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PERT/CPM for Project Scheduling & Management
Basically, CPM (Critical Path Method) and PERT (Programme Evaluation
Review Technique) are project management techniques, which have
been created out of the need of Western industrial and military
establishments to plan, schedule and control complex projects.
1.1 Brief History of CPM/PERT
CPM/PERT or Network Analysis as the technique is sometimes called,
developed along two parallel streams, one industrial and the other
CPM 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 computation
was designed for the UNIVAC-I computer. The first test was made
in 1958, when CPM was applied to the construction of a new chemical
plant. In March 1959, the method was applied to a maintenance
shut-down at the Du Pont works in Louisville, Kentucky. Unproductive
time was reduced from 125 to 93 hours.
PERT was devised in 1958 for the POLARIS missile program by the
Program Evaluation Branch of the Special Projects office of the
U.S.Navy, helped by the Lockheed Missile Systems division and
the Consultant firm of Booz-Allen & Hamilton. The calculations
were so arranged so that they could be carried out on the IBM
Naval Ordinance Research Computer (NORC) at Dahlgren, Virginia.
1.2 Planning, Scheduling & Control
Planning, Scheduling (or organising) and Control are considered
to be basic Managerial functions, and CPM/PERT has been rightfully
accorded due importance in the literature on Operations Research
and Quantitative Analysis.
Far more than the technical benefits, it was found that PERT/CPM
provided a focus around which managers could brain-storm and put
their ideas together. It proved to be a great communication medium
by which thinkers and planners at one level could communicate
their ideas, their doubts and fears to another level. Most important,
it became a useful tool for evaluating the performance of individuals
There are many variations of CPM/PERT which have been useful
in planning costs, scheduling manpower and machine time. CPM/PERT
can answer the following important questions:
How long will the entire project take to be completed? What are
the risks involved?
Which are the critical activities or tasks in the project which
could delay the entire project if they were not completed on time?
Is the project on schedule, behind schedule or ahead of schedule?
If the project has to be finished earlier than planned, what
is the best way to do this at the least cost?
1.3 The Framework for PERT and CPM
Essentially, there are six steps which are common to both the
techniques. The procedure is listed below:
- Define the Project and all of it’s significant activities
or tasks. The Project (made up of several tasks) should have
only a single start activity and a single finish activity.
- Develop the relationships among the activities. Decide which
activities must precede and which must follow others.
- Draw the "Network" connecting all the activities.
Each Activity should have unique event numbers. Dummy arrows
are used where required to avoid giving the same numbering to
- 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 Key Concept used by CPM/PERT is that a small set of activities,
which make up the longest path through the activity network control
the entire project. If these "critical" activities could
be identified and assigned to responsible persons, management
resources could be optimally used by concentrating on the few
activities which determine the fate of the entire project.
Non-critical activities can be replanned, rescheduled and resources
for them can be reallocated flexibly, without affecting the whole
Five useful questions to ask when preparing an activity network
- Is this a Start Activity?
- Is this a Finish Activity?
- What Activity Precedes this?
- What Activity Follows this?
- What Activity is Concurrent with this?
Some activities are serially linked. The second activity can
begin only after the first activity is completed. In certain cases,
the activities are concurrent, because they are independent of
each other and can start simultaneously. This is especially the
case in organisations which have supervisory resources so that
work can be delegated to various departments which will be responsible
for the activities and their completion as planned.
When work is delegated like this, the need for constant feedback
and co-ordination becomes an important senior management pre-occupation.
1.4 Drawing the CPM/PERT Network
Each activity (or sub-project) in a PERT/CPM Network is represented
by an arrow symbol. Each activity is preceded and succeeded by
an event, represented as a circle and numbered.
At Event 3, we have to evaluate two predecessor activities -
Activity 1-3 and Activity 2-3, both of which are predecessor activities.
Activity 1-3 gives us an Earliest Start of 3 weeks at Event 3.
However, Activity 2-3 also has to be completed before Event 3
can begin. Along this route, the Earliest Start would be 4+0=4.
The rule is to take the longer (bigger) of the two Earliest Starts.
So the Earliest Start at event 3 is 4.
Similarly, at Event 4, we find we have to evaluate two predecessor
activities - Activity 2-4 and Activity 3-4. Along Activity 2-4,
the Earliest Start at Event 4 would be 10 wks, but along Activity
3-4, the Earliest Start at Event 4 would be 11 wks. Since 11 wks
is larger than 10 wks, we select it as the Earliest Start at Event
4.We have now found the longest path through the network. It
will take 11 weeks along activities 1-2, 2-3 and 3-4. This is
the Critical Path.
1.5.3 The Backward Pass - Latest Finish Time Rule
To make the Backward Pass, we begin at the sink or the final
event and work backwards to the first event.
At Event 3 there is only one activity, Activity 3-4 in the backward
pass, and we find that the value is 11-7 = 4 weeks. However at
Event 2 we have to evaluate 2 activities, 2-3 and 2-4. We find
that the backward pass through 2-4 gives us a value of 11-6 =
5 while 2-3 gives us 4-0 = 4. We take the smaller value
of 4 on the backward pass.
1.5.4 Tabulation & Analysis of Activities
We are now ready to tabulate the various events and calculate
the Earliest and Latest Start and Finish times. We are also now
ready to compute the SLACK or TOTAL FLOAT, which is defined as
the difference between the Latest Start and Earliest Start.
- The Earliest Start is the value in the rectangle near the
tail of each activity
- The Earliest Finish is = Earliest Start + Duration
- The Latest Finish is the value in the diamond at the head
of each activity
- The Latest Start is = Latest Finish - Duration
There are two important types of Float or Slack. These are Total
Float and Free Float.
TOTAL FLOAT is the spare time available when
all preceding activities occur at the earliest possible
times and all succeeding activities occur at the latest
- Total Float = Latest Start - Earliest Start
Activities with zero Total float are on the Critical Path
FREE FLOAT is the spare time available when all
preceding activities occur at the earliest possible times
and all succeeding activities occur at the earliest possible
When an activity has zero Total float, Free float will also be
There are various other types of float (Independent, Early Free,
Early Interfering, Late Free, Late Interfering), and float can
also be negative. We shall not go into these situations at present
for the sake of simplicity and be concerned only with Total Float
for the time being.
Having computed the various parameters of each activity, we are
now ready to go into the scheduling phase, using a type of bar
chart known as the Gantt Chart.
There are various other types of float (Independent, Early Free,
Early Interfering, Late Free, Late Interfering), and float can also
be negative. We shall not go into these situations at present for
the sake of simplicity and be concerned only with Total Float for
the time being. Having computed the various parameters of each activity,
we are now ready to go into the scheduling phase, using a type of
bar chart known as the Gantt Chart.
1.5.5 Scheduling of Activities Using a Gantt Chart
Once the activities are laid out along a Gantt Chart (Please
see chart below), the concepts of Earliest Start & Finish, Latest
Start & Finish and Float will become very obvious.
Activities 1-3 and 2-4 have total float of 1 week each, represented
by the solid timeline which begins at the latest start and ends
at the latest finish. The difference is the float, which gives
us the flexibility to schedule the activity.
For example, we might send the staff on leave during that one
week or give them some other work to do. Or we may choose to start
the activity slightly later than planned, knowing that we have
a week’s float in hand. We might even break the activity in the
middle (if this is permitted) for a week and divert the staff
for some other work, or declare a National or Festival holiday
as required under the National and Festival Holidays Act.
These are some of the examples of the use of float to schedule
an activity. Once all the activities that can be scheduled are
scheduled to the convenience of the project, normally reflecting
resource optimisation measures, we can say that the project has
A Social Project manager is faced with a project with the following
Work Team to live in Village
Research Team to do survey
results of survey
Mother & Child Health Program
Rural Credit Programme
out Immunisation of Under Fives
- Draw the arrow diagram, using the helpful numbering of the
activities, which suggests the following logic:
- Unless the Social Work team lives in the village, the Mother
and Child Health Programme cannot be started due to ignorance
and superstition of the villagers
- The Analysis of the survey can obviously be done only after
the survey is complete.
- Until rural survey is done, the Rural Credit Programme cannot
- Unless Mother and Child Programme is established, the Immunisation
of Under Fives cannot be started
- - Calculate the Earliest and Latest Event Times
- - Tabulate and Analyse the Activities
- - Schedule the Project Using a Gantt Chart
3. The PERT (Probabilistic) Approach
So far we have talked about projects, where there is high certainty
about the outcomes of activities. In other words, the cause-effect
logic is well known. This is particularly the case in Engineering
However, in Research & Development projects, or in Social Projects
which are defined as "Process Projects", where learning is an
important outcome, the cause-effect relationship is not so well
In such situations, the PERT approach is useful, because it can
accommodate the variation in event completion times, based on
an expert’s or an expert committee’s estimates.
For each activity, three time estimates are taken
The Duration of an activity is calculated using the following formula:
- The Most Optimistic
- The Most Likely
- The Most Pessimistic
Where te is the Expected time, to is the
Optimistic time, tm is the most probable activity time
and tp is the Pessimistic time.
It is not necessary to go into the theory behind the formula.
It is enough to know that the weights are based on an approximation
of the Beta distribution.
The Standard Deviation, which is a good measure of the variability
of each activity is calculated by the rather simplified formula:
Variance is the Square of the Standard Deviation.
4. PERT Calculations for the Social Project
In our Social Project, the Project Manager is now not so certain
that each activity will be completed on the basis of the single
estimate he gave. There are many assumptions involved in each
estimate, and these assumptions are illustrated in the three-time
estimate he would prefer to give to each activity.
In Activity 1-3, the time estimates are 3,12 and 21. Using our
PERT formula, we get:
|| The Standard Deviation (s.d.) for this activity
is also calculated using
the PERT formula
We calculate the PERT event times and other details as below
for each activity:
5. Estimating Risk
Having calculated the s.d. and the Variance, we are ready to
do some risk analysis. Before that we should be aware of two of
the most important assumptions made by PERT.
- The Beta distribution is appropriate for calculation of activity
- Activities are independent, and the time required to complete
one activity has no bearing on the completion times of it’s
successor activities in the network. The validity of this assumption
is questionable when we consider that in practice, many activities
5.1. Expected Length of a Project
PERT assumes that the expected length of a project (or a sequence
of independent activities) is simply the sum of their separate
Thus the summation of all the te's along the critical
path gives us the length of the project.
Similarly the variance of a sum of independent activity times
is equal to the sum of their individual variances.
In our example, the sum of the variance of the activity times
along the critical path, VT is found to be equal to (9+16) = 25.
The square root VT gives us the standard deviation of the project
length. Thus, ST=Ö 25=5. The higher the standard deviation, the
greater the uncertainty that the project will be completed on
the due date.
Although the te's are randomly distributed, the average
or expected project length Te approximately follows a Normal Distribution.
Since we have a lot of information about a Normal Distribution,
we can make several statistically significant conclusions from
A random variable drawn from a Normal Distribution has 0.68 probability
of falling within one standard deviation of the distribution average.
Therefore, there is a 68% chance that the actual project duration
will be within one standard deviation, ST of the estimated average
length of the project, te.
In our case, the te = (12+16) = 28 weeks and the ST
= 5 weeks. Assuming te to be normally distributed,
we can state that there is a probability of 0.68 that the project
will be completed within 28 ± 5 weeks, which is to say, between
23 and 33 weeks.
Since it is known that just over 95% (.954) of the area under
a Normal Distribution falls within two standard deviations, we
can state that the probability that the project will be completed
within 28 ± 10 is very high at 0.95.
5.2. Probability of Project Completion by Due Date
Now, although the project is estimated to be completed within
28 weeks (te=28) our Project Director would like to
know what is the probability that the project might be completed
within 25 weeks (i.e. Due Date or D=25).
For this calculation, we use the formula for calculating Z, the
number of standard deviations that D is away from te.
By looking at the following extract from a standard normal
table, we see that the probability associated with a Z of
-0.6 is 0.274. This means that the chance of the project
being completed within 25 weeks, instead of the expected
28 weeks is about 2 out of 7. Not very encouraging.
On the other hand, the probability that the project will be completed
within 33 weeks is calculated as follows:
The probability associated with Z= +1 is 0.84134. This
is a strong probability, and indicates that the odds are
16 to 3 that the project will be completed by the due date.
If the probability of an event is p, the odds for its occurrence
are a to b, where:
Wiest, Jerome D., and Levy, Ferdinand K., A Management
Guide to PERT/CPM, New Delhi: Prentice-Hall of India Private Limited,
Render, Barry and Stair Jr., Ralph M. - Quantitative
Analysis for Management, Massachusetts: Allyn & Bacon Inc., 1982,
Freund, John E., Modern Elementary Statistics, New Delhi:
Prentice-Hall of India Private Limited, 1979