Project management - CPM & PERT

Project Management - CPM/PERT
Dr. M Varaprasada Rao
DEAN - ACADEMICS
GIET RAJAHMUNDRY

Dr. Varaprasada Rao GGSESTC 2
What exactly is a project?
PM 1 – A building supervisor is in-charge for construction of a retail
development in the centre of Rajahmundry. There are 26 retail units
and a super market in the complex. The main responsibilities are to co-
ordinate the work of the various contractors to ensure that the project is
completed to specification, within budget and on time.
PM 2 – Dr. Rao directing a team of research scientists. They are
running trials on a new analgesic drug on behalf of a pharmaceutical
company. It is the responsibility to design the experiments and make
sure that proper scientific and legal procedures are followed, so that the
results can be subjected to independent statistical analysis.
PM 3- The international aid agency which employs me is sending me to
New Delhi to organize the introduction of multimedia resources at a
teachers’ training college. My role is quite complex. I have to make
sure that appropriate resources are purchased- and in some cases
developed within the college. I also have to encourage the acceptance
of these resources by lecturers and students within the college.03/13/18

Project is not defined by the type of outcome it is set up to achieve
PM 1 – A building supervisor is in-charge for construction of a retail
development in the centre of Rajahmundry. There are 26 retail units
and a super market in the complex. The main responsibilities are to co-
ordinate the work of the various contractors to ensure that the project is
completed to specification, within budget and on time.
PM 2 – Dr. Rao directing a team of research scientists. They are
running trials on a new analgesic drug on behalf of a pharmaceutical
company. It is the responsibility to design the experiments and make
sure that proper scientific and legal procedures are followed, so that the
results can be subjected to independent statistical analysis.
PM 3- The international aid agency which employs me is sending me to
New Delhi to organize the introduction of multimedia resources at a
teachers’ training college. My role is quite complex. I have to make
sure that appropriate resources are purchased- and in some cases
developed within the college. I also have to encourage the acceptance
of these resources by lecturers and students within the college.03/13/18
A shopping complex
A new drug
A new method of teaching students

Dr. Varaprasada Rao GIET 4
Characteristic of a project
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.
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
Project
03/13/18

Dr. Varaprasada Rao GIET 5
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 our National Event ‘Mythri’
– introducing new products into market
03/13/18

LET US LEARN THE TECHNIQUES
• My dear Students please look at the following
links at home and try to understand the basic
information and we will proceed further in the
next session on Monday.
• https://www.youtube.com/watch?v=vUMGvpsb8dc
(Only up to 11.58 Minutes)
• https://www.slideshare.net/annaprasad/project-mana
(Only 36 slides out of 145 slides)
03/13/18 Dr. Varaprasada Rao GIET 6

What is project management
• 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
03/13/18

Project Management Process
• Project planning - Project scheduling - Project control
• Project team
– made up of individuals from various areas and departments within a
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 expected result
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 work items
• Responsibility Assignment Matrix
– shows who is responsible for work in a project
03/13/18

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
03/13/18

Work Breakdown Structure for Computer Order
Processing System Project
Work Breakdown Structure for Computer Order
Processing System Project
03/13/18

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
03/13/18

Project Scheduling and Control Techniques
Gantt Chart
Critical Path Method (CPM)
Program Evaluation and Review Technique (PERT)
03/13/18

Graph or bar chart with a bar for each project activity that shows
passage of time
Provides visual display of project scheduleProvides visual display of project schedule
Gantt Chart
03/13/18

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)
03/13/18

• 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
Project Network
03/13/18

Project Network
• Network analysis is the general name given to certain specific
techniques which can be used for the planning, management and
control of projects
• Use of nodes and arrows
Arrows An arrow leads from tail to head directionally
– Indicate ACTIVITY, a time consuming effort that is required to perform a
part of the work.
Nodes  A node is represented by a circle
- Indicate EVENT, a point in time where one or more activities start and/or
finish.
• Activity
– A task or a certain amount of work required in the project
– Requires time to complete
– Represented by an arrow
• Dummy Activity
– Indicates only precedence relationships
– Does not require any time of effort
03/13/18

03/13/18 Dr. Varaprasada Rao GGSESTC 17

AOA Project Network for House
3
2 0
1
3
1 1
1
1 2 4 6 7
3
5
Lay
foundation
Design house
and obtain
financing
Order and
receive
materials
Dummy
Finish
work
Select
carpet
Select
paint
Build
house
AON Project Network for House
1
3
2
2
4
3
3
1 5
1
6
1
7
1Start
Design house and
obtain financing
Order and receive
materials
Select paint
Select carpet
Lay foundations Build house
Finish work
03/13/18

Situations in network diagram
A
B
C
A must finish before either B or C can start
A
B
C both A and B must finish before C can start
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 can start
03/13/18

Concurrent Activities
2 3
Lay foundationLay foundation
Order materialOrder material
(a)(a) Incorrect precedenceIncorrect precedence
relationshiprelationship
(b)(b) Correct precedenceCorrect precedence
relationshiprelationship
3
42
DummyDummy
LayLay
foundationfoundation
Order materialOrder material
11
22 00
03/13/18

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 ?
03/13/18

For clarity, this list is kept to a minimum by specifying only
immediate relationships, that is relationships involving activities
that "occur near to each other in time".
03/13/18

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?
03/13/18

CPM calculation
• 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
03/13/18

Forward Pass
• 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
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
LS = minimum LS of immediate predecessors
Backward Pass
03/13/18

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
03/13/18

CPM Example:
• CPM Network
a, 6a, 6
f, 15f, 15
b, 8b, 8
c, 5c, 5
e, 9e, 9
d, 13d, 13
g, 17g, 17 h, 9h, 9
i, 6i, 6
j, 12j, 12
03/13/18

CPM Example
• ES and EF Times
a, 6a, 6
f, 15f, 15
b, 8b, 8
c, 5c, 5
e, 9e, 9
d, 13d, 13
g, 17g, 17 h, 9h, 9
i, 6i, 6
j, 12j, 12
0 6
0 8
0 5
03/13/18

CPM Example
• ES and EF Times
a, 6a, 6
f, 15f, 15
b, 8b, 8
c, 5c, 5
e, 9e, 9
d, 13d, 13
g, 17g, 17 h, 9h, 9
i, 6i, 6
j, 12j, 12
0 6
0 8
0 5
5 14
8 21
6 23
6 21
03/13/18

CPM Example
• ES and EF Times
a, 6a, 6
f, 15f, 15
b, 8b, 8
c, 5c, 5
e, 9e, 9
d, 13d, 13
g, 17g, 17 h, 9h, 9
i, 6i, 6
j, 12j, 12
0 6
0 8
0 5
5 14
8 21 21 33
6 23
21 30
23 29
6 21
Project’s EF = 33Project’s EF = 33
03/13/18

CPM Example
• LS and LF Times
a, 6a, 6
f, 15f, 15
b, 8b, 8
c, 5c, 5
e, 9e, 9
d, 13d, 13
g, 17g, 17
h, 9h, 9
i, 6i, 6
j, 12j, 12
0 6
0 8
0 5
5 14
8 21
21 33
6 23
21 30
23 29
6 21
21 33
27 33
24 33
03/13/18

CPM Example
• LS and LF Times
a, 6a, 6
f, 15f, 15
b, 8b, 8
c, 5c, 5
e, 9e, 9
d, 13d, 13
g, 17g, 17
h, 9h, 9
i, 6i, 6
j, 12j, 12
0 6
0 8
0 5
5 14
8 21
21 33
6 23
21 30
23 29
6 21
4 10
0 8
7 12
12 21
21 33
27 33
8 21
10 27
24 33
9 24
03/13/18

CPM Example
• Float
a, 6a, 6
f, 15f, 15
b, 8b, 8
c, 5c, 5
e, 9e, 9
d, 13d, 13
g, 17g, 17
h, 9h, 9
i, 6i, 6
j, 12j, 12
0 6
0 8
0 5
5 14
8 21 21 33
6 23
21 30
23 29
6 21
3 9
0 8
7 12
12 21
21 33
27 33
8 21
10 27
24 33
9 24
3 4
3
3
4
0
0
7
7
0
03/13/18

CPM Example
• Critical Path
a, 6a, 6
f, 15f, 15
b, 8b, 8
c, 5c, 5
e, 9e, 9
d, 13d, 13
g, 17g, 17 h, 9h, 9
i, 6i, 6
j, 12j, 12
03/13/18

PERT
• 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
Mean (expected time): te =
tp + 4 tm + to
6
Variance: Vt =σ 2
=
tp - to
6
2
03/13/18

PERT analysis
• Draw the network.
• Analyze the paths through the network and find the critical path.
• 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.
03/13/18

Probability computation
Determine probability that project is completed within specified
time
Z =
x - µ
σ
where µ = tp = project mean time
σ = project standard mean time
x = (proposed ) specified time
03/13/18

Normal Distribution of Project Time
µ = tp Timex
Zσ
Probability
03/13/18

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 5 703/13/18

PERT Example
AA
DD
CC
BB
FF
EE
GG
II
HH
KK
JJ
PERT Network
03/13/18

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
03/13/18

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 *
03/13/18

PERT Example
Vpath = VA + VC + VF + VI + VK
= 4/9 + 0 + 1/9 + 1 + 4/9
= 2
σpath = 1.414
z = (24 - 23)/σ = (24-23)/1.414 = .71
From the Standard Normal Distribution table:
P(z < .71) = .5 + .2612 = .7612
03/13/18

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 critical path
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, the03/13/18

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
03/13/18

Activity crashingActivitycost
Activity time
Crashing activity
Crash
time
Crash
cost
Normal Activity
Normal
time
Normal
cost
Slope = crash cost per unit time
03/13/18

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
cost
time
Direct cost
Indirect
cost
Total project costMin total cost =
optimal project
time
03/13/18

Project Crashing example
11
11
22
22
88
44
11
22
33
44 55
44
66
44
77
44
03/13/18

Time Cost data
Activity Normal
time
Normal
cost Rs
Crash
time
Crash
cost Rs
Allowable
crash time
slope
1
2
3
4
5
6
7
12
8
4
12
4
4
4
3000
2000
4000
50000
500
500
1500
7
5
3
9
1
1
3
5000
3500
7000
71000
1100
1100
22000
5
3
1
3
3
3
1
400
500
3000
7000
200
200
7000
75000 110700
03/13/18

1
12
2
8
3
4 5
4
6
4
7
4
R400
R500
R3000
R7000
R200
R200
R70012
4
Project duration = 36
From…..
To…..
1
7
2
8
3
4 5
4
6
4
7
4
R400
R500
R3000
R7000
R200
R200
R70012
4
Project
duration = 31
Additional cost = R2000
03/13/18

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
•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?
CPM/PERT can answer the following important
questions:
03/13/18

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
03/13/18

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.)
03/13/18

Practice Example
A social project manager is faced with a project with the following
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.03/13/18

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
03/13/18

Re-cap
Please try to understand various systems now

ACTIVITY ON NODE
03/13/18 65Dr. Varaprasada Rao GGSESTC

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
Predecessor
Duration
(weeks)
A Develop product specifications None 4
B Design manufacturing process A 6
C Source & purchase materials A 3
D Source & purchase tooling & equipment B 6
E Receive & install tooling & equipment D 14
F Receive materials C 5
G Pilot production run E & F 2
H Evaluate product design G 2
I Evaluate process performance G 3
J Write documentation report H & I 4
K Transition to manufacturing J 203/13/18 66

Step 2- Diagram the Network for
Cables By ITD
03/13/18 67

Step 3 (a)- Add Deterministic Time Estimates
and Connected Paths
03/13/18 68

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
03/13/18 69

ACTIVITY ON ARROW

PERT

Revisiting Cables By ITD Using Probabilistic Time
Estimates
Activity Description
Optimistic
time
Most likely
time
Pessimistic
time
A Develop product specifications 2 4 6
B Design manufacturing process 3 7 10
C Source & purchase materials 2 3 5
D Source & purchase tooling & equipment 4 7 9
E Receive & install tooling & equipment 12 16 20
F Receive materials 2 5 8
G Pilot production run 2 2 2
H Evaluate product design 2 3 4
I Evaluate process performance 2 3 5
J Write documentation report 2 4 6
K Transition to manufacturing 2 2 203/13/18 75

Using Beta Probability Distribution to
Calculate Expected Time Durations
• A typical beta distribution is shown below, note that it has
definite end points
• The expected time for finishing each activity is a weighted
average
( )
6
cpessimistilikelymost4optimistic
timeExp.
++
=03/13/18 76

Calculating Expected Task Times
Activity
Optimistic
time
Most likely
time
Pessimistic
time
Expected
time
A 2 4 6 4
B 3 7 10 6.83
C 2 3 5 3.17
D 4 7 9 6.83
E 12 16 20 16
F 2 5 8 5
G 2 2 2 2
H 2 3 4 3
I 2 3 5 3.17
J 2 4 6 4
K 2 2 2 2
( )
6
4 cpessimistilikelymostoptimistic
timeExpected
++
=
03/13/18 77

Network Diagram with Expected
Activity Times
03/13/18 78

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
03/13/18 79

PROBABILITY IN PERT

Estimating the Probability of
Completion Dates
• Using probabilistic time estimates offers the advantage of predicting the
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 distribution is:
– where p=pessimistic activity time estimate
o=optimistic activity time estimate
2
2
6
op
σ 




 −
=
03/13/18 81

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
K 2 2 2 0.00
03/13/18 82

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
03/13/18 83

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:
Where DT = the specified completion date
EFPath = the expected completion time of the path





 −
=
−
= 2
Pσ
EFD
timestandardpath
timeexpectedpathtimespecified
z
PT
pathofvarianceσ 2
Path =
03/13/18 84

Example: Calculating the probability of
finishing the project in 48 weeks
• Use the z values in Appendix B to determine probabilities
• 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
1.52
4.82
weeks44.66weeks48
z =




 −
=
03/13/18 85

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”
03/13/18 86

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)
03/13/18 87

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 1,20003/13/18 88

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
03/13/18 89

A convenient analytical and visual technique of PERT and
CPM prove extremely valuable in assisting the managers in
managing the projects.
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; 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.

Project management generally consists of three phases.
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.

COMPONENTS of PERT/CPM NETWORK
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, (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,
(Figure 8.2).

Merge and Burst Events
One or more activities can start and end simultaneously at an
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 as succeeding activities.
Activities A and B precede activities C and D
respectively.

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 Figure 8.5.

ERRORS TO BE AVOIDED IN CONSTRUCTING A
NETWORK
a. Two activities starting from a tail event
must not have a same end event. To ensure
this, it is absolutely necessary to introduce a
dummy activity, as shown in Figure 8.6.
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 Figure 8.7.
c. In a network, there should be only one
start event 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.

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 end event.

Some conventions of network diagram are shown in Figure 8.10
(a), (b), (c), (d) below:

PROCEDURE FOR NUMBERING THE 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 Table 8.1, below.

CRITICAL PATH ANALYSIS
The critical path for any network is the longest path through the entire
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 the project.
Thus if the project is to be completed in that shortest time, all
activities on the critical path must be started as soon as possible.
These activities are called critical activities.
If the project has to be completed ahead of the schedule, then the time
required for at least one of the critical activity must be reduced.
Further, any delay in completing the critical activities will increase the
project duration.

The activity, which does not lie on the critical path, is called non-critical
activity.
These non-critical activities may have some slack time.
The slack is the amount of time by which the start of an activity may be
delayed without affecting the overall completion time of the project.
But a critical activity has no slack.
To reduce the overall project time, it would require more resources (at
extra cost) to reduce the time taken by the critical activities to complete.

Scheduling of Activities: Earliest Time (TE) and Latest
Time(TL)
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
activities terminating at the event can be completed.
Forward Pass Computations (to calculate Earliest, Time TE)
Step 1: Begin from the start event and move towards the end
event.
Step 2: Put TE = 0 for the start event.
Step 3: Go to the next event (i.e node 2) if there is an incoming activity for
event 2, add calculate TE of previous event (i.e event 1) and activity time.
Note: If there are more than one incoming activities, calculate TE for all incoming
activities and take the maximum value. This value is the TE for event 2.
Step 4: Repeat the same procedure from step 3 till the end event.

Backward Pass Computations (to calculate Latest Time TL)
Procedure :
Step 1: Begin from end event and move towards the start
event. Assume that the direction of arrows is reversed.
Step 2: Latest Time TL for the last event is the earliest
time. TE of the last event.
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 TE value.
Step 4: Repeat the same procedure from step 2 till the
start event.

DETERMINATION OF FLOAT AND SLACK
TIMES
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 the overall project completion time.
tij = duration of activity
TE = earliest expected time
TL = latest allowable time
ESij = earliest start time of the activity
EFij = earliest finish time of the activity
LSij = latest start time of the activity
LFij = latest finish time of the activity
Total Float TFij: The total float of an activity is the difference between
the latest start time and the earliest start time of that activity.
TFij = LS ij – ESij ....................(1)
or
TFij = (TL – TE) – tij …………..(ii)03/13/18 103Dr. Varaprasada Rao GGSESTC

Free Float FFij: The time by which the completion of an activity can
be delayed from its earliest finish time without affecting the
earliest start time of the succeeding activity is called free float.
FF ij = (Ej – Ei) – tij ....................(3)
FFij = Total float – Head event
slack
Independent Float IFij: 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 ....................
(4)
IFij = Free float – Tail event slack
Where tail event slack = Li – Ei
The negative value of independent float is considered to be zero.

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.
An activity is said to be critical only if both the conditions are satisfied.
1. TL – TE = 0
2. TLj – tij – TEj = 0
Example :
A project schedule has the following characteristics as shown in Table
i. Construct PERT network.
ii. Compute TE and TL for
each activity.
iii. Find the critical path.

(i) From the data given in the problem, the activity network is
constructed as shown in Figure given below

(ii) To determine the critical path, compute the earliest time TE
and latest time TL for each of the activity of the project. The
calculations of TE and TL are as follows:,
To calculate TE for all activities
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
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, 1)
= 1 day
TL2 = TL4 – t2, 4 = 10 – 1 = 9
TL1 = Min (TL2 – t1, 2 and TL3 – t1, 3)
= Min (9 – 4 and 1 – 1) = 0

(iii) From the Table 8.6, we observe that the
activities 1 – 3, 3 – 5, 5 – 7,7 – 8 and 8 – 10 are
critical activities as their floats are zero.

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 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 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 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.03/13/18 110

Taking all these time estimates into consideration, the expected time
of an activity is arrived at.
The average or mean (ta) value of
the activity duration is given by,
The variance of the activity
time is calculated using the
formula,
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 for Project Duration
Probability of completing the project
within the scheduled time is,

An R & D project has a list of tasks to be performed whose time estimates
are given in the Table 8.11, as follows.
Example
a. Draw the project network.
b. Find the critical path.
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.

Time expected for each activity is
calculated using the formula (5):
Similarly, the expected time is
calculated for all the activities.
The variance of activity time is
calculated using the formula (6).
Similarly, variances of all the activities
are calculated.

calculate the time earliest
(TE) and time Latest (TL)
for all the activities.
Construct a network diagram:
From the network diagram Figure 8.24, the critical path
is identified as
1-4, 4-6, 6-7, with a project duration of 22 days.

The probability of completing the project within 19 days is given by, P (Z< Z0)
To find 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 probability of 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.

COST ANALYSIS
The two important components of any activity are the cost and time.
Cost is directly proportional to time and vice versa.
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 cost of project and total time required to complete it.
Normal time:
Normal time is the time required to complete
the activity at normal conditions and cost.
Crash time:
Crash time is the shortest possible activity
time; crashing more than the normal time
will increase the direct cost.
Cost Slope
Cost slope is the increase in cost per unit of
time saved by crashing. A linear cost curve
is shown in Figure 8.27.

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 cost of the activity?
Example
Incremental Cost or Cost Slope
It means, if one day is reduced we have to spend Rs. 100/- extra
per day.
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 the activities.
Step3: Find the critical path and other paths.
Step 4: Find the slope for all activities and rank them in ascending order.

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 the activity, else go to the next highest ranked 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 other
data.
If the indirect cost is Rs. 20 per day, crash the activities to find the minimum
duration of the project and the project cost associated.
03/13/18
118

From the data provided in the table, draw the network diagram (Figure 8.28)
and find the critical path.
Solution
From the diagram, we observe that the
critical path is 1-2-5 with project duration of
14 days
The cost slope for all activities and their
rank is calculated as shown in Table 8.14

The available paths of the network are listed down in Table 8.15
indicating the sequence of crashing (see Figure 8.29).
The sequence of crashing
and the total cost involved
is given in Table 8.16 Initial
direct cost = sum of all
normal costs given
= Rs. 490.00

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.
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) + (2 x
20) = 770
10 x 20 =
200
970

Assignment

a. Draw the project network diagram.
b. Calculate the length and variance of the critical path.
c. What is the probability that the jobs on the critical path can be
completed in 41 days?

Dr. M Varaprasada Rao
Director.ggsestc@gmail.com
03/13/18

Project management - CPM & PERT

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