CPM and PERT introduction and problems

1. CPM / PERT TECHNIQUES
Lesson Content Prepared by Dr. Mamatha S Upadhya.
Introduction
Any project involves planning, scheduling and controlling a number of interrelated activities with
use of limited resources, namely, men, machines, materials, money and time. The projects may be
extremely large and complex such as construction of a power plant, a highway, a shopping
complex, ships and aircraft, introduction of new products and research and development projects.
It is required that managers must have a dynamic planning and scheduling system to produce the
best possible results and also to react immediately to the changing conditions and make necessary
changes in the plan and schedule. A convenient analytical and visual technique of PERT and CPM
prove extremely valuable in assisting the managers in managing the projects.
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. PERT stands for Project Evaluation and Review Technique
developed during 1950’s. The technique was developed and used in conjunction with the planning
and designing of the Polaris missile project. CPM stands for Critical Path Method which was
developed by DuPont Company and applied first to the construction projects in the chemical
industry. Though both PERT and CPM techniques have similarity in terms of concepts, the basic
difference is, PERT is used for analysis of project scheduling problems. CPM has single time
estimate and PERT has three time estimates for activities and uses probability theory to find the
chance of reaching the scheduled time
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
USED IN: Production management - for the jobs of repetitive in nature where the activity time
estimates can be predicted with considerable certainty due to the existence of past experience.

2. In PERT activities are shown as a network of precedence relationships using activity-on- arrow
network construction
– Multiple time estimates
– Probabilistic activity times
USED IN: Project management - for non-repetitive jobs (research and development work),
where the time and cost estimates tend to be quite uncertain. This technique uses probabilistic time
estimates.
Benefits of PERT/CPM
Useful at many stages of project management
Mathematically simple
Give critical path and slack time
Provide project documentation
Useful in monitoring costs
Limitations of PERT/CPM
Clearly defined, independent and stable activities
Specified precedence relationships
Over emphasis on critical paths
Applications of CPM / PERT
These methods have been applied to a wide variety of problems in industries and have found
acceptance even in government organizations. These include
Construction of a dam or a canal system in a region
Construction of a building or highway
Maintenance or overhaul of airplanes or oil refinery
Space flight
Cost control of a project using PERT / COST
Designing a prototype of a machine Development of supersonic planes
Basic Steps in PERT / CPM
Project scheduling by PERT / CPM consists of three phases

3. Planning: Planning involves setting the objectives of the project. Identifying various activities to
be performed and determining the requirement of resources such as men, materials, machines, etc.
The cost and time for all the activities are estimated, and a network diagram is developed showing
sequential interrelationships (predecessor and successor) between various activities during the
planning stage.
Scheduling: Based on the time estimates, the start and finish times for each activity are worked
out by applying forward and backward pass techniques, critical path is identified, along with the
slack and float for the non-critical paths.
Controlling: Controlling refers to analyzing and evaluating the actual progress against the plan.
Reallocation of resources, crashing and review of projects with periodical reports are carried out.
The Framework for PERT and CPM
Essentially, there are six steps which are common to both the techniques. The procedure is listed
below:
I. Define the Project and all of its significant activities or tasks. The Project (made
up of several tasks) should have only a single start activity and a single finish activity.
II. Develop the relationships among the activities. Decide which activities must precede and
which must follow others.
III. 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
two activities.
IV. Assign time and/or cost estimates to each activity
V. Compute the longest time path through the network. This is called the critical path.
VI. Use the Network to help plan, schedule, and monitor and control the project.
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 earlier completion of the entire project.

4. Non-critical activities can be replanned, rescheduled and resources for them can be reallocated
flexibly, without affecting the whole project.
Five useful questions to ask when preparing an activity network are:
Is this a Start Activity?
Is this a Finish Activity?
What Activity Precedes this?
What Activity Follows this?
What Activity is Concurrent with this?
Network Diagram Representation
Network Diagram is a pictorial presentation of the various events and activities concerning of a
project.
PERT / CPM networks contain two major components
i. Activities, and
ii. Events
Activity: An activity represents an action and consumption of resources (time, money, energy)
required to complete a portion of a project. Activity is represented by an arrow as shown below:
Event: An event (or node) will always occur at the beginning and end of an activity. The event
has no resources and is represented by a circle. The ith
event and jth
event are the tail event and
head event respectively.
Predecessor activity – Activities that must be completed immediately prior to the start of another
activity are called predecessor activities.

5. Successor activity – Activities that cannot be started until one or more of other activities are
completed but immediately succeed them are called successor activities.
In the following figure, activities A and B precede activities C and D succeeding respectively.
Dummy activity – Two or more activities cannot be identified by the same beginning and ending
events. In such situation dummy activity is introduced. Dummy Activity An imaginary activity
which does not consume any resource and time is called a dummy activity. Dummy activities are
simply used to represent a connection between events in order to maintain a logic in the network.
It is represented by a dotted line in a network, as shown below.

6. 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.

7. RULES FOR CONSTRUCTING A NETWORK:
1. No single activity can be represented more than once in a network. The length of an arrow
has no significance.
2. The event numbered 1 is the start event and an event with highest number is the end event.
Before an activity can be undertaken, all activities preceding it must be completed. That
is, the activities must follow a logical sequence (or – interrelationship) between activities.
3. In assigning numbers to events, there should not be any duplication of event numbers in a
network.
4. Dummy activities must be used only if it is necessary to reduce the complexity of a
network.
5. A network should have only one start event and one end event. Some conventions of
network diagram are shown below:

8. PROCEDURE FOR NUMBERING EVENTS USING FULKERSON’S RULE
Step1: Number the start or initial event as 1.
Step2: From event 1, strike off all outgoing activities. This would have made one or more events
as initial events (event which do not have incoming activities). Number that event as 2.
Step3: Repeat step 2 for event 2, event 3 and till the end event. The end event must have the
highest number.
Example 1: Draw a network for a house construction project. The sequence of activities with their
predecessors are given in following table:

9. Solution:
The network diagram in the above figure shows the procedure relationship between the activities.
Activity A (preparation of house plan), has a start event 1 as well as an ending event 2. Activity B
(Construction of house) begins at event 2 and ends at event 3. The activity B cannot start until
activity A has been completed. Activities C and D cannot begin until activity B has been
completed, but they can be performed simultaneously. Similarly, activities E and F can start only
after completion of activities C and D respectively. Both activities E and F finish at the end of
event 6.
Critical activity: An activity is said to be critical, if a delay in its start cause a further delay in the
completion of the entire project.

10. Critical Path: The sequence of critical activities in a network which determines the duration of a
project is called the critical path. It is the longest path in the network, from the starting event to
the ending event and defines the minimum time required to complete the project.
In the network it is denoted by a double line and identifies all the critical activities of the project.
Critical Path Method (CPM)
Step1 : List all jobs and then draw arrow (network) diagram. Each job is indicated by an arrow
with the direction of the arrow showing the sequence of jobs. The length of arrow has no
significance.
Step2 : Indicate the normal time (tij) for each activity (i,j) above the arrow, which is deterministic.
Step 3: Calculate the earliest start time (Es) and Earliest finish time (Ef )
Forward pass computations yield the earliest start and the earliest finish time for each activity.
Let zero be the starting time for the project.
Earliest finish time (Ef ) of an activity is the earliest starting time +activity time
Ef = Es + tij
Earliest event time for event j is the maximum of the earliest finish time of all the activities ending
at that event.
Ej =Max (Ei+tij )
The completed E values are put over the respective rectangle representing each event.
Step4 : Calculate the latest finish and latest start time through Backward pass computations. Now
we have to calculate the latest start (Ls) time and latest finish (Lf) time for each of the activity. We
do this by adopting a backword pass through network , i.e. starting at the last activity and working
back word to the first activity.
Latest Start Ls = Latest finish time -Expected activity time
= Lf-tij
If an activity has two or more preceding activities then Lf will be the smallest of Ls of two or more
activitity. The computed Ls values are put over the respective in each event.
Step 5: Tabulate the various times namely normal time, earliest time and latest time on the arrow
diagram.
Step 6: Determine the total float for each activity by taking the difference between the earliest start
and the latest start time.

11. (Tf )ij =(Latest Start Ls - Earliest start Es) for each activity (i, j).
Activities are called critical activities if their total float is 0. Identify such activities in the network
by double line arrows. (Which gives the crtitical path).
Step 7: Calculate total duration of the project.
Example1: A project schedule has the following characteristics.
(1) Construct a network diagram
(2) Compute the earliest event time and latest event time
(3) Determine the crtitical path and total project duration
(4) Compute total and free float for each activity.
Solution : First we construct the network with the given constraints.

12. Hence, we have the following critical path 1-3- 5 -7- 8 -10 with the total project duration of 22
days.
Example 2: A small maintenance project consists of the following jobs, whose precedence
relationships are given below.
Job 1-2 1-3 2-3 2-5 3-4 3-6 4-5 4-6 5-6 6-7
Duration
(days)
15 15 3 5 8 12 1 14 3 14
1. Draw an arrow diagram representing the project
2. Find the total float for each activity
3. Find the critical path and the total project duration
Solution:

13. Critical path is given by 1-2-3-4-6-7. The total time taken for project completion is 54 days.

14. Example 3: A project consists of a series of tasks labelled A,B…..H,I with the following
constraints, Find the crtitical path and minimum time of completion of the project.
Activity A B C D E F G H I
Preceding
Activity
- - - A A B,D C B F,G
Time (days) 23 8 20 16 24 18 19 4 10

15. Critical path 1-2-3-5-6, the total project duration is 67 days.
Project Evaluation and Review Technique (PERT)
In the critical path method, the time estimates are assumed to be known with certainty. In certain
projects like research and development, new product introductions, it is difficult to estimate the
time of various activities. Hence PERT is used in such projects with a probabilistic method using
three time estimates for an activity, rather than a single estimate, as shown in Figure

16. Optimistic time tO: It is the shortest time taken to complete the activity. It means that if everything
goes well then there is more chance of completing the activity within this time.
Most likely time tm: It is the normal time taken to complete an activity, if the activity were
frequently repeated under the same conditions.
Pessimistic time tp: It is the longest time that an activity would take to complete. It is the worst
time estimate that an activity would take if unexpected problems are faced.
Taking all these time estimates into consideration, the expected time of an activity is arrived at.,
Expected time – It is the average time an activity will take if it were to be repeated on large
number of times and is based on the assumption that the activity time follows Beta distribution,
this is given by
𝑡𝑒 =
𝑡0 + 4𝑡𝑚 + 𝑡𝑝
6
The variance for the activity is given by 𝜎2 = [
𝑡𝑝−𝑡0
6
]
2
Probability for Project Duration The main objective of the analysis through PERT is to find the
completion date for a particular event within the specified date Ts, given by P (Z≤ 𝐷) where,
D=
𝐷𝑢𝑒 𝑑𝑎𝑡𝑒− 𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑑𝑎𝑡𝑒 𝑜𝑓 𝑐𝑜𝑚𝑝𝑙𝑒𝑡𝑖𝑜𝑛
√𝑃𝑟𝑜𝑗𝑒𝑐𝑡 𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒
Here, Z stands for standard normal variable.
PERT PROCEDURE

17. Step1: Draw the project network
Step2 : Compute the expected duration of each activity using the formula
𝑡𝑒 =
𝑡0+4𝑡𝑚+𝑡𝑝
6
Also calculate the expected variance 𝜎2
= [
𝑡𝑝−𝑡0
6
]
2
of each activity
Step3: Find the critical path and identify the critical activities
Step 4: Compute the project length variance 𝜎2, which is the sum of the variance of all the
critical activities and hence, find the standard deviation of the project length 𝜎 .
Step 6: Calculate the standard normal variable 𝑍 =
𝑇𝑠−𝑇𝑒
𝜎
Where Ts is the schedule time to complete the project.
Te =Normal expected project length duration
Using the normal curve, we can estimate the probability of completing the project within a
specified time.
Example1: The following table shows the jobs of a network along with their time estimates.
Job 1-2 1-6 2-3 2-4 3-5 4-5 6-7 5-8 7-8
a(days) 1 2 2 2 7 5 5 3 8
m(days) 7 5 14 5 10 5 8 3 17
b(days) 13 14 26 8 19 17 29 9 32
Draw the project network and find the probability of the project completing in 40 days.
Solution: First we calculate the expected time and standard deviation for each activity.

19. Expected project duration =36 days.
Critical path 1-2-3-5-8
Project length variance, 𝜎2=4+16+4+1=25
Std. Deviation 𝜎 = 5
The probability that the project will be completed in 40 days is given by P (Z≤ 𝐷)
D=
𝐷𝑢𝑒 𝑑𝑎𝑡𝑒− 𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑑𝑎𝑡𝑒 𝑜𝑓 𝑐𝑜𝑚𝑝𝑙𝑒𝑡𝑖𝑜𝑛
√𝑃𝑟𝑜𝑗𝑒𝑐𝑡 𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒
D=
𝑇𝑠−𝑇𝑒
𝜎
=
40−36
5
= 0.8
Area under the normal curve for the region Z≤ 0.8
P(Z ≤ 0.8 )=0.5 +∅(0.8) =0.7881=78.81%
Conclusion: If the project is performed 100 times, under the same conditions, there
will be 78.81 occasions for this job to be completed in 40 days.

20. Example2. A small project is composed of seven activities, whose time estimates
are listed in the table as follows:
You are required to:
1.Draw the project network
2.Find the expected duration and variance of each activity.
3.What is the probability that the project will be completed,
(i) 4 weeks earlier than expected?
(ii) Not more than 4 weeks later than expected?
(iii) If the project due date is 19 weeks, what is the probability of meeting the due
date?
Solution:

23. Example3. Consider the following project, Find the probability of completing the project by 18
weeks.

25. Example 4. The following table shows the jobs of a network along with their time estimates. The
time estimates are in days.

27. Example 5. Assuming that the expected times are normally distributed, find the probability of
meeting the scheduled time as given for the network. Scheduled project completion time is 30
days. Also find the date on which the project manager can complete the project with a probability
of 0.90

28. Solution:

29. DIFFERENCE BETWEEN PERT AND CPM
1. PERT is probabilistic in nature. It acknowledges and considers the variability in completion
times of activities and, in turn, the project. Accordingly, it is useful for analyzing project
scheduling problems with uncertain completion times of the activities involved. CPM is
deterministic in nature. It is most appropriately used in projects in which activity durations
are known with certainty. Not only the amount of time needed to complete various facets
of the project but also the amount of resources required for performing each of the activities
are assumed to be known and certain.
2. PERT is useful for projects that are new, non-repetitive, or which involve research and
development while CPM is of great value for projects that are repetitive and standardized,
like those involving construction activities.
3. PERT focuses primarily on time element and attaches lesser significance to the cost. CPM
puts strong emphasis on cost and specifically considers the time-cost relationship and
trade-off.

30. 4. From (1) and (3) it is evident that whereas variation in the project time is inherent in
projects when PERT is used, the time is systematically varied using additional resources
where CPM is employed.
5. PERT is event oriented so that probabilities of reaching various events by certain dates
may be calculated through event variances. CPM is activity oriented.