Project management techniques like CPM and PERT are used to plan and schedule projects. CPM involves creating a network diagram of all the tasks in a project with their time estimates and dependencies. The critical path is identified as the longest path through the network that determines the minimum project duration. PERT is similar but accounts for uncertainty in time estimates by using three time estimates per task - optimistic, most likely and pessimistic - to calculate the expected duration using beta distribution. Both techniques are useful for project scheduling and tracking progress against the plan.

1. *Project Management - CPM/PERT
Sushant Waghmare
CPM and PERT 1

2. A project is an endeavour involving a connected sequence of
activities and a range of resources, which is designed to achieve a
specific outcome and which operates within a time frame, cost and
quality constraints and which is often used to introduce change.
Characteristic of a project
 A unique, one-time operational activity or effort
 Requires the completion of a large number of
interrelated activities
 Established to achieve specific objective
 Resources, such as time and/or money, are limited
 Typically has its own management structure
 Need leadership
Sushant Waghmare
CPM and PERT 2
*Project

3. Sushant Waghmare
CPM and PERT 3
*Examples
– constructing houses, factories, shopping malls,
athletic stadiums or arenas
– developing military weapons systems, aircrafts,
new ships
– launching satellite systems
– constructing oil pipelines
– developing and implementing new computer
systems
– planning concert, football games, or basketball
tournaments
– introducing new products into market

4. • The application of a collection of tools and techniques to direct
the use of diverse resources towards the accomplishment of a
unique, complex, one time task within time, cost and quality
constraints.
• Its origins lie in World War II, when the military
authorities used the techniques of operational research to
plan the optimum use of resources.
• One of these techniques was the use of networks to represent a
system of related activities
Sushant Waghmare
CPM and PERT 4
*Project Management

5. Sushant Waghmare
CPM and PERT 5
*Project Management Process
• Project planning - Project scheduling - Projectcontrol
• Project team
– made up of individuals from various areas and departments withina
company
• Matrix organization
– a team structure with members from functional areas, depending on skills
required
• Project Manager
– most important member of project team
• Scope statement
– a document that provides an understanding, justification, and expectedresult
of a project
• Statement of work
– written description of objectives of a project
• Organizational Breakdown Structure
– a chart that shows which organizational units are responsible for workitems
• Responsibility Assignment Matrix
– shows who is responsible for work in a project

6. Sushant Waghmare
CPM and PERT 6
*Work breakdown structure
• A method of breaking down a project into individual elements (
components, subcomponents, activities and tasks) in a
hierarchical structure which can be scheduled and cost
• It defines tasks that can be completed independently of other
tasks, facilitating resource allocation, assignment of
responsibilities and measurement and control of the project
• It is foundation of project planning
• It is developed before identification of dependencies and
estimation of activity durations
• It can be used to identity the tasks in the CPM and PERT

7. Work Breakdown Structure for Computer Order
Processing System Project
Sushant Waghmare
CPM and PERT 7

8. Sushant Waghmare
CPM and PERT 8
*Project Planning
• Resource Availability and/or Limits
– Due date, late penalties, early completion incentives
– Budget
• Activity Information
– Identify all required activities
– Estimate the resources required (time) to complete each
activity
– Immediate predecessor(s) to each activity needed to create
interrelationships

9. Sushant Waghmare
CPM and PERT 9
*Project Scheduling and Control Techniques
• Gantt Chart
• Critical Path Method (CPM)
• Program Evaluation and Review Technique (PERT)

10. Graph or bar chart with a bar for each project activity that shows
passage of time
Provides visual display of project schedule
Sushant Waghmare
CPM and PERT 10
*Gantt Chart

11. Sushant Waghmare
CPM and PERT 11
History of CPM/PERT
• Critical Path Method (CPM)
– E I Du Pont de Nemours & Co. (1957) for construction of new
chemical plant and maintenance shut-down
– Deterministic task times
– Activity-on-node network construction
– Repetitive nature of jobs
• Project Evaluation and Review Technique (PERT)
– U S Navy (1958) for the POLARIS missile program
– Multiple task time estimates (probabilistic nature)
– Activity-on-arrow network construction
– Non-repetitive jobs (R & D work)

12. • Event
– Signals the beginning or ending of an activity
– Designates a point in time
– Represented by a circle (node)
• Network
– Shows the sequential relationships among activities using nodes
and arrows
Activity-on-node (AON)
nodes represent activities, and arrows show precedence
relationships
Activity-on-arrow (AOA)
arrows represent activities and nodes are events for points in
time Sushant Waghmare
CPM and PERT 12
*Project Network

13. Sushant Waghmare
CPM and PERT 13

14. Sushant Waghmare
CPM and PERT 14

15. Sushant Waghmare
CPM and PERT 15

16. Sushant Waghmare
CPM and PERT 16
*
PERT / CPM networks contain two majorcomponents
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 byan arrow, (Figure 8.1).
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, (Figure8.2).

17. Merge and BurstEvents
Oneor moreactivitiescan start and end simultaneouslyatan
event (Figure 8.3 a, b).
Preceding and Succeeding Activities
Activities performed before given events are known as
preceding activities (Figure 8.4), and activities performed after
a given event are known assucceeding activities.
Activities A and B precedeactivities C and D
respectively.
Sushant Waghmare
CPM and PERT 17

18. 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, see Figure8.5.
Sushant Waghmare
CPM and PERT 18

19. Sushant Waghmare
CPM and PERT 19
a. Twoactivities starting from a tailevent
must not have a same end event. To ensure
this, it is absolutely necessary to introduce a
dummy activity, as shown in Figure8.6.
b.Looping error should not be formed in a
network, as it represents performance of
activities repeatedly in a cyclic manner, as
shown below in Figure8.7.
c.In a network, there should be only one
startevent and one ending event as shown
below, in Figure 8.8.
d.The direction of arrows should
flow from left to right avoiding
mixing of direction as shown in
Figure 8.9.

20. RULES IN 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 endevent.
Sushant Waghmare
CPM and PERT 20

21. *
Sushant Waghmare
CPM and PERT 21

22. Sushant Waghmare
CPM and PERT 22

23. Sushant Waghmare
CPM and PERT 23

24. Sushant Waghmare
CPM and PERT 24

25. Sushant Waghmare
CPM and PERT 25

26. Dr. V
araprasada Rao GGSESTC 24
*AOA Project Network for
House
2 0
1
1 2 4 6 7
3
5
Lay
foundation
3
Design house
and obtain
financing
1
Order and
receive
materials
Dummy
Finish
work
1
Select
carpet
Select
paint
Build
house
3
1
AON Project Network for House
Start 1
3
Design house and
obtain financing
3
1
Order and receive
materials
5
1
Select paint
6
1
Select carpet
Lay foundations
2
2
Build house
4
3 Finish work
7
1
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CPM and PERT 26

27. *Situations in network diagram
A
B
A must finish before either B or C can start
C
A
B
C both A and B must finish before C canstart
D
C
B
A
both A and C must finish before either of B
or D can start
A
C
B
D
Dummy
A must finish before B can start
both A and C must finish before D canstart
Sushant Waghmare
CPM and PERT

28. Sushant Waghmare
CPM and PERT 28
*Concurrent Activities
2 3
Lay foundation
Order material
(a) Incorrect precedence
relationship
(b) Correct precedence
relationship
3
4
2
Dummy
Lay
foundation
Order material
1
2 0

29. Sushant Waghmare
CPM and PERT 29
*Network example
Illustration of network analysis of a minor redesign of a product and
its associated packaging.
The key question is: How long will it take to complete this project ?

30. *
occur near to each other in time".
Sushant Waghmare
CPM and PERT 30

31. Dr. V
araprasada Rao GGSESTC 29
CPM and PERT
*Questions to prepare 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?
Sushant Waghmare
31

32. Sushant Waghmare
CPM and PERT
*CPM
calculation
30
• Path
– A connected sequence of activities leading from
the starting event to the ending event
• Critical Path
– The longest path (time); determines the project
duration
• Critical Activities
– All of the activities that make up the critical path
32

33. *Forward
Pass
LS = minimum LS of immediate predecessors
• Earliest Start Time (ES)
– earliest time an activity can start
– ES = maximum EF of immediate predecessors
• Earliest finish time (EF)
– earliest time an activity can finish
– earliest start time plus activity time
EF= ES + t
Backward Pass
Latest Start Time (LS)
Latest time an activity can start without delaying critical path
time
LS= LF - t
Latest finish time (LF)
latest time an activity can be completed without delaying
critical path time
5/10/2016 Dr. Varaprasada Rao GGSESTC 31
Sushant Waghmare
CPM and PERT 33

34. Sushant Waghmare
CPM and PERT 34
*CPM
analysis
• Draw the CPM network
• Analyze the paths through the network
• Determine the float for each activity
– Compute the activity’s float
float = LS - ES = LF - EF
– Float is the maximum amount of time that this activity can be
delay in its completion before it becomes a critical activity,
i.e., delays completion of the project
• Find the critical path is that the sequence of activities and events
where there is no “slack” i.e.. Zero slack
– Longest path through a network
• Find the project duration is minimum project completion time

35. Sushant Waghmare
CPM and PERT 35
*CPM
Example:
a, 6
• CPM Network
f, 15
b, 8
c, 5
e, 9
d, 13
g, 17 h, 9
i, 6
j, 12

36. Sushant Waghmare
CPM and PERT 36
*CPM
Example
• ES and EF Times
a, 6
f, 15
b, 8
c, 5
e, 9
d, 13
g, 17 h, 9
i, 6
j, 12
0 6
0 8
0 5

37. Sushant Waghmare
CPM and PERT 37
*CPM
Example
• ES and EF Times
a, 6
f, 15
b, 8
c, 5
e, 9
d, 13
g, 17 h, 9
i, 6
j, 12
0 6
0 8
0 5
5 14
8 21
6 23
6 21

38. Sushant Waghmare
CPM and PERT 38
*CPM
Example
• ES and EF Times
a, 6
f, 15
b, 8
c, 5
d, 13
g, 17 h, 9
i, 6
j, 12
0 6
0 8
8 21 21 33
6 23
21 30
23 29
6 21
e, 9
0 5
Project’s EF = 33
5 14

39. aprasada Rao
CPM and PERT 39
*CPM
Example
• LS and LF Times
a, 6
f, 15
b, 8
c, 5
e, 9
d, 13
g, 17
h, 9
i, 6
j, 12
0 6
Dr.Var GGSESTC
0 8
0 5
5 14
21 33
21 33
23 29
27 33
8 21
6 23
21 30
24 33
6 21
Sushant Waghmare

40. aprasada Rao
CPM and PERT 40
*CPM
Example
• LS and LF Times
a, 6
f, 15
b, 8
c, 5
e, 9
d, 13
g, 17
h, 9
i, 6
j, 12
0 6
4 10
Dr.Var GGSESTC
0 8
0 8
0 5
7 12 5 14
12 21
21 33
21 33
23 29
27 33
8 21
8 21
6 23
10 27
21 30
24 33
6 21
9 24
Sushant Waghmare

41. da Rao GGS
CPM and PERT 41
*CPM
Example
• Float
a, 6
b, 8
d, 13
g, 17
f, 15
h, 9
i, 6
j, 12
0 6
3
3 9
c, 5
e, 9
Dr.Var ESTC
6 23
4
10 27
6 21
3
9 24
21 30
3
24 33
23 29
4
27 33
21 33
0
21 33
0 8
0
0 8
0 5
7
7 12 5 14
7
12 21
aprasa
8 21
0
8 21
Sushant Waghmare

42. Sushant Waghmare
CPM and PERT 42
*CPM
Example
• Critical Path
a, 6
f, 15
b, 8
d, 13
g, 17 h, 9
i, 6
j, 12
c, 5
e, 9

43. Dr. Varaprasada Rao GGSESTC 41
*PE
RT
• PERT is based on the assumption that an activity’s duration
follows a probability distribution instead of being a single value
• Three time estimates are required to compute the parameters of
an activity’s duration distribution:
– pessimistic time (tp ) - the time the activity would take if
things did not go well
– most likely time (tm ) - the consensus best estimate of the
activity’s duration
– optimistic time (to ) - the time the activity would take if things
did go well
e
Mean (expected time): t =
t + 4 t + t
p m o
6
Variance: Vt = 2 =
tp -to
6
2
5/10/2016 Sushant Waghmare
CPM and PERT 43

44. Sushant Waghmare
CPM and PERT 44
*PERT
analysis
• Draw the network.
• Analyze the paths through the network and find the criticalpath.
• The length of the critical path is the mean of the project duration
probability distribution which is assumed to be normal
• The standard deviation of the project duration probability
distribution is computed by adding the variances of the critical
activities (all of the activities that make up the critical path)and
taking the square root of that sum
• Probability computations can now be made using the normal
distribution table.

45. Sushant Waghmare
CPM and PERT 45
*Probability
computation
Z =
Determine probability that project is completed within specified time
x - 

where  = tp = project meantime
 = project standard mean time
x = (proposed ) specified time

46. Sushant Waghmare
CPM and PERT 46
*Normal Distribution of Project
Time
Time
 = tp x
Z
Probability

47. Sushant Waghmare
CPM and PERT 47
*PERT
Example
Immed. Optimistic Most Likely Pessimistic
Activity Predec. Time (Hr.) Time (Hr.) Time (Hr.)
A -- 4 6 8
B -- 1 4.5 5
C A 3 3 3
D A 4 5 6
E A 0.5 1 1.5
F B,C 3 4 5
G B,C 1 1.5 5
H E,F 5 6 7
I E,F 2 5 8
J D,H 2.5 2.75 4.5
K G,I 3 5 7

48. A
D
C
B
F
E
G
I
H
K
J
Sushant Waghmare
CPM and PERT 48
* PERT Example

49. Sushant Waghmare
CPM and PERT 49
*PERT
Example
Activity Expected Time Variance
A 6 4/9
B 4 4/9
C 3 0
D 5 1/9
E 1 1/36
F 4 1/9
G 2 4/9
H 6 1/9
I 5 1
J 3 1/9
K 5 4/9

50. Sushant Waghmare
CPM and PERT 50
*PERT
Example
Activity ES EF LS LF Slack
A 0 6 0 6 0 *critical
B 0 4 5 9 5
C 6 9 6 9 0 *
D 6 11 15 20 9
E 6 7 12 13 6
F 9 13 9 13 0 *
G 9 11 16 18 7
H 13 19 14 20 1
I 13 18 13 18 0 *
J 19 22 20 23 1
K 18 23 18 23 0 *

51. PERT Example
Vpath = VA + VC + VF + VI + VK
= 4/9 + 0 + 1/9 + 1 + 4/9
= 2
Sushant Waghmare
CPM and PERT 51
path = 1.414
z = (24 - 23)/  (24-23)/1.414 = .71
From the Standard Normal Distribution table:
P(z < .71) = .5 + .2612 = .7612

52. Sushant Waghmare
CPM and PERT 52

53. *PROJECT
COST
Sushant Waghmare
CPM and PERT 53

54. Sushant Waghmare
CPM and PERT 54
*Cost consideration in
project
• Project managers may have the option or requirement to crash the
project, or accelerate the completion of the project.
• This is accomplished by reducing the length of the critical path(s).
• The length of the critical path is reduced by reducing the duration
of the activities on the critical path.
• If each activity requires the expenditure of an amount of money to
reduce its duration by one unit of time, then the project manager
selects the least cost critical activity, reduces it by one time unit,
and traces that change through the remainder of the network.
• As a result of a reduction in an activity’s time, a new criticalpath
may be created.
• When there is more than one critical path, each of the critical
paths must be reduced.
• If the length of the project needs to be reduced further, the
process is repeated.

55. Sushant Waghmare
CPM and PERT 55
*Project
Crashing
• Crashing
– reducing project time by expending additional resources
• Crash time
– an amount of time an activity is reduced
• Crash cost
– cost of reducing activity time
• Goal
– reduce project duration at minimum cost

56. Dr. Varaprasada Rao GGSESTC
*Activity
crashing
Activity time
53
Crashing activity
Crash
time
Crash
cost
NormalActivity
Normal
time
Normal
cost
Slope = crash cost per unit time
5/10/2016 Sushant Waghmare
CPM and PERT 56

57. Dr. Varaprasada Rao GGSESTC 54
*Time-Cost
Relationship
 Crashing costs increase as project duration decreases
 Indirect costs increase as project duration increases
 Reduce project length as long as crashing costs are less than
indirect costs
Time-Cost Tradeoff
time
Direct cost
Total project cost
Indirect
cost
Min total cost =
optimal project
time
5/10/2016 Sushant Waghmare
CPM and PERT 57

58. Sushant Waghmare
CPM and PERT 58
*Project Crashing
example
1
12
2
8
4
12
3
4 5
4
6
4
7
4

59. Sushant Waghmare
CPM and PERT 59
*Time Cost
data
Activity Normal
time
Normal
cost Rs
Crash
time
Crash
cost Rs
Allowable
crash time
slope
1 12 3000 7 5000 5 400
2 8 2000 5 3500 3 500
3 4 4000 3 7000 1 3000
4 12 50000 9 71000 3 7000
5 4 500 1 1100 3 200
6 4 500 1 1100 3 200
7 4 1500 3 22000 1 7000
75000 110700

60. Dr. V
araprasada Rao GGSESTC 57
1
12
R400
R500
2
8
3
4
R3000
5
4
R200
6
4
R200
R700
7
4
R7000
4
12
*
From…..
To…..
1
7
R400
R500
2
8
3
4
R3000
5
4
R200
6
4
R200
R700
7
4
R7000
4
12
Project
duration =31
Additional cost = R2000
5/10/2016 Sushant Waghmare
CPM and PERT 60

61. 58
*Benefits of
CPM/PERT
• Useful at many stages of project management
• Mathematically simple
• Give critical path and slack time
• Provide project documentation
• Useful in monitoring costs
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 whichcould
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 thebest
way 5
t/
o1
0
/
d2
0
o1
6
this at the leastcD
or
.Vsatr?aprasadaRao GGSESTC Sushant Waghmare
CPM and PERT 61

62. Sushant Waghmare
CPM and PERT 62
* Limitations to
CPM/PERT
• Clearly defined, independent and stable activities
• Specified precedence relationships
• Over emphasis on critical paths
• Deterministic CPM model
• Activity time estimates are subjective and depend on judgment
• PERT assumes a beta distribution for these time estimates, but
the actual distribution may be different
• PERT consistently underestimates the expected project
completion time due to alternate paths becoming critical
To overcome the limitation, Monte Carlo simulations can be
performed on the network to eliminate the optimistic bias

63. Sushant Waghmare
CPM and PERT 63
* Computer Software
for Project Management
• Microsoft Project (Microsoft Corp.)
• MacProject (Claris Corp.)
• PowerProject (ASTA Development Inc.)
• Primavera Project Planner (Primavera)
• Project Scheduler (Scitor Corp.)
• Project Workbench (ABT Corp.)

64. Sushant Waghmare
CPM and PERT 64
*Practice
Example
A social project manager is faced with a project with thefollowing
activities:
Activity Description Duration
Social work team to live in village 5w
Social research team to do survey 12w
Analyse results of survey 5w
Establish mother & child health program 14w
Establish rural credit programme 15w
Carry out immunization of under fives 4w
Draw network diagram and show the critical path.
Calculate project duration.

65. Sushant Waghmare
CPM and PERT 65
*Practice
problem
Activity Description Duration
1-2 Social work team to live in village 5w
1-3 Social research team to do survey 12w
3-4 Analyse results of survey 5w
2-4 Establish mother & child health program 14w
3-5 Establish rural credit programme 15w
4-5 Carry out immunization of under fives 4w
3
1
2
4
5

66. Re-cap
Please try to understand various systems now

67. *ACTIVITY ON
NODE
Sushant Waghmare
CPM and PERT 67

68. Dr. Varaprasada Rao GGSESTC
*Step 1-Define the Project: Cables By ITD is bringing a new
product on line to be manufactured in their current facility in
existing space. The owners have identified 11 activities and their
precedence relationships. Develop an AON for the project.
Activity Description
Immediate Duration
Predecessor (weeks)
A Developproduct specifications None 4
B Designmanufacturing process A 6
C Source&purchasematerials A 3
D Source&purchasetooling&equipment B 6
E Receive&installtooling&equipment D 14
F Receivematerials C 5
G Pilotproduction run E&F 2
H Evaluateproduct design G 2
I Evaluateprocess performance G 3
J Writedocumentationreport H&I 4
K5/10/201T6ransitiontomanufacturing J 65 2
Sushant Waghmare
CPM and PERT 68

69. *Step 2- Diagram the Network
for
*Cables By ITD
Sushant Waghmare
CPM and PERT 69

70. * Step 3 (a)- Add Deterministic
Time Estimates and Connected
Paths
Sushant Waghmare
CPM and PERT 70

71. Sushant Waghmare
CPM and PERT 71
* Step 3 (a) (Con’t):
Calculate the Project
Completion Times
• The longest path (ABDEGIJK) limits the
project’s duration (project cannot finish in less
time than its longest path)
• ABDEGIJK is the project’s critical path
Paths Path duration
ABDEGHJK 40
ABDEGIJK 41
ACFGHJK 22
ACFGIJK 23

72. *ACTIVITY ON
ARROW
Sushant Waghmare
CPM and PERT 72

73. Sushant Waghmare
CPM and PERT 73

74. Sushant Waghmare
CPM and PERT 74

75. Sushant Waghmare
CPM and PERT 75

76. *PER
T
Sushant Waghmare
CPM and PERT 76

77. Dr. Varaprasada Rao GGSESTC
* Revisiting Cables By ITD
Using Probabilistic Time
Estimates
Activity Description
O
p
tim
isti
c time
Mostlikely
time
P
essim
isti
c time
A Developproductspecifications 2 4 6
B Designmanufacturingprocess 3 7 10
C Source&purchasematerials 2 3 5
D Source&purchasetooling&equipment 4 7 9
E Receive&installtooling&equipment 12 16 20
F Receivematerials 2 5 8
G Pilotproductionrun 2 2 2
H Evaluateproductdesign 2 3 4
I Evaluateprocessperformance 2 3 5
J Writedocumentationreport 2 4 6
2 2
K 5
/
1
0
T/
2
r0
a1
n6
sitiontomanufacturing 742
Sushant Waghmare
CPM and PERT 77

78. *Using Beta Probability
Distribution to Calculate
Expected Time Durations
• A typical beta distribution is shown below, note that ithas
definite end points
• The expected time for finishing each activity is a weighted
average
6
optimistic 4most likelypessimistic
Ex
5/p
10/.
20t
16
ime  Dr. V
araprasada Rao GGSESTC 75
Sushant Waghmare
CPM and PERT 78

79. Dr. Varaprasada Rao GGSESTC
*Calculating Expected Task
Times
Activity
Optimistic
time
Most likely
time
Pessimistic
time
Expected
time
A
B
C
D
E
F
G
H
I
5/10
J 16
/20
K
2
3
2
4
12
2
2
2
2
2
2
4
7
3
7
16
5
2
3
3
4
2
6
10
5
9
20
8
2
4
5
6
2
4
6.83
3.17
6.83
16
5
2
3
3.17
4
2 76
Expected time 
optimistic 4most likely pessimistic
6
Sushant Waghmare
CPM and PERT 79

80. * Network Diagram with
Expected Activity
Times
Sushant Waghmare
CPM and PERT 80

81. Sushant Waghmare
CPM and PERT 81
* Estimated
Path Durations
through the
Network
• ABDEGIJK is the expected critical path &
the project has an expected duration of 44.83
weeks
Activities on paths Expected duration
ABDEGHJK 44.66
ABDEGIJK 44.83
ACFGHJK 23.17
ACFGIJK 23.34

82. *PROBABILITY IN
PERT
Sushant Waghmare
CPM and PERT 82

83. Sushant Waghmare
CPM and PERT 83
* Estimating the
Probability of
Completion Dates
• Using probabilistic time estimates offers the advantage of predictingthe
probability of project completion dates
• We have already calculated the expected time for each activity by making
three time estimates
• Now we need to calculate the variance for each activity
• The variance of the beta probability distributionis:
– where p=pessimistic activity time estimate
o=optimistic activity time estimate
6
σ 2

 
 p  o 
2
 

84. Dr.Varapra sada Rao GGSESTC
*Project Activity
Variance
Activity Optimistic Most Likely Pessimistic Variance
A 2 4 6 0.44
B 3 7 10 1.36
C 2 3 5 0.25
D 4 7 9 0.69
E 12 16 20 1.78
F 2 5 8 1.00
G 2 2 2 0.00
H 2 3 4 0.11
I 2 3 5 0.25
J 2 4 6 0.44
2016
K 2 2 2 0.0
81
0
5/10/ Sushant Waghmare
CPM and PERT 84

85. Sushant Waghmare
CPM and PERT 85
* Variances of
Each Path
through the
Network
Path
Number
Activities on
Path
Path Variance
(weeks)
1 A,B,D,E,G,H,J,k 4.82
2 A,B,D,E,G,I,J,K 4.96
3 A,C,F,G,H,J,K 2.24
4 A,C,F,G,I,J,K 2.38

86. Sushant Waghmare
CPM and PERT 86
* Calculating the Probability of
Completing the Project in Less Than a
Specified Time
• When you know:
– The expected completion time
– Its variance
• You can calculate the probability of completing the project in“X”
weeks with the following formula:


 

2
σP
path standard time
z 
specifiedtime  path expectedtime  DT  EFP 
Where DT = the specified completiondate
EFPath = the expected completion time of thepath
σPath
2
 variance of path

87. Sushant Waghmare
CPM and PERT 87
* Example: Calculating the
probability of finishing the
project in 48 weeks
• Use the z values in Appendix B to determineprobabilities
• e.g. probability for path 1 is
Path
Number
Activities on Path Path Variance
(weeks)
z-value Probability of
Completion
1 A,B,D,E,G,H,J,k 4.82 1.5216 0.9357
2 A,B,D,E,G,I,J,K 4.96 1.4215 0.9222
3 A,C,F,G,H,J,K 2.24 16.5898 1.000
4 A,C,F,G,I,J,K 2.38 15.9847 1.000
4.82
  1.52

 48 weeks  44.66 weeks
z  


88. Sushant Waghmare
CPM and PERT 88
* Reducing Project
Completion
Time
• Project completion times may need to be
shortened because:
– Different deadlines
– Penalty clauses
– Need to put resources on a new project
– Promised completion dates
• Reduced project completion time is
“crashing”

89. Sushant Waghmare
CPM and PERT 89
*Reducing Project Completion
Time
–
• Crashing a project needs to balance
– Shorten a project duration
– Cost to shorten the project duration
• Crashing a project requires you to know
– Crash time of each activity
– Crash cost of each activity
Crash cost/duration = (crash cost-normal cost)/(normal time – crash time)

90. Dr.Va raprasada Rao GGSESTC
*Reducing the Time of a Project
(crashing)
Activity Normal
Time (wk)
Normal
Cost
Crash
Time
Crash
Cost
Max. weeks
of reduction
Reduce cost
per week
A 4 8,000 3 11,000 1 3,000
B 6 30,000 5 35,000 1 5,000
C 3 6,000 3 6,000 0 0
D 6 24,000 4 28,000 2 2,000
E 14 60,000 12 72,000 2 6,000
F 5 5,000 4 6,500 1 1500
G 2 6,000 2 6,000 0 0
H 2 4,000 2 4,000 0 0
I 3 4,000 2 5,000 1 1,000
J 4 4,000 2 6,400 2 1,200
5/1
K
0/2016
2 5,000 2 5,000 0
8
0
7
Sushant Waghmare
CPM and PERT 90

91. Sushant Waghmare
CPM and PERT 91
*Crashing Example: Suppose the Cables By ITD
project manager wants to reduce the new product project
from 41 to 36 weeks.
• Crashing Costs are considered to be linear
• Look to crash activities on the critical path
• Crash the least expensive activities on the critical path first
(based on cost per week)
– Crash activity I from 3 weeks to 2 weeks 1000
– Crash activity J from 4 weeks to 2 weeks 2400
– Crash activity D from 6 weeks to 4 weeks 4000
– Recommend Crash Cost 7400

92. Sushant Waghmare
CPM and PERT 92
*
PERT
CPM
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 missileproject.
CPM stands for Critical Path Method which was
developed by DuPont Company and applied first to the
construction projects in the chemicalindustry.
Though both PERT and CPM techniques have similarity in terms of
concepts, the basic difference is; CPM has single time estimate and PERT
has three time estimates for activities and uses probability theory to find
the chance of reaching the scheduledtime.

93. Sushant Waghmare
CPM and PERT 93
*
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 theplanning 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
reviewof projectswith periodical reportsarecarried out.

94. Sushant Waghmare
CPM and PERT 94
*
* FULKERSON'S RULE
Step1: Numberthe startor initial eventas 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). Numberthateventas 2.
Step3: Repeat step 2 for event 2, event 3 and till the end event. The end
event must have the highestnumber

95. Sushant Waghmare
CPM and PERT 95
*
Thecritical path forany network is the longestpath through theentire
network.
Since all activities must be completed to complete the entire project, the
length of the critical path is also the shortest time allowable for
completion of theproject.
Thus if the project is to be completed in that shortest time, all activities
on thecritical path must bestarted as soonas possible.
These activitiesare called critical activities.
If the project has to be completed ahead of the schedule, then the time
required forat leastoneof thecritical activity must be reduced.
Further, any delay in completing the critical activities will increase the
projectduration.

96. The activity, which does not lie on the critical path, is called non-critical
activity.
These non-critical activities may have someslack time.
The slack is the amount of time by which the start of an activity may be
delayedwithoutaffecting theoverall completion timeof theproject.
Butacritical activity has noslack.
To reduce the overall project time, it would require more resources (at
extracost) toreduce the time taken by thecritical activities tocomplete.
Sushant Waghmare
CPM and PERT 96

97. Sushant Waghmare
CPM and PERT 97
*
Before the critical path in a network is determined, it is necessary to
find the earliest and latest time of each event to know the earliest
expected time (TE) at which the activities originating from the event
can be started and to know the latest allowable time (TL) at which
activitiesterminating at theeventcan becompleted.
Forward Pass Computations (to calculate Earliest, Time TE)
Step 1: Begin from thestarteventand move towards theend event.
Step 2: PutTE = 0 forthestartevent.
Step 3: Go to the next event (i.e node 2) if there is an incoming activity for
event 2, add calculate TE of previousevent (i.eevent 1) and activitytime.
Note: If there are more than one incoming activities, calculate TE for all incoming
activitiesand take the maximum value. Thisvalue is theTE forevent 2.
Step 4: Repeat the same procedure from step 3 till theend event.

98. Backward Pass Computations (tocalculate LatestTimeTL)
Sushant Waghmare
CPM and PERT 98
*
Step 1: Begin from end event and move towards the start
event. Assume that thedirectionof arrows is reversed.
Step 2: Latest Time TL for the last event is the earliest
time. TE of the lastevent.
Step 3: Go to the next event, if there is an incoming activity, subtract
the value of TL of previous event from the activity duration time. The
arrived value is TL for that event. If there are more than one incoming
activities, take the minimum TEvalue.
Step 4: Repeat the same procedurefrom step 2 till the
startevent.

99. *
As discussed earlier, the non – critical activities have some slack
or float. The float of an activity is the amount of time available by
which it is possible to delay its completion time without
extending theoverall projectcompletiontime.
tij = duration of activity
TE = earliest expectedtime
TL = latest allowabletime
ESij = earliest start time of the activity
EFij = earliest finish time of theactivity
LSij = latest start time of the activity
LFij = latest finish time of theactivity
Total Float TFij: The total floatof an activity is thedifference between
the lateststart timeand theearlieststart timeof thatactivity.
TFij = LS ij – ESij ....................(1)
or
TFij = (TL – TE)– tij …………..(ii)
5/10/2016 102
Dr. Varaprasada Rao GGSESTC Sushant Waghmare
CPM and PERT 99

100. Free Float FFij: The time by which the completion of an activity can
be delayed from its earliest finish time without affecting the
earlieststarttimeof thesucceedingactivity iscalled free float.
Sushant Waghmare
CPM and PERT
*
*
*FFij = Total float – Head event
*Independent FlsolaatckIFij:The amount of time by which
the start of an activity can be delayed without
affecting the earliest start time of any immediately
following activities, assuming that the preceding
activity has finished at its latest finish time.
*IF ij = (Ej – Li) – tij
100
....................(4) Where tail event slack = Li –Ei
IFij = Free float – Tail eventslack
The negativevalueof independentfloat isconsidered to be zero.
100

101. Critical Path:
After determining the earliest and the latest scheduled times for various
activities, the minimum time required to complete the project is
calculated. In a network, among various paths, the longest path which
determines the total time duration of the project is called the critical path.
The following conditions must be satisfied in locating the critical path of a
network.
Anactivity is said to becritical only if both theconditionsaresatisfied.
1. TL – TE = 0
2. TLj – tij – TEj = 0
Example :
A projectschedule has the following characteristicsas shown in Table
i. Construct PERT network.
ii. Compute TE and TLfor
eachactivity.
iii. Find the critical path.
Sushant Waghmare
CPM and PERT 101
101

102. *
Sushant Waghmare
CPM and PERT 102
102

103. Sushant Waghmare
CPM and PERT
*
*
*To calculate TL for all
activities
*TL10 = TE10 = 22
*TL9 = TE10 – t9,10 =
22 – 7 = 15 TL8 = TE10 –
t8, 10 = 22 – 5 =17
*TL7 = TE8 – t7, 8 = 17 – 2 = 15
*TL6 = TE8 – t6, 8 = 17 – 1 =16
*TL5 = min (TE6 – t5, 6 and TE7
– t5,7)
*= min (16 – 4 and 15 –8) =
min (12,7)
*= 7 days
* TL4 = TL9 – t4, 9 = 15 – 5 =10
* TL3 = min (TL4 – t3, 4 and TL5 – t3,
5 )
* = min (10 – 1 and 7 – 6) = min (9,
Tocalculate TE for allactivities
TE1 = 0
TE2 = TE1 + t1, 2 = 0 + 4 = 4
TE3 = TE1 + t1, 3 = 0 + 1 =1
TE4 = max (TE2 + t2, 4 and TE3 + t3,4)
= max (4 + 1 and 1 + 1) = max (5,2)
= 5 days
TE5 = TE3 + t3, 6 = 1 + 6 = 7
TE6 = TE5 + t5, 6 = 7 + 4 = 11
TE7 = TE5 + t5, 7 = 7 + 8 = 15
TE8 = max (TE6 + t6, 8 and TE7 + t7, 8)
= max (11 + 1 and 15 + 2) = max (12, 17)
= 17 days
TE9 = TE4 + t4, 9 = 5 + 5 = 10
TE10 = max (TE9 + t9, 10 and TE8 + t8,10)
= max (10 + 7 and 17 + 5) = max (17,22)
= 22 days
103
103

104. Sushant Waghmare
CPM and PERT 104
104

105. *
Sushant Waghmare
CPM and PERT 105
105

106. PROJECT EVALUATION 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 variousactivities.
Hence PERT is used in such projects with a probabilistic method using three time
estimates foran activity, rather than a singleestimate, as shown in Figure 8.22.
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 thistime.
Most likely timetm:
It is the normal time taken to complete an activity,
if the activity were frequently repeated under the
sameconditions.
Pessimistic time tp:
It is the longest time that an activity would take to
5/10/2016
worst time estimate that an
if unexpected problems are
109
complete. It is the
activity would take
faced.
Dr. Varaprasada Rao GGSESTC
Sushant Waghmare
CPM and PERT 106

107. Sushant Waghmare
CPM and PERT
*
*
Theaverage or mean (ta) valueof
theactivityduration isgiven by,
Thevarianceof theactivitytime
is calculated using theformula,
Probability for ProjectDuration
The probability of completing the project
within the scheduled time (Ts) or contracted
time may be obtained by using the standard
normal deviate where Te is the expected time
of project completion.
Probability of completing theproject
within the scheduled timeis,
107
107

108. An R & D project has a list of tasks to be performed whose timeestimatesare
given in the Table 8.11, asfollows.
Sushant Waghmare
CPM and PERT
*
a. Draw the projectnetwork.
b. Find the criticalpath.
c. Find the probability that the project is completed in 19 days. If the
probability is less than 20%, find the probability of completing it in 24
days.
108
108

109. Sushant Waghmare
CPM and PERT
*
The variance of activity time is
calculated using the formula (6).
Similarly, variances of all theactivities
arecalculated.
109
109

110. Sushant Waghmare
CPM and PERT
*
calculate the time earliest
(TE) and time Latest (TL)
for all theactivities.
From the network diagram Figure 8.24, the critical path
is identified as
1-4, 4-6, 6-7, with a projectdurationof 22 days.
110
110

111. Sushant Waghmare
CPM and PERT
*
Tofind Z0 ,
we know, P (Z <Z Network Model 0) = 0.5 – z (1.3416) (from normal tables, z (1.3416) = 0.4099)
= 0.5 – 0.4099
= 0.0901
= 9.01% Thus, the probabilityof completing the R & D project in 19 days is 9.01%.
Since the probability of completing the project in 19 days is less than 20% As in question, we
find the probability of completing it in 24 days.
111
111

112. Sushant Waghmare
CPM and PERT
*
The two importantcomponentsof anyactivityare thecostand time.
Cost is directly proportional to timeand viceversa.
For example, in constructing a shopping complex, the expected time of completion can be
calculated using the time estimates of various activities. But if the construction has to be finished
earlier, it requires additional cost to complete the project. We need to arrive at a time/cost trade-
off between total costof project and total time required tocomplete it.
Normal time:
Normal time is the time required to complete
the activity at normal conditions andcost.
Crash time:
Crash time is the shortest possible activity
time; crashing more than the normal time
will increase the directcost.
Cost Slope
Cost slope is the increase in cost per unit of
time saved by crashing. A linear cost curve
is shown in Figure8.27.
112
112

113. Sushant Waghmare
CPM and PERT
*
An activity takes 4 days to complete at a normal cost of Rs. 500.00. If it is
possible to complete the activity in 2 days with an additional cost of Rs.
700.00, what is the incremental costof theactivity?
Incremental Cost or CostSlope
It means, if oneday is reduced we have tospend Rs. 100/- extra
perday.
Project Crashing
Procedure for crashing
Step1: Draw the network diagram and mark the Normal time and Crash time.
Step2: Calculate TE and TL for all theactivities.
Step3: Find the critical path and other paths.
Step 4: Find the slope for all activities and rank them in ascending order.
113
113

114. Step 5: Establish a tabular column with required field.
Step 6: Select the lowest ranked activity; check whether it is a critical activity. If
so,crash theactivity, elsego to the next highestranked activity.
Note: The critical path must remain critical while crashing.
Step 7: Calculate the total cost of project for each crashing
Step 8: Repeat Step 6 until all the activities in the critical path are fully
crashed.
Example
The following Table
8.13 gives the activities
of a construction
project and otherdata.
117
If the indirect cost is Rs. 20 perday, crash theactivities to find the minimum
duration of the projectand the projectcostassociated.
5/10/2016
Dr. Varaprasada Rao GGSESTC
Sushant Waghmare
CPM and PERT 114

115. From thedata provided in the table, draw the networkdiagram (Figure 8.28)
and find the criticalpath.
*
From the diagram, we observe that the
critical path is 1-2-5 with project duration of
14 days
Thecost slope forall activities and theirrank
iscalculated as shown in Table 8.14
5/10/2016 Dr. Varaprasada Rao GGSESTC 118
Sushant Waghmare
CPM and PERT 115

116. Theavailable pathsof the network are listed down in Table 8.15
indicating the sequence of crashing (see Figure8.29).
The sequence of crashing
and the total cost involved
is given in Table 8.16Initial
direct cost = sum of all
normal costsgiven
= Rs. 490.00
Sushant Waghmare
CPM and PERT 116
116

117. It is not possible to crash more than 10 days, as all
the activities in the critical path are fully crashed.
Hence the minimum project duration is 10 days
with the total cost of Rs.970.00.
Sushant Waghmare
CPM and PERT 117
Activity
Crashed
Project
Duration
Critical Path Direct Cost in (Rs.) Indirect
Cost in
(Rs.)
Total
Cost in
(Rs.)
- 14 1-2-5 490 14 x 20 =
280
770
1 – 2(2)
2 – 5(2)
2 – 4(1)
3 – 4(2)
10 1 – 2 – 5
1 – 3 – 4 – 5
1 – 2 – 4 – 5
490 + (2 x 15) + (2 x
100) + (1 x 10) + (2x
20) = 770
10 x 20 =
200
970
117

118. *
Sushant Waghmare
CPM and PERT 118
118

119. Sushant Waghmare
CPM and PERT 119
119

120. Sushant Waghmare
CPM and PERT 120
120

121. a. Draw the project networkdiagram.
b. Calculate the lengthand variance of thecritical path.
c. What is the probabilitythat the jobson thecritical path can be
completed in 41 days?
Sushant Waghmare
CPM and PERT 121
121

122. *
Sushant Waghmare
CPM and PERT 122
122