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Operational Research Level 4Operational Research Level 4
Prepared byPrepared by T.M.J.A.CoorayT.M.J.A.CoorayDepartment ofDepartment of MathematicsMathematics
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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: develo recedenceBe able to: develo recedencerelationship diagram which showsrelationship diagram which showsse uential relationshi of ro ect activitiesse uential relationshi of ro ect activities
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Popular tool for project schedulingPopular tool for project scheduling Gra h with bar re resentin time for eachGra h with bar re resentin time for each
tasktask
Also shows slack for activitiesAlso shows slack for activities amoun o me ac v y can e e ayeamoun o me ac v y can e e aye
without delaying project)without delaying project)
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20 4 106 8Month
Activity
Design house andobtain financing
Lay foundation
Order and receivematerials
Build house
Select paint
Select carpet
Finish work
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31 5 7 9
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Critical Path Method (CPM)Critical Path Method (CPM) -- u on em ng onu on em ng on-- anan
-- deterministic task timesdeterministic task times -- -- --
Pro ect Evaluation & Review Techni uePro ect Evaluation & Review Techni ue(PERT)(PERT)
-- U.S. Navy, Booz, Allen & HamiltonU.S. Navy, Booz, Allen & Hamilton -- multiple task time estimates( probabilistic)multiple task time estimates( probabilistic)
-- activityactivity--onon--arrow network construction (AOA)arrow network construction (AOA)
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In AON, nodes represent activities &In AON, nodes represent activities &arrows show recedence relationshi sarrows show recedence relationshi s In AOA, arrows represent activities &In AOA, arrows represent activities &
nodes are events for oints in timenodes are events for oints in timeAn event is the completion or beginningAn event is the completion or beginning
of an activitof an activitA dummy shows precedence for twoA dummy shows precedence for two
activities with same start & end nodesactivities with same start & end nodes
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Network consists of branches & nodesNetwork consists of branches & nodes
Branch
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1 32Construct forms Pour concrete
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Consider the following table which describesConsider the following table which describesthe activities to be done to build a housethe activities to be done to build a house
ActivityActivity predecessors Durationpredecessors Duration
Design house and obtain financingDesign house and obtain financing --
BB Lay foundationLay foundation A 2A 2Order and receive materialsOrder and receive materials
DD Build houseBuild house B,C 3B,C 3
Select paintSelect paint ,,
FF Select carpetSelect carpet E 1E 1
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n s wor n s wor ,,
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3
DummyLayLay foundation
422 3
Order materialr er ma er a
Incorrect Correctprecedence
relationshipprecedence
relationship
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3Lay foundation Dummy
1 2 4 6 73
2
1
3 1Desi n house Order and
Finish
work
Build
house
5
1
and obtain
financing
receive
materialsSelect
carpet
Select
paint
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A path is a sequence of connected activitiesA path is a sequence of connected activities
running from the start to the end node in arunning from the start to the end node in a
The critical path is the path with the longestThe critical path is the path with the longesturation in t e networuration in t e networ
time of the critical path (under normaltime of the critical path (under normalcircumstances)circumstances)
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path1:path1: 11--22--33--44--66--773 + 2 + 0 + 3 + 1 = 9 months; the critical path3 + 2 + 0 + 3 + 1 = 9 months; the critical path
path2:path2: 11--22--33--44--55--66--77
3 + 2 + 0 + 1 + 1 + 1 = 8 months3 + 2 + 0 + 1 + 1 + 1 = 8 monthspa :pa : -- -- -- --
3 + 1 + 3 + 1 = 83 + 1 + 3 + 1 = 8 monthsmonths
-- -- -- -- --
3 + 1 + 1 + 1 + 1 = 73 + 1 + 1 + 1 + 1 = 7 monthsmonths
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(Housebuilding example)(Housebuilding example)
ESES -- earliest time activity can startearliest time activity can start
network to determine ES timesnetwork to determine ES times= ac v y me= ac v y me
ESESijij = maximum (EF= maximum (EFii)) i
EFEFijij = ES= ESijij + t+ tijij ESES1212 = 0= 0
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EFEF1212 = ES= ES1212 + t+ t1212 = 0 + 3 = 3 months= 0 + 3 = 3 months
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--ESES
2323
= max (EF= max (EF22
) = 3 months) = 3 months
-- ESES4646 = max (EF= max (EF44) = max (5,4) = 5 months) = max (5,4) = 5 months
-- EFEF4646 = ES= ES4646 + t+ t4646 = 5 + 3 = 8 months= 5 + 3 = 8 months
-- EFEF6767 =9 months, the project duration=9 months, the project duration
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LSLS -- latest time activity can be startedlatest time activity can be started
without delaying the projectwithout delaying the project
Backward pass starts at end of network toBackward pass starts at end of network to
determine LS timesdetermine LS times LFLF -- latest time activity can be completedlatest time activity can be completed
without delaying the projectwithout delaying the project LSLSijij = LF= LFijij -- ttijij LFLFijij = minimum (LS= minimum (LSjj))
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If a deadline is not given take LF of theIf a deadline is not given take LF of the
project to be EF of the last activityproject to be EF of the last activity 6767 ==
LSLS6767 = LF= LF6767 -- tt6767 = 9= 9 -- 1 = 8 months1 = 8 months 5656 66
LSLS5656 = LF= LF5656 -- tt5656 = 8= 8 -- 1 = 7 months1 = 7 months
2424 44 ,,monthsmonths
LSLS = LF= LF -- tt = 5= 5 -- 1 = 4 months1 = 4 months
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3ES=3, EF=5LS=3, LF=5
= , =
LS=5, LF=5
= = ES=8, EF=9
1 2 4 6 73
2
1
3 1ES=0, EF=3
LS=5, LF=8 LS=8, LF=9
ES=3, EF=4
5
1LS=0, LF=3 ES=6, EF=7
LS=7, LF=8
LS=4, LF=5
ES=5, EF=6
= =,
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Activit SlackActivit Slack
-- --
Activities on critical path have ES = LS &Activities on critical path have ES = LS &==
SSijij = LS= LSijij -- ESESijij
ijij == ijij -- ijij
SS = LS= LS -- ESES = 4= 4 -- 3 = 1 month3 = 1 month
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Total slack/float or Slack of an activityTotal slack/float or Slack of an activity
TotalTotal slack/slack/ floatfloat meansmeans thethe amountamount ofof
withoutwithout affectingaffecting thethe entireentire projectprojectcomp e oncomp e on meme..
TheThe activitactivit onon aa iveniven athath shareshare thethemaximummaximum possiblepossible slackslack of of thethe activityactivity
.. SumSum of of thethe possiblepossible slacksslacks of of thethe
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ac v esac v es cancan nono exceeexcee ee max mummax mumslackslack alongalong thatthat pathpath..
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an activity which does not affect itsan activity which does not affect itsimme iate successors.imme iate successors.
FSFSijij = ES= ESjj EFEFijijFF34=ES4FF34=ES4--EF34 =5EF34 =5--5=05=0FF24=ES4FF24=ES4--ES24=5ES24=5--4=14=1
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ActivityActivity ESES LSLS EFEF LFLF Slack (S) Free slackSlack (S) Free slack
11--2*2* 00 00 33 33 0 00 0
22--33 33 33 55 55 0 00 0
--
33--4*4* 55 55 55 55 0 00 0
44--55 55 66 66 77 1 01 0
44--6*6* 55 55 88 88 0 00 0
55--66 66 77 77 88 1 11 1
--
* Critical path* Critical path
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0 2 4 6 8 10
Activity
Design house and
Lay foundation
Order and receivema er a s
Build house
Select aint
Select carpet
Finish work
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1 3 5 7 9
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Reflect uncertainty of activity timesReflect uncertainty of activity times Beta distribution is used in PERTBeta distribution is used in PERT
a + 4m + bt =
b - a
6
(Variance: 2 =
where
2
a = optimistic estimate
m = most likely time estimate
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b = pessimistic time estimate
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(tim
e)
(tim
e)
ba m btam
ime)
t
P(
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m = t ba
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Equipment testingand modification
DummyEquipmentinstallation
Finaldebugging
1 83 5 9
System
Training
TestingSystem
Testing
development
changeover
Job
Orientation
oston
recruitingtraining
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Activit InformationActivit InformationTime estimates (wks) Mean Time Variance
Activity a m b t 2
1 - 2 6 8 10 8 .44
1 - 3 3 6 9 6 1.001 - 4 1 3 5 3 .44
2 - 5 0 0 0 0 .00
2 - 6 2 4 12 5 2.78
3 - 5 2 3 4 3 .11
4 - 5 3 4 5 4 .11
4 - 7 2 2 2 2 .00
5 - 8 3 7 11 7 1.785 - 7 2 4 6 4 .44
7 - 8 0 0 0 0 .00
-
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.
8 - 9 1 10 13 9 4.00
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Early And Late TimesEarly And Late Times
Activity t ES EF LS LF S FS?
11 -- 22 88 0.440.44 00 88 11 99 11
11 -- 33 66 1.001.00 00 66 00 66 00
--
22 -- 55 00 0.000.00 88 88 99 99 11
22 -- 66 55 2.782.78 88 1313 1616 2121 88-- ..
44 -- 55 44 0.110.11 33 77 55 99 22
44 -- 77 22 0.000.00 33 55 1414 1616 1111
-- ..55 -- 77 44 0.440.44 99 1313 1212 1616 33
77 -- 88 00 0.000.00 1313 1313 1616 1616 33
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-- ..
88 -- 99 99 4.004.00 1616 2525 1616 2525 00
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( )ES=8, EF=13LS=16 LF=21
( )ES=8, EF=8LS=9, LF=9
( )
E = , EF=
LS=1, LF=98
0
5
4( )
ES=13, EF=17
LS=21 LF=25
1 83 5 9= =
( )ES=0, EF=6
LS=0, LF=6 93
( )ES=9, EF=16
LS=9, LF=16= =
( ),
LS=6, LF=9
ES=0, EF=3 04 ( )ES=9, EF=13
LS=12, LF=16
( ),
LS=16 LF=25
LS=2, LF=53 4
2
( )= , =
LS=5, LF=9( )
,
LS=16 LF=16
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( )ES=3, EF=5LS=14, LF=16
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Project variance is theProject variance is the sum of the variancessum of the variancesalong the critical pathalong the critical path
22 22 22 22 221313 3535 5858 8989
= 1.00 +0.11 + 1.78 + 4.00= 1.00 +0.11 + 1.78 + 4.00
= 6.89 weeks= 6.89 weeks
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completed (project completion time is )completed (project completion time is )within a s ecified eriod of timewithin a s ecified eriod of time
wherewhere
== ==
Z =x -
pp = project standard deviation= project standard deviation
Z = number of standard deviations of x fromZ = number of standard deviations of x from
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Probability),(~ X
=z
Z
= t Timex
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Standard Normal Distribution OfStandard Normal Distribution Of
transformed Project Timetransformed Project TimeProbability
)1,0(~ NZ
Z
=0 Timez
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What is the probability that the project isWhat is the probability that the project is
completed within 30 weeks?completed within 30 weeks?
P(XP(X 30) = ?30) = ?22 = 6.89 weeks= 6.89 weeks
== 6.896.89 = 2.62 weeks= 2.62 weeks
Z = xZ = x -- =30=30 -- 25 = 1.9125 = 1.91
P ZP Z 1.91 = ?1.91 = ?
2.62
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Determining Probability
From Z Value
Z 0.00 0.01 .. 04 0.09. . .
+0.4719
. . . .
1.9 0.4713 0.4767
.. .. ..
P( x < 30 weeks) = 0.50+ 04719
= 0.9719
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= 25 Time (weeks)x = 30
What is the probability that the project willWhat is the probability that the project will
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What is the probability that the project willWhat is the probability that the project will
22 - 25 -32.62 2.62=Z = = -1.14
P(Z< -1.14) = 0.1271
P( x< 22 weeks) = 0.1271
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Useful at many stages of projectUseful at many stages of projectmana ementmana ement Mathematically simpleMathematically simple
Gives critical path & slack timeGives critical path & slack time
Provides project documentationProvides project documentation Useful in monitorin costsUseful in monitorin costs
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Networks generated provide valuableNetworks generated provide valuableproject documentation and graphicallyproject documentation and graphicallypoint out who is responsible for variouspoint out who is responsible for various
project activitiesproject activitiesApplicable to a wide variety of projectsApplicable to a wide variety of projects andand
industriesindustriesUseful in monitoring not only schedules, butUseful in monitoring not only schedules, but
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Assumes clearly defined, independent,Assumes clearly defined, independent,& stable activities& stable activities Specified precedence relationshipsSpecified precedence relationships
beta distributionbeta distribution
u v m mu v m m
OverOver--emphasis on critical pathemphasis on critical path
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