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. 5/10/2016


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. 5/10/2016


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

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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

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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

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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

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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

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Work Breakdown Structure for Computer Order

Processing System Project

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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

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Project Scheduling and Control Techniques

Gantt Chart

Critical Path Method (CPM)

Program Evaluation and Review Technique (PERT)

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Graph or bar chart with a bar for each project activity that shows

passage of time

Provides visual display of project schedule

Gantt Chart

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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)

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• 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

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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 5/10/2016

5/10/2016 Dr. Varaprasada Rao GGSESTC 16


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 1 Start

Design house and

obtain financing

Order and receive

materials Select paint

Select carpet

Lay foundations Build house

Finish work

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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

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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

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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 ?

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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".

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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?

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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

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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

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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

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CPM Example:

• CPM Network

a, 6

f, 15

b, 8

c, 5

e, 9

d, 13

g, 17 h, 9

i, 6

j, 12

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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

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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

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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 21 33

6 23 21 30

23 29

6 21

Project’s EF = 33

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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

0 8

0 5

5 14

8 21 21 33

6 23

21 30

23 29

6 21

21 33

27 33

24 33

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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

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

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CPM Example • Float

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 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

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CPM Example

• Critical Path

a, 6

f, 15

b, 8

c, 5

e, 9

d, 13

g, 17 h, 9

i, 6

j, 12

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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 =2 =

tp - to

6

2

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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.

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Probability computation

Determine probability that project is completed within specified time

Z = x -

where = tp = project mean time

= project standard mean time

x = (proposed ) specified time

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Normal Distribution of Project Time

= tp Time x

Z

Probability

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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 5/10/2016


PERT Example

A

D

C

B

F

E

G

I

H

K

J

PERT Network

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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

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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 *

5/10/2016


PERT Example

Vpath = VA + VC + VF + VI + VK

= 4/9 + 0 + 1/9 + 1 + 4/9

= 2

path = 1.414

z = (24 - 23)/(24-23)/1.414 = .71

From the Standard Normal Distribution table:

P(z < .71) = .5 + .2612 = .7612

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PROJECT COST


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, the process is repeated. 5/10/2016


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

5/10/2016


Activity crashing

Activity time

Crashing activity

Crash

time

Crash

cost

Normal Activity

Normal

time

Normal

cost

Slope = crash cost per unit time

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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

Indirect

cost

Total project cost Min total cost =

optimal project

time

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Project Crashing example

1 12

2

8

4 12

3

4 5

4

6

4

7

4

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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

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1

12

2

8

3

4 5

4

6

4

7

4

R400

R500

R3000

R7000

R200

R200

R700 12

4 Project duration = 36

From…..

To…..

1

7

2

8

3

4 5

4

6

4

7

4

R400

R500

R3000

R7000

R200

R200

R700 12

4

Project

duration = 31

Additional cost = R2000

5/10/2016


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:

5/10/2016


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

5/10/2016


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.)

5/10/2016


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. 5/10/2016


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

5/10/2016

Re-cap

Please try to understand various systems now


ACTIVITY ON NODE

5/10/2016 64 Dr. 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 DescriptionImmediate

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 25/10/2016 65


Step 2- Diagram the Network for

Cables By ITD

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Step 3 (a)- Add Deterministic Time Estimates

and Connected Paths

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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

5/10/2016 68

ACTIVITY ON ARROW



PERT


Revisiting Cables By ITD Using Probabilistic Time

Estimates

Activity DescriptionOptimistic

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 25/10/2016 74


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

cpessimistilikelymost 4optimistic timeExp.

5/10/2016 75


Calculating Expected Task Times

ActivityOptimistic

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 cpessimistilikelymost optimistictime Expected

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Network Diagram with Expected

Activity Times

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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

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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

opσ

5/10/2016 80


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 5/10/2016 81


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

5/10/2016 82


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

2Pσ

EFD

time standard path

time expected pathtime specifiedz

PT

path of varianceσ 2Path

5/10/2016 83


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

1.524.82

weeks 44.66weeks 48z

5/10/2016 84


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”

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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)

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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

K 2 5,000 2 5,000 0 0 5/10/2016 87


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

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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. 5/10/2016 92 Dr. Varaprasada Rao GGSESTC

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)


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. 5/10/2016 109


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.

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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

[email protected]

5/10/2016

project management cpm-pert

Education