration on the critical path.

17. Activity Floats
 Given the estimates of activity time and event slacks, activity floats can be
calculated. There are three measures of float, total float, free float, and
independent float.
 The total float of an activity is the extra time available to complete the
activity if it is started as early as possible, without delaying the completion
of the project. The total float represents the float under the most favourable
conditions.
 The free float of an activity is the extra time available to complete the
activity when the activity is started at the earliest occurrence time (EOT) of
the preceding event and completed by the EOT of its succeeding event.
 The independent float of an activity is the extra time available to complete
the activity when the activity is started at the LOT of its preceding event and
completed by the EOT of its succeeding event. The independent float
represents the float under the most adverse conditions.

18. Floats
More generally, floats may be represented by the following equations:
TF (i,j) = LOT (j) – EOT (i) – d(i, j) (22.8)
FF (i, j) = EOT (j) – EOT (i) – d(i, j) (22.9)
IF (i, j) = EOT (j) – LOT (i) – d (i, j) (22.10)
Where TF (i, j) = total float of activity (i, j)
LOT (j) = latest occurrence time for event j
EOT (i) = earliest occurrence time of event i
d (i, j) = duration of activity (i, j)
FF(i, j) = free float of activity (i,j)
EOT (j) = earliest occurrence time of event j
IF (i,j) = independent float of activity (i, j)
LOT (i) = latest occurrence time of event i

19. Early Start Schedule
The early start schedule refers to the schedule in which all activities
start as early as possible. In this schedule (i) all events occur at their
earliest because all activities start at their earliest starting time and
finish at their earliest finishing time, (ii) there may be time lags
between the completion of certain activities and the occurrence of
events which these activities lead to; and (iii) all activities emanating
from an event begin at the same time.
The early start schedule suggests a cautious attitude towards the
project and a desire to minimise the possibility of delay. It provides a
greater measure of protection against uncertainties and adverse
circumstances. Such a schedule, however, calls for an earlier
application of resources.

20. Late Start Schedule
The late start schedule refers to the schedule arrived at when all
activities are started as late as possible. In this schedule (i) all events
occur at their latest because all activities start at their latest finishing
time; (ii) some activities may start after a time lag subsequent to the
occurrence of the preceding events; and (iii) all activities leading to an
event are completed at the same time
The late start schedule reflects a desire to commit resources late-as
late as possible. However, such a schedule provides no elbow room in
the wake of adverse developments. Any unanticipated delay results in
increased project duration

21. PERT Model
 Variability in PERT analysis is measured by variance or its square
root, standard deviation
 The steps involved in calculating the standard deviation of the
duration of critical path are as follows:
1. Determine the standard deviation of the duration of each
activity on the critical path.
2. Determine the standard deviation of the total duration of the
critical path on the basis of information obtained in step 1.

22. Standard Deviation of the Duration of an Activity
For determining the standard deviation of the duration of any activity
we require the entire probability distribution of the activity distribution.
We,however, have only three values from this distribution: tp , tm, and
to. In PERT analysis, a simplification is used in calculating the standard
deviation. It is estimated by the formula.
tp - to
6
Where  = standard deviation
tp = pessimistic time
to = optimistic time
Variance is obtained by squaring 
 =

23. Probability of Completion by a Specified Date
Armed with information about mean (T) and standard deviation
() for critical path duration, which is normally distributed, we
can compute the probability of completion by a specified date
(D) as follows:
D - T
1. Find Z =

2. Obtain cumulative probability up to Z by looking at the
probability distribution of the standard normal variate.

24. CPM Model
The PERT model was developed for projects characterised by
uncertainty and the CPM model was developed for projects which are
relatively risk-free. While both the approaches begin with the
development of the network and a focus on the critical path, the PERT
approach is ‘probabilistic’ and the CPM approach is ‘deterministic’.
This does not, however, mean that in CPM analysis we work with
single time estimates. In fact, the principal focus of CPM analysis is on
variations in activity times as a result of changes in resource
assignments. These variations are planned and related to resource
assignments and are not caused by random factors beyond the control
of management as in the case of PERT analysis. The main thrust of
CPM analysis is on time-cost relationships and it seeks to determine the
project schedule which minimises total cost.

25. Assumptions Underlying CPM Analysis
The usual assumptions underlying CPM analysis are:
1. The costs associated with a project can be divided into two components:
direct costs and indirect costs. Direct costs are incurred on direct material
and direct labour. Indirect costs consist of overhead items like indirect
supplies, rent, insurance, managerial services, etc.
2. Activities of the project can be expedited by crashing which involves
employing more resources.
3. Crashing reduces time but enhances direct costs because of factors like
overtime payments, extra payments, and wastage. The relationship between
time and direct activity cost can be reasonably approximated by a downward
sloping straight line.
4. Indirect costs associated with the project increase linearly with project
duration.

26. Procedure
The bulk of CPM analysis is concerned with the relationship between total
direct cost and project duration. The procedure used in this respect is
generally as follows:
1. Obtain the critical path in the normal network. Determine the project
duration and direct cost.
2. Examine the cost-time slope of activities on the critical path obtained
and crash the activity which has the least slope
3. Construct the new critical path after crashing as per step 2. Determine
project duration and cost
4. Repeat steps 2 and 3 till activities on the critical path (which may
change every time) are crashed.

27. Network Cost System
The techniques of PERT and CPM discussed earlier are essentially time-oriented.
They seek to answer questions like:
 What is the most desirable time schedule of activities?
 How much time would it take, on an average, to complete the project?
 What is the probability of completing the project in a specified time? (It may be
noted that in CPM analysis, the primary objective is to develop a time schedule
in view of time-cost tradeoff and not to develop a detailed budget of costs,
activity-wise and time-wise. Similarly, when scheduling is done under funds
constraint, the primary objective is to establish a time schedule consistent with
funds constraint and not to develop a system for cost planning and control)
Such analysis largely overlooks the cost aspect which is usually as important as
the time aspect and sometimes even more. To provide a vehicle for cost
planning and control of projects, the network cost system was developed. This
represents a very useful supplement to the traditional time-oriented network
analysis. Let us look at cost projection and cost analysis and control under the
network cost system.

28. Cost Projection
 The basic principles of the network cost system is fairly simple:
costs are planned, measured, analysed, and controlled in terms of
project activities.
 Once costs are estimated in terms of activities, cost projections can
be made for any chosen schedule. For cost projection it is usually
assumed that the expenditure for any activity is incurred evenly
over the duration of that activity.

29. Analysis and Control of costs
As the project progresses the following may be measured / estimated
periodically for purposes of monitoring and control.
1. Costs incurred to date
2. Budgeted costs to date
3. Value of work done to date
Budgeted costs x Percentage work accomplished
4. Cost over-run (under-run) to date
Actual cost - Value of work completed
Value of work completed
5. Time over-run (under-run) to date
x 100

30. Time and Cost, Budget and Actual
1 2 3 4 5 6 7 8
Months
60
50
40
30
20
10
Rs in
million
A-Budgeted cost
C-Value of work
B- Actual cost
1 2 3 4 5 6 7 8

31. D
E
% Over-run
% Under-run
Months
behind
Months
ahead
100
50
2
1
1
0

32. SUMMARY
 For proper planning, scheduling and control of the activities of a project, given their
interrelationships and constraints on the availability of resources, network techniques
have been found quite useful.
 There are two basic network techniques: PERT and CPM. PERT is applied mostly to
projects characterised by uncertainty: its orientation is probabilistic. CPM is applied to
projects which are relatively risk-free; its orientation is deterministic. Widely diverse
projects are amenable to analysis by PERT and CPM.
 The steps involved in PERT analysis are: (i) development of project network, (ii) time
estimation, (iii) determination of critical path, event slacks, and activity floats.(iv)
development of project schedule, and (v) calculation of the variability of project
duration and the probability of completion in a given time.
 The network diagram is constructed in terms of activities and events. An activity is a
definite task, job or function to be performed in a project. An event is a specific point
of time indicating the beginning or end of one or more activities.
 The rules to be observed in constructing the network diagram are: (i) Each activity
must have a preceding and succeeding event. (ii) Each event should have a distinct
number. (iii) There should be no loops in the project network, (iv) Not more than one
activity can have the same preceding and succeeding events.

33.  To ensure that each activity is uniquely numbered it may be necessary to introduce
dummy activities. A dummy activity is an imaginary activity which can be
accomplished in zero time and which does not consume resources. A dummy activity
may also be used to represent a constraint necessary to show the proper relationship
between activities.
 Once the logic and detail of the network have been established, time estimates may be
assigned to each activity. Generally, three time values are obtained for each activity:
optimistic time (to), most likely time (tm), and pessimistic time (tp). The average time
(te) is obtained by the formula:
to + 4 tm + tp
6
 Once the network diagram with single time estimates has been developed, the
following computational procedure may be employed for determining the critical
path/s, event slacks, and activity floats: (i) Calculate the earliest occurrence time for
each event. (ii) Calculate the latest occurrence time for each event. (iii) Calculate the
slack of each event. (iv) Obtain the critical and slack paths. (iv) Compute the activity
floats.
te =

34.  The early start schedule refers to the schedule in which all activities start as early a
possible. The late start schedule reflects a desire to commit resources late-as late as
possible
 In real life situations there may be restrictions on the availability of resources. In the
face of these, various schedules may have to be considered to find out which one is
most appropriate in the light of these restrictions.
 Variability in PERT analysis is measured by variance or its square root, standard
deviation.
 The steps involved in calculating the standard deviation of the duration of the critical
path are as follows:(i) Determine the standard deviation of the duration of each activity
on the critical path. (ii) Determine the standard deviation of the total duration of the
critical path on the basis of information obtained in step (i)
 Armed with information about the mean and standard deviation of critical path
duration , we can compute the probability of completion by specified date
 The usual assumptions underlying CPM analysis are : (i) The costs associated with a
project can be divided into two components: direct costs and indirect costs. (ii)
Activities of the project can be expedited by crashing which involves employing more
resources. (iii) Crashing reduces time but enhances direct costs. (iv) Indirect costs
associated with the project increase linearly with project duration.

35.  Given the above assumptions, CPM analysis seeks to examine the consequences of
crashing on total cost (direct cost plus indirect cost). The procedure used in this respect
is generally as follows: (i) Obtain the critical path in normal network. Determine the
project duration and direct cost. (ii) Examine the cost-time slope of activities on the
critical path obtained and crash the activity which has the least slope. (iii) Construct
the new critical path crashing as per step (ii). Determine project duration and cost. (iv)
Repeat steps (ii) and (iii) till all activities on the critical path (which may change every
time) are crashed.
 To provide a vehicle for cost planning and control of projects, the network cost system
was developed. This represents a very useful supplement to the traditional time-
oriented network analysis
 The basic principle of network cost system is fairly simple: costs are planned,
measured, analysed, and controlled in terms of project activities
 As the project progresses, the following may be measured / estimated periodically for
purposes of monitoring and control : (i) costs incurred to-date. (ii) budgeted costs to-
date. (iii) value of work done to-date, (iv) cost over-run (under-run) to –date, and (v)
time over-run (under-run) to-date.