UNIT-VIUNIT-VI

Project Management Project Management (PERT/CPM)(PERT/CPM)

INDEX INDEX UNIT 6 PPT SLIDESUNIT 6 PPT SLIDES

S.NO. TOPIC LECTURE NO.

1. Project Management ( PERT/CPM) L1

2. Network Analysis L2

3. Programme Evaluation and Review Technique L3

4. (PERT), Critical Path Method ( CPM ), Identify L4

5. Critical Path, Probability of Completing the Project L5

6. Within given time, Project Cost Analysis L6

7. Project Crashing, (simple problems) L7

44

Project PlanningProject Planning

• Given:Given:– Statement of workStatement of work

• written description of goalswritten description of goals• work & time frame of projectwork & time frame of project

– Work Breakdown StructureWork Breakdown Structure

• Be able to: develop precedence Be able to: develop precedence relationship diagram which shows relationship diagram which shows sequential relationship of project activitiessequential relationship of project activities

55

Gantt ChartGantt Chart

• Popular tool for project schedulingPopular tool for project scheduling

• Graph with bar representing time for each Graph with bar representing time for each tasktask

• Provides visual display of project scheduleProvides visual display of project schedule

• Also shows slack for activitiesAlso shows slack for activities– (amount of time activity can be delayed (amount of time activity can be delayed

without delaying project)without delaying project)

66

20 4 106 8

31 5 7 9

Month

Activity

Design house andobtain financing

Lay foundation

Order and receivematerials

Build house

Select paint

Select carpet

Finish work

A Gantt ChartA Gantt Chart

77

CPM/PERTCPM/PERT• Critical Path Method (CPM)Critical Path Method (CPM)

– - DuPont & Remington-Rand (1956)- DuPont & Remington-Rand (1956)– - deterministic task times- deterministic task times– - activity-on-node network construction (AON)- activity-on-node network construction (AON)

• Project Evaluation & Review Technique Project Evaluation & Review Technique (PERT)(PERT)

– - U.S. Navy, Booz, Allen & Hamilton- U.S. Navy, Booz, Allen & Hamilton– - multiple task time estimates( probabilistic)- multiple task time estimates( probabilistic)– - activity-on-arrow network construction (AOA)- activity-on-arrow network construction (AOA)

88

Network ConstructionNetwork Construction

• In AON, nodes represent activities & In AON, nodes represent activities & arrows show precedence relationshipsarrows show precedence relationships

• In AOA, arrows represent activities & In AOA, arrows represent activities & nodes are events for points in timenodes are events for points in time

• An event is the completion or beginning of An event is the completion or beginning of an activityan activity

• A dummy shows precedence for two A dummy shows precedence for two activities with same start & end nodesactivities with same start & end nodes

99

The Project NetworkThe Project Network

Network consists of branches & nodes Network consists of branches & nodes

1 32

Branch

Node

1010

Simplified Project NetworkSimplified Project Network

1 32Construct forms Pour concrete

1111

Consider the following table which describes Consider the following table which describes the activities to be done to build a house the activities to be done to build a house

and its sequenceand its sequence

Activity Activity predecessors predecessors DurationDuration

AA Design house and obtain financingDesign house and obtain financing - - 33BB Lay foundation Lay foundation A 2A 2CC Order and receive materialsOrder and receive materials A 1A 1DD Build houseBuild house B,C 3B,C 3EE Select paint Select paint B,C 1B,C 1FF Select carpetSelect carpet E 1E 1GG Finish work Finish work D,F 1D,F 1

1212

Concurrent ActivitiesConcurrent Activities

4

3

2

DummyLay foundation

2 3

Lay foundation

Order materialOrder material

Incorrect precedence relationship

Correct precedence relationship

1313

Project Network For A HouseProject Network For A House

1 2 4 6 7

3

5

32

0

1

31

1

1

Lay foundation

Design house and obtain financing

Order and receive materials

Dummy

Finish work

Select carpet

Select paint

Build house

1414

Critical PathCritical Path

• A path is a sequence of connected A path is a sequence of connected activities running from the start to the end activities running from the start to the end node in a networknode in a network

• The critical path is the path with the The critical path is the path with the longest duration in the networklongest duration in the network

• A project cannot be completed in less A project cannot be completed in less than the time of the critical path (under than the time of the critical path (under normal circumstances)normal circumstances)

1515

All Possible PathsAll Possible Paths

path1: path1: 1-2-3-4-6-71-2-3-4-6-73 + 2 + 0 + 3 + 1 = 9 months; the critical path3 + 2 + 0 + 3 + 1 = 9 months; the critical path

path2: path2: 1-2-3-4-5-6-71-2-3-4-5-6-73 + 2 + 0 + 1 + 1 + 1 = 8 months3 + 2 + 0 + 1 + 1 + 1 = 8 months

path3: path3: 1-2-4-6-71-2-4-6-73 + 1 + 3 + 1 = 83 + 1 + 3 + 1 = 8 months months

path4: path4: 1-2-4-5-6-71-2-4-5-6-73 + 1 + 1 + 1 + 1 = 73 + 1 + 1 + 1 + 1 = 7 months months

1616

Early TimesEarly Times(House building example)(House building example)

• ESES - earliest time activity can start - earliest time activity can start• Forward pass starts at beginning of Forward pass starts at beginning of

network to determine ES timesnetwork to determine ES times• EF = ES + activity timeEF = ES + activity time

– ESESijij = maximum (EF = maximum (EFii))

– EFEFijij = ES = ESijij + t + tijij

– ESES1212 = 0 = 0

– EFEF1212 = ES = ES1212 + t + t1212 = 0 + 3 = 3 months = 0 + 3 = 3 months

ij

1717

Computing Early Times Computing Early Times

-ES-ES2323 = max (EF = max (EF22) = 3 months) = 3 months

-- ES ES4646 = max (EF = max (EF44) = max (5,4) = 5 months) = max (5,4) = 5 months

-- EF EF4646 = ES = ES4646 + t + t46 46 = 5 + 3 = 8 months= 5 + 3 = 8 months

-- EF EF6767 =9 months, the project duration =9 months, the project duration

1818

Late TimesLate Times

• LSLS - latest time activity can be started - latest time activity can be started without delaying the projectwithout delaying the project

• Backward pass starts at end of network to Backward pass starts at end of network to determine LS timesdetermine LS times

• LFLF - latest time activity can be completed - latest time activity can be completed without delaying the projectwithout delaying the project– LSLSijij = LF = LFijij - t - tijij

– LFLFijij = minimum (LS = minimum (LSjj))

1919

Computing Late TimesComputing Late Times

– If a deadline is not given take LF of the If a deadline is not given take LF of the project to be EF of the last activityproject to be EF of the last activity

– LFLF6767 = 9 months = 9 months– LSLS6767 = LF = LF6767 - t - t67 67 = 9 - 1 = 8 months= 9 - 1 = 8 months– LFLF5656 = minimum (LS = minimum (LS66) = 8 months) = 8 months– LSLS5656 = LF = LF5656 - t - t56 56 = 8 - 1 = 7 months= 8 - 1 = 7 months– LFLF2424 = minimum (LS = minimum (LS44) = min(5, 6) = 5 months) = min(5, 6) = 5 months– LSLS2424 = LF = LF2424 - t - t24 24 = 5 - 1 = 4 months= 5 - 1 = 4 months

2020

Project cost analysis:-Project cost analysis:-

1 2 4 6 7

3

5

32

0

13

11

1

ES=0, EF=3

LS=0, LF=3

ES=3, EF=5

LS=3, LF=5

ES=5, EF=5

LS=5, LF=5

ES=5, EF=8

LS=5, LF=8

ES=6, EF=7

LS=7, LF=8

ES=8, EF=9

LS=8, LF=9

ES=3, EF=4

LS=4, LF=5

ES=5, EF=6

LS=6, LF=7

2121

Activity SlackActivity Slack

• Slack is defined as the LS-ES or LF-EFSlack is defined as the LS-ES or LF-EF• Activities on critical path have ES = LS & EF Activities on critical path have ES = LS & EF

= LF (slack is 0)= LF (slack is 0)

• Activities not on critical path have slackActivities not on critical path have slack– SSijij = LS = LSijij - ES - ESijij

– SSijij = LF = LFijij - EF - EFijij

– SS24 24 = LS= LS2424 - ES - ES24 24 = 4 - 3 = 1 month= 4 - 3 = 1 month

2222

Total slack/float or Slack of an activityTotal slack/float or Slack of an activity• Total slack/ float means the amount of Total slack/ float means the amount of

time that an activity can be delayed time that an activity can be delayed without affecting the entire project without affecting the entire project completion time. completion time.

• The activity on a given path share the The activity on a given path share the maximum possible slack of the activity maximum possible slack of the activity along that path according to its share.along that path according to its share.

• Sum of the possible slacks of the activities Sum of the possible slacks of the activities can not exceed the maximum slack along that can not exceed the maximum slack along that path.path.

2323

Free slack of an activityFree slack of an activity

• This is the maximum possible delay of This is the maximum possible delay of an activity which does not affect its an activity which does not affect its immediate successors.immediate successors.

• This is evaluated asThis is evaluated as

• FSFSijij = ES = ESj j – EF– EFijij

2424

Activity Slack DataActivity Slack Data

ActivityActivity ESES LSLS EFEF LFLF Slack (S) Free slack Slack (S) Free slack1-2*1-2* 00 00 33 33 0 0 0 02-32-3 33 33 55 55 0 0 0 02-42-4 3 3 44 44 55 1 1 1 13-4*3-4* 55 55 55 55 0 0 0 04-54-5 55 66 66 77 1 0 1 04-6*4-6* 55 55 88 88 0 0 0 05-65-6 66 77 77 88 1 1 1 16-7*6-7* 88 88 99 99 0 0 0 0

* Critical * Critical pathpath

2525

Probability of completing the project Probability of completing the project within given time:-within given time:-

• Activity

• Design house and• obtain financing

• Lay foundation

• Order and receive• materials

• Build house

• Select paint

• Select carpet

• Finish work

0 2 4 6 8 10

1 3 5 7 9

2626

Probabilistic Time EstimatesProbabilistic Time Estimates

• Reflect uncertainty of activity timesReflect uncertainty of activity times

• Beta distribution is used in PERTBeta distribution is used in PERT

b - a6(

)Variance: 2 =

a = optimistic estimatem = most likely time estimateb = pessimistic time estimate

where,

2Mean (expected time): a + 4m + b

6t =

2727

Example Beta DistributionsExample Beta Distributions

m = t ba

P (

tim

e)

b a

P (

tim

e)

m bta

P (

tim

e)

m t

2828

PERT ExamplePERT Example

1

2

4

6

83 5 9

7

Manual Testing

Dummy

System Training

Dummy

System Testing

Orientation

Position recruiting

System development

Equipment installation

Equipment testing and modification

Final debugging

System changeover

Job training

2929

Activity InformationActivity Information

1 - 2 6 8 10 8 .441 - 3 3 6 9 6 1.001 - 4 1 3 5 3 .442 - 5 0 0 0 0 .002 - 6 2 4 12 5 2.783 - 5 2 3 4 3 .114 - 5 3 4 5 4 .114 - 7 2 2 2 2 .005 - 8 3 7 11 7 1.785 - 7 2 4 6 4 .447 - 8 0 0 0 0 .006 - 9 1 4 7 4 1.008 - 9 1 10 13 9 4.00

Time estimates (wks)Mean Time VarianceActivity a m bt 2

3030

Early And Late TimesEarly And Late Times

1 - 21 - 2 88 0.440.44 00 88 11 99 111 - 31 - 3 66 1.001.00 00 66 00 66 001 - 41 - 4 33 0.440.44 00 33 22 55 222 - 52 - 5 00 0.000.00 88 88 99 99 112 - 6 2 - 6 55 2.782.78 88 1313 1616 2121 883 - 5 3 - 5 33 0.110.11 66 99 66 99 004 - 54 - 5 44 0.110.11 33 77 55 99 22

4 - 74 - 7 22 0.000.00 33 55 1414 1616 11115 - 85 - 8 77 1.781.78 99 1616 99 1616 005 - 75 - 7 44 0.440.44 99 1313 1212 1616 337 - 87 - 8 00 0.000.00 1313 1313 1616 1616 336 - 96 - 9 44 1.001.00 1313 1717 2121 2525 888 - 98 - 9 99 4.004.00 1616 2525 1616 2525 00

Activity t 2 ES EF LS LF S FS?

3131

Network With TimesNetwork With Times

1

2

4

6

83 5 9

7

( )ES=8, EF=8

LS=9, LF=9

( )ES=6, EF=9

LS=6, LF=9

( )

ES=3, EF=5

LS=14, LF=16

( )ES=0, EF=3

LS=2, LF=5

( )ES=0, EF=6

LS=0, LF=6

( )ES=0, EF=8

LS=1, LF=9

3

80

5

4

4

7

0

2

93

6

( )ES=3, EF=7

LS=5, LF=9

4 ( )

ES=9, EF=13

LS=12, LF=16

( )

ES=9, EF=16

LS=9, LF=16

( )

ES=13, EF=13

LS=16 LF=16

( )

ES=16, EF=25

LS=16 LF=25

( )

ES=13, EF=17

LS=21 LF=25

( )

ES=8, EF=13

LS=16 LF=21

3232

Project VarianceProject Variance

Project variance is the Project variance is the sum of the variances sum of the variances along the critical pathalong the critical path

2 2 = = 2 2 2 2

2 2 2 2

= 1.00 +0.11 + 1.78 + 4.00= 1.00 +0.11 + 1.78 + 4.00

= 6.89 weeks= 6.89 weeks

3333

Probabilistic Network AnalysisProbabilistic Network Analysis

Determine the probability that a project is Determine the probability that a project is completed (project completion time is )completed (project completion time is )

within a specified period of timewithin a specified period of time

wherewhere

= t= tpp = project mean time = project mean time

= project standard deviation= project standard deviation

x = project time (random variable)x = project time (random variable)

Z = number of standard deviations of x from Z = number of standard deviations of x from

the mean (standardized random variable) the mean (standardized random variable)

Z = x -

3434

Normal Distribution Of Project TimeNormal Distribution Of Project Time

= tpTimex

Z

Probability),(~ 2NX

x

z

3535

Standard Normal Distribution Of Standard Normal Distribution Of transformed Project Timetransformed Project Time

=0 Timez

Z

Probability)1,0(~ NZ

3636

Probabilistic Analysis ExampleProbabilistic Analysis ExampleWhat is the probability that the project is completed within 30 weeks?What is the probability that the project is completed within 30 weeks?

P(XP(X 30) = ? 30) = ? 2 2 = 6.89 weeks = 6.89 weeks

== 6.896.89 = 2.62 weeks = 2.62 weeks

Z = x - Z = x - =30 - 25 = 1.91=30 - 25 = 1.91

P(Z P(Z 1.91) = ? 1.91) = ?

2.62

3737

+0.4719

Determining Probability From Z Value

Z 0.00 0.01 .. 04 …0.09

1.1 0.3643 0.3665 0.3729

1.9 0.4713 … 0.4767

......

...

= 25 Time (weeks)

x = 30

P( x < 30 weeks) = 0.50+ 04719

= 0.9719

3838

What is the probability that the project will be What is the probability that the project will be completed within 22 weeks?completed within 22 weeks?

= 25 x=28

Time (weeks)x = 22

P( x< 22 weeks) = 0.1271

22 - 25 -3 2.62 2.62

P(Z< -1.14) = 0.1271

=Z =

= -1.14

3939

Benefits of PERT/CPMBenefits of PERT/CPM

• Useful at many stages of project Useful at many stages of project managementmanagement

• Mathematically simpleMathematically simple

• Uses graphical displaysUses graphical displays

• Gives critical path & slack timeGives critical path & slack time

• Provides project documentationProvides project documentation

• Useful in monitoring costsUseful in monitoring costs

4040

Advantages of PERT/CPM

Networks generated provide valuable project Networks generated provide valuable project documentation and graphically point out documentation and graphically point out who is responsible for various project who is responsible for various project activitiesactivities

Applicable to a wide variety of projects Applicable to a wide variety of projects and industriesand industries

Useful in monitoring not only schedules, but Useful in monitoring not only schedules, but costs as wellcosts as well

4141

• Assumes clearly defined, independent, Assumes clearly defined, independent, & stable activities& stable activities

• Specified precedence relationshipsSpecified precedence relationships

• Activity times (PERT) follow Activity times (PERT) follow beta distributionbeta distribution

• Subjective time estimatesSubjective time estimates

• Over-emphasis on critical pathOver-emphasis on critical path

Limitations of PERT/CPMLimitations of PERT/CPM

4242

Identifying Critical path:-Identifying Critical path:-

1 2 4 6 7

3

5

32

0

1

31

1

1

Lay foundation

Design house and obtain financing

Order and receive materials

Dummy

Finish work

Select carpet

Select paint

Build house

Project crashing:-Project crashing:-

• When the two methods like work study, trade off and other possible ones fail, we go for crashing.

• Crashing includes: Normal cost Normal Time Crash cost Crash Time Direct cost Indirect cost optimization cost