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1.040/1.401 – Project Management
Probabilistic Planning Part I
Samuel Labi and Fred MoavenzadehSamuel Labi and Fred Moavenzadeh
Department of Civil and Environmental EngineeringDepartment of Civil and Environmental EngineeringMassachusetts Institute of TechnologyMassachusetts Institute of Technology
March 19, 2007
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Project Management PhasesProject Management Phases
FEASIBILITY CLOSEOUTDEVELOPMENT OPERATIONS DESIGNDESIGNPLANNING PLANNING
Project Evaluation
Project Finance
Project Organization
Project Cost Estimation
Project Planning and Scheduling
Recall …
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Outline for this Lecture
Deterministic systems decision-making – the general picture
Probabilistic systems decision-making – the general picture– the special case of project planning
Why planning is never deterministic
Simplified examples of deterministic, probabilistic planning- everyday life- project planning
Illustration of probabilistic project planning
PERT – Basics, terminology, advantages, disadvantages, example
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Deterministic Systems Planning and Decision-making
DeterministicAnalysis of
Engineering System
Input variables and their FIXED VALUES
Examples: queuing systems, network systems, etc.
Examples: costs, time duration, quality, interest rates, etc.
FIXED VALUES of the Outputs (system performance criteria, etc.)
VALUE OF combined output (index, utility or value) representing multiple performance measures of the system
A SINGLE evaluation outcome
Alt. 1 Alt. k Alt. n …
…
…
Final evaluation result and decision
Influence of deterministic inputs on the outputs of engineering systems --the general picture
X*
Y*
Z*
O1*
O2*
OM*
O*COMBO
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But engineering systems are never deterministic!Why?
For sample, for Project Planning Systems …Variations in planning input parametersBeyond control of project managerCategories of the variation factors● Natural (weather -- good and bad, natural disasters, etc.) ● Man-made (equipment breakdowns, strikes, new technology,
worker morale, poor design, site problems, interest rates, etc.)
Combined effect of input factor variation is a variation in the outputs (costs, time, quality)
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Why planning is never deterministic -- II
Probabilistic Analysis of
Engineering System
fX
Input variables and their probability distributions
Examples: queuing systems, network systems, etc.
Examples: costs, time durations, quality, interest rates, etc.
f(O1)
Outputs (system performance criteria, etc.) and their probability distributionsProbability distributions for individual outputs
Probability distribution for combined output (index, utility or value) representing multiple performance measures
Discrete probability distribution for evaluation outcome
PAlt. i is the probability that an alternative turns out to be most desirable or most critical
PAlt. i
Alt. 1 Alt. 2 Alt. n …
…
…
Variability of final evaluation result and
decision
Influence of stochastic inputs on the outputs of engineering systems -- the general picture
fY
fZ
f(O2)
f(OM)
f(OCOMBO)
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Why planning is never deterministic -- III
f
Input variable and its probability distribution
Probability distribution?
What exactly do you mean by that?
See survey for dinner durations (times) of class members
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Probability Distribution for your Dinner Times:
00.050.1
0.150.2
0.250.3
0.35
0 2.5 7.5 12.5 17.5 22.5 27.5 32.5
Time Spent for Dinner (minutes)
Prob
abili
ty
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00.050.1
0.150.2
0.250.3
0.35
0 2.5 7.5 12.5 17.5 22.5 27.5 32.5
Time Spent for Dinner (minutes)
Prob
abili
ty
Probability that a randomly selected student spends less than T* minutes for dinner is:
… less than T* = P(T < T*) = *)(* ZZPTTP <=⎟⎠⎞
⎜⎝⎛ −
<−
σμ
σμ
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00.050.1
0.150.2
0.250.3
0.35
0 2.5 7.5 12.5 17.5 22.5 27.5 32.5
Time Spent for Dinner (minutes)
Prob
abili
ty
Probability that a randomly selected student spends less than T* minutes for dinner is:
… less than T1 = P(T < T*) =
P(T)
T
f(Z)
Z
T*
Z*
Mean = μStd dev = σ
Mean = 0Std dev = 1*)(* ZZPTTP <=⎟
⎠⎞
⎜⎝⎛ −
<−
σμ
σμ
μ = 0
μ0
Standard Normal Transformation
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00.050.1
0.150.2
0.250.3
0.35
0 2.5 7.5 12.5 17.5 22.5 27.5 32.5
Time Spent for Dinner (minutes)
Prob
abili
ty
Probability that a randomly selected student spends …
… less than T1 = P(T < T*) =
… more than T1 = P(T > T*) =
P(T)
T
f(Z)
Z
T*
Z*
Mean = μStd dev = σ
Mean = 0Std dev = 1
*)(* ZZPTTP <=⎟⎠⎞
⎜⎝⎛ −
<−
σμ
σμ
μ = 0
μ0
Standard Normal Transformation
*)(* ZZPTTP >=⎟⎠⎞
⎜⎝⎛ −
>−
σμ
σμ
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00.050.1
0.150.2
0.250.3
0.35
0 2.5 7.5 12.5 17.5 22.5 27.5 32.5
Time Spent for Dinner (minutes)
Prob
abili
ty
Probability that a randomly selected student spends …
… less than T* = P(T < T*) =
… more than T* = P(T > T*) =
… between T*1 and T*2 = P(T*1 < T < T*2) =
P(T)
T
f(Z)
Z
T*
Z*
Mean = μStd dev = σ
Mean = 0Std dev = 1
*)(* ZZPTTP <=⎟⎠⎞
⎜⎝⎛ −
<−
σμ
σμ
μ = 0
μ0
Standard Normal Transformation
)( *2
*1
*2
*1 ZZZPTTTP <<=⎟⎟
⎠
⎞⎜⎜⎝
⎛ −<
−<
−σμ
σμ
σμ
*)(* ZZPTTP >=⎟⎠⎞
⎜⎝⎛ −
>−
σμ
σμ
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Probability is the area under the probability distribution/density curves)
Probability can be found using any one of three ways:
- coordinate geometry
- calculus
- statistical tables
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Like-wise, we can build probability distributions for project planning parameters by …
- using historical data from past projects, OR
- computer simulation
And thus we can find the probability that project durations falls within a certain specified range
00.050.1
0.150.2
0.250.3
0.35
0 2.5 7.5 12.5 17.5 22.5 27.5 32.5
Time Spent for Dinner (minutes)
Prob
abili
ty
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Probabilistic planning of project management systems can involve uncertainties in:● Need for an Activity (need vs. no need)● Durations
» Activity durations» Activity start-times and end-times
● Cost of activities● Quality of Workmanship and materials● Etc.
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Probabilistic planning of project management systems can involve uncertainties in:● Need for an Activity (need vs. no need)● Durations
» Activity durations» Activity start-times and end-times
● Cost of activities● Quality of Workmanship and materials● Etc.
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Probabilistic Analysis of Project
Planning
Input variable and its probability distribution
Project planning system
Project activity durations
Outputs (system performance criteria, etc.) and their probability distributions
Probability distributions for the output
In this case, …“Output” is the identification of the critical path for the project.
Frequency distribution or probability distribution for evaluation outcome
PAlt.i is the probability that a given path turns out to be the critical path
PAlt. i
Path 1 Path 2 Path n …
Variability of final evaluation result and
decision
Influence of stochastic inputs -- the specific picture of project planning
fX f(O1)
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SAM Waking up and meditating
START = 7AM
Duration 1hr FINISH =8AM
ActivityStart Time
Activity Duration Finish Time
KEY
SAM Bathing, Breakfast,, Reading, Transport to MIT, etc.START =8AM
Duration 5hrs FINISH = 1 PM
US Meeting in Class
For this LectureSTART= 1PM
Duration1.5hr FINISH= 2:30
YOU Preparing for Classes, etc.
START= 7 AM
Duration 1 hr FINISH= 8AM
YOU showing up at Other Classes
START= 8AM
Duration 5 hr FINISH=1 PM
YOU missing the lectureSTART=
1 PMDuration1.5hr FINISH=
2:30
Perfectly deterministicProbabilistic planning: An example in everyday life
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SAM Waking up and meditating
LS = 7AM
ES = 6AM
Duration = 1 hour
EF = 7AMLF = 8AM
Activity
Latest Start
Earliest Start
Duration of
Activity
Earliest Finish
Latest Finish
KEY
SAM Bathing, Breakfast,, Reading, Transport to MIT, etc.LS = 8AM
ES =7AM
Duration = 5 hrs
EF = NOONLF = 1 PM
US Meeting in Class
For this Lecture
LS = 1 PM
ES = 12:55
Duration = 1.5 hrs
EF = 2:25LF = 2:30
YOU Preparing for Classes, etc.
LS = 7AM
ES = 6:45
Duration = 1 hr
EF = 7:45LF = 8AM
YOU showing up at Other Classes
LS = 8AM
ES = 7:45
Duration = 5 hour
EF = 12:45LF = 1 PM
YOU missing the lecture
LS = 7AM
ES = 6AM
Duration = 1 hour
EF = 7AMLF = 8AM
Partly deterministic, Partly probabilisticProbabilistic planning: An example in everyday life
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SAM Waking up and meditating
LS = 7AM
ES = 6AM
Duration EF = 7AMLF = 8AM
Activity
Latest Start
Earliest Start
Duration of
Activity
Earliest Finish
Latest Finish
KEY
SAM Bathing, Breakfast,, Reading, Transport to MIT, etc.LS = 8AM
ES =7AM
Duration EF = NOONLF = 1 PM
US Meeting in Class
For this Lecture
LS = 1 PM
ES = 12:55
Duration EF = 2:25LF = 2:30
YOU Preparing for Classes, etc.
LS = 7AM
ES = 6:45
Duration EF = 7:45LF = 8AM
YOU showing up at Other Classes
LS = 8AM
ES = 7:45
Duration EF = 12:45LF = 1 PM
YOU missing the lecture
LS = 7AM
ES = 6AM
Duration EF = 7AMLF = 8AM
Fully probabilistic
μ = 1hrσ = 0.25hr
μ = 5 hrσ = 0.6hr
μ =1.5 hrσ = 0.31hr
μ = 1 hrσ = 0.15 hr
μ = 5 hrσ = 0.35 hr
μ = 1hrσ = 0.2 hr
Probabilistic planning: An example in everyday life
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SAM Waking up and meditating
LS = 7AM
ES = 6AM
Duration EF = 7AMLF = 8AM
Activity
Latest Start
Earliest Start
Duration of
Activity
Earliest Finish
Latest Finish
KEY
SAM Bathing, Breakfast,, Reading, Transport to MIT, etc.LS = 8AM
ES =7AM
Duration EF = NOONLF = 1 PM
US Meeting in Class
For this Lecture
LS = 1 PM
ES = 12:55
Duration EF = 2:25LF = 2:30
YOU Preparing for Classes, etc.
LS = 7AM
ES = 6:55
Duration EF = 7:55LF = 8AM
YOU showing up at Other Classes
LS = 8AM
ES = 7:55
Duration EF = 12:55LF = 1 PM
YOU missing the lecture
LS = 7AM
ES = 6AM
Duration EF = 7AMLF = 8AM
Fully probabilistic
μ = 1hrσ = 0.25hr
μ = 5 hrσ = 0.6hr
μ =1.5 hrσ = 0.31hr
μ = 1 hrσ = 0.15 hr
μ = 5 hrσ = 0.35 hr
μ = 1hrσ = 0.2 hr
Probabilistic planning: An example in everyday life
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Probabilistic planning: An example in Project Management
Activity NameEarly StartLate Start
DurationEarly FinishLate Finish
KEY
Activity RMonth 0 4
months Month 4
Month 0
Month 4
Activity GMonth 4
3 months
Month 10
Month 7
Month 7
Activity JMonth 10
5 months
Month 15
Month 10
Month 15
Activity WMonth 15
2 months
Month 17
Month 15
Month 17
Activity AMonth 4 4
months Month 15
Month 11
Month 8
Activity CMonth 4 6
months Month 10
Month 4
Month 10
Activity MMonth 10 3
months Month 15
Month 12
Month 13
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Activity NameEarly StartLate Start
Duration(O-M-P)
Early Finish
Late Finish
KEY
Activity RMonth 0
4 Months(3-4-8) Mont
h 4Month 0
Month 4
Activity GMonth 4
3 Months(2-3-5) Month
10Month 7
Month 7
Activity JMonth 10
5 Months(2-5-6) Month
15Month 10
Month 15
Activity WMonth 15
2 Months(1-2-4) Month
17Month 15
Month 17
Activity AMonth 4
4 Months(3-4-5) Month
15Month 11
Month 8
Activity CMonth 4
6 Months(3-6-7) Month
10Month 4
Month 10
Activity MMonth 10
3Months(1-3-5) Month
15Month 12
Month 13
O: Optimistic (earliest time)M: Most probable timeP: Pessimistic (latest time)
Probabilistic planning: An example in Project Management
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Probabilistic planning involving activity durations An Illustration
Activity RMonth 0
4 Months(3-4-8) Mont
h 4Month 0
Month 4
Activity GMonth 4
3 Months(2-3-5) Month
10Month 7
Month 7
Activity JMonth 10
5 Months(2-5-6) Month
15Month 10
Month 15
Activity WMonth 15
2 Months(1-2-4) Month
17Month 15
Month 17
Activity AMonth 4
4 Months(3-4-5) Month
15Month 11
Month 8
Activity CMonth 4
6 Months(3-6-7) Month
10Month 4
Month 10
Activity MMonth 10
3Months(1-3-5) Month
15Month 12
Month 13
Activity
Opt
imis
tic
Mos
t Lik
ely
Pess
imis
tic
R 3 4 8A 3 4 5G 2 3 5C 3 6 7J 2 5 6M 1 3 5W 1 2 4
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Probabilistic planning involving activity durations An Illustration
Activity RMonth 0
4 Months(3-4-8) Mont
h 4Month 0
Month 4
Activity GMonth 4
3 Months(2-3-5) Month
10Month 7
Month 7
Activity JMonth 10
5 Months(2-5-6) Month
15Month 10
Month 15
Activity WMonth 15
2 Months(1-2-4) Month
17Month 15
Month 17
Activity AMonth 4
4 Months(3-4-5) Month
15Month 11
Month 8
Activity CMonth 4
6 Months(3-6-7) Month
10Month 4
Month 10
Activity MMonth 10
3Months(1-3-5) Month
15Month 12
Month 13
Activity
Opt
imis
tic
Mos
t Lik
ely
Pess
imis
tic
R 3 4 8A 3 4 5G 2 3 5C 3 6 7J 2 5 6M 1 3 5W 1 2 4
Let’s say we have lots of data on the durations of each activity. Such data is typically from:
- Historical records (previous projects)
- Computer simulation (Monte Carlo)
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Probabilistic planning involving activity durations An Illustration
Activity RMonth 0
4 Months(3-4-8) Mont
h 4Month 0
Month 4
Activity GMonth 4
3 Months(2-3-5) Month
10Month 7
Month 7
Activity JMonth 10
5 Months(2-5-6) Month
15Month 10
Month 15
Activity WMonth 15
2 Months(1-2-4) Month
17Month 15
Month 17
Activity AMonth 4
4 Months(3-4-5) Month
15Month 11
Month 8
Activity CMonth 4
6 Months(3-6-7) Month
10Month 4
Month 10
Activity MMonth 10
3Months(1-3-5) Month
15Month 12
Month 13
Activity
Opt
imis
tic
Mos
t Lik
ely
Pess
imis
tic
Mean Standard Dev. Variance
R 3 4 8 4.50 0.833 0.69444A 3 4 5 4.00 0.333 0.11111G 2 3 5 3.17 0.500 0.25000C 3 6 7 5.67 0.667 0.44444J 2 5 6 4.67 0.667 0.44444M 1 3 5 3.00 0.667 0.44444W 1 2 4 2.17 0.500 0.25000
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Calculate the Expected Duration of each path, and Identify the Critical Path on the basis of the mean only:
Path:Deterministic
Duration:Probabilistic
Duration:R-A-W 10 10.67R-G-J-W 14 14.50R-C-J-W 17 17.00 Critical PathR-C-M-W 15 15.33
Activity
Opt
imis
tic
Mos
t Lik
ely
Pess
imis
tic
Mean Standard Dev. Variance
R 3 4 8 4.50 0.833 0.69444A 3 4 5 4.00 0.333 0.11111G 2 3 5 3.17 0.500 0.25000C 3 6 7 5.67 0.667 0.44444J 2 5 6 4.67 0.667 0.44444M 1 3 5 3.00 0.667 0.44444W 1 2 4 2.17 0.500 0.25000
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Calculate the Expected Duration of each path, and Identify the Critical Path on the basis of both the mean and the std dev:
Activity
Opt
imis
tic
Mos
t Lik
ely
Pess
imis
tic
Mean Standard Dev. Variance
R 3 4 8 4.50 0.833 0.69444A 3 4 5 4.00 0.333 0.11111G 2 3 5 3.17 0.500 0.25000C 3 6 7 5.67 0.667 0.44444J 2 5 6 4.67 0.667 0.44444M 1 3 5 3.00 0.667 0.44444W 1 2 4 2.17 0.500 0.25000
0
5
10
15
20
25
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Mea
n D
urat
ion
Path R-A-
W
Path R-G-J-
W
Path R-C-J-
W
Path R-C-M-
W
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0
5
10
15
20
25
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Mea
n D
urat
ion
Path R-A-W
Path R-G-J-W
Path R-C-J-W Path
R-C-M-W
Critical Path
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0
5
10
15
20
25
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Mea
n D
urat
ion
Critical path here can be considered as that with:- Longest duration (mean)- Greatest variation (stdev)
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0
5
10
15
20
25
30
35
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Mean
Dur
ation
Consider the following hypothetical project paths:
Path P-Q-R
Path P-F-W-R
Path P-H-W-R Path
P-H-M-R
On the basis of mean duration only, Path P-H-W-R is the critical path
On the basis of the variance of durations only, Path P-F-W-R is the critical path
How would you decide the critical path on the basis of both mean duration and variance of durations?
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Better way to identify critical path is using the amount of slack in each
path (see later slides)
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Another Example of Probabilistic Project Scheduling
- Monte Carlo Simulation- Similar activity structure as before, but Start and End activities are dummies (zero durations).
See Excel Sheet Attached
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Benefits of Probabilistic Project Planning
Helps identify likely critical paths in situations where there is great uncertainty
Helps ascertain the likelihood (probability) that overall project duration will fall within a given range
Helps establish a scale of “criticality”among the project activities
Discussed in previous slides
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Benefits of Probabilistic Project Planning
Helps identify likely critical paths in situations where there is great uncertainty
Helps ascertain the likelihood (probability) that overall project duration will fall within a given range
Helps establish a scale of “criticality”among the project activities
Discussed in subsequent slides
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Probabilistic planning …
Is it ever used in real-life project management?
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A Tool for Stochastic Planning: PERT
Program Evaluation and Review Technique (PERT)- Need for PERT arose during the Space Race, in the
late fifties- Developed by Booz-Allen Hamilton for US Navy, and Lockheed Corporation
- Polaris Missile/Submarine Project- R&D Projects- Time Oriented- Probabilistic Times- Assumes that activity durations are Beta distributed
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PERT Parameters
Optimistic duration a
Most Likely duration m
Pessimistic duration b
Expected duration
Standard deviation
Variance
( )6
4212
31_ bmabamd ++
=⎥⎦⎤
⎢⎣⎡ ++=
v s= 2
sb a
=−6
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Steps in PERT Analysis
Obtain a, m and b for each activity
Compute Expected Activity Duration d=te
Compute Variance v=s2
Compute Expected Project Duration D=Te
Compute Project Variance V=S2 as Sum of Critical Path Activity Variance
In Case of Multiple Critical Path Use the One with the Largest Variance
Calculate Probability of Completing the Project
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PERT Example
Activity Predecessor a m b d vA - 1 2 4 2.17 0.25B - 5 6 7 6.00 0.11C - 2 4 5 3.83 0.25D A 1 3 4 2.83 0.25E C 4 5 7 5.17 0.25F A 3 4 5 4.00 0.11G B,D,E 1 2 3 2.00 0.11
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PERT Example
Finding the Standard Deviation of the duration of a given path comprising Activities C, E, and G.
TS V C V E V G
S
e =
= + += + +=
==
11
0 25 0 25 0 11110 6111
0 61110 7817
2 [ ] [ ] [ ]. . ..
..
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PERT AnalysisFinding probability that project duration is less than some valueExample: probability that project ends before 10 months
( )
( )( )
P T T P T
P zT
S
P z
P zP z
d
e
≤ = ≤
= ≤−⎛
⎝⎜⎞⎠⎟
= ≤−⎛
⎝⎜⎞⎠⎟
= ≤ −
= − ≤= −==
( )
..
..
.
10
10
10 110 78171 2793
1 1 27931 0 89970 100310%
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Probability that the project will end before 13 months
( )
( )
P T P z
P z
≤ = ≤−⎛
⎝⎜⎞⎠⎟
= ≤=
1313 110 7817
2 55850 9948
..
.
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Probability that the project will have a duration between 9 and 11.5 months
( ) ( )( ) ( )
( ) ( )( ) ( )[ ]
[ ]
P T T T P T
P T P T
P z P z
P z P z
P z P z
L U≤ ≤ = ≤ ≤
= ≤ − ≤
= ≤−⎛
⎝⎜⎞⎠⎟ − ≤
−⎛⎝⎜
⎞⎠⎟
= ≤ − ≤ −
= ≤ − − ≤
= − −= −=
9 15
115 9115 110 7817
9 110 7817
0 6396 2 5585
0 6396 1 2 55850 7389 1 0 99480 7389 0 00520 7337
.
.. .
. .
. .. .. ..
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PERT Advantages
Includes Variance
Assessment of Probability of Achieving a Goal
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PERT Disadvantages
Data intensive - Very Time Consuming
Validity of Beta Distribution for Activity Durations
Only one Critical Path considered
Assumes independence between activity durations
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PERT Monte Carlo Simulation
Determine the Criticality Index of an Activity
Used 10,000 Simulations, Now from 1000 to 400 Have Been Reported as Giving Good Results
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PERT Monte Carlo Simulation Process
Set the Duration Distribution for Each Activity
Generate Random Duration from Distribution
Determine Critical Path and Duration Using CPM
Record Results
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Example Network
FF44
BB66
GG22 EndEndStartStart
AA2.172.17
CC3.833.83
DD2.832.83
EE5.175.17
AA
FFBB
EE
CC
DD GG EndEndStartStart
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Monte Carlo Simulation Example
Statistics for Example Activities
Activity
OptimisticTime,
a
Most LikelyTime,
m
PessimisticTime,
b
ExpectedVal ue,
d
Stan dardDeviation ,
sA 2 5 8 5 1B 1 3 5 3 0.66C 7 8 9 8 0.33D 4 7 10 7 1E 6 7 8 7 0.33F 2 4 6 4 0.66G 4 5 6 5 0.33
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Monte Carlo Simulation Example
Activity DurationSummary of Simulation Runs for Example Project
RunNumber A B C D E F G
CriticalPath
CompletionTime
1 6.3 2.2 8.8 6.6 7.6 5.7 4.6 A-C-F-G 25.42 2.1 1.8 7.4 8.0 6.6 2.7 4.6 A-D-F-G 17.43 7.8 4.9 8.8 7.0 6.7 5.0 4.9 A-C-F-G 26.54 5.3 2.3 8.9 9.5 6.2 4.8 5.4 A-D-F-G 25.05 4.5 2.6 7.6 7.2 7.2 5.3 5.6 A-C-F-G 23.06 7.1 0.4 7.2 5.8 6.1 2.8 5.2 A-C-F-G 22.37 5.2 4.7 8.9 6.6 7.3 4.6 5.5 A-C-F-G 24.28 6.2 4.4 8.9 4.0 6.7 3.0 4.0 A-C-F-G 22.19 2.7 1.1 7.4 5.9 7.9 2.9 5.9 A-C-F-G 18.910 4.0 3.6 8.3 4.3 7.1 3.1 4.3 A-C-F-G 19.7
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Project Duration Distribution
1717
101099
88
77
66
55
44
33
22
11
001818 1919 2020 2121 2222 2323 2424 2525 2626 2727 2828 2929
Freq
uenc
yFr
eque
ncy
Project LengthProject Length
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Probability Computations from Monte Carlo results
( ) =≤ *TTP Number of Times Project Finished in Less Than or Equal to T* uTotal Number of Replications
( ) =≥ *TTP Number of Times Project Finished in More Than or Equal to T*Total Number of Replications
ETC.
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Criticality Index for an Activity
Definition:Proportion of Runs in which the Activity is in the Critical Path
100%
0%
80%
20%
0%
100% 100%
0%
20%
40%
60%
80%
100%
A B C D E F G
Activity
Criticality
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Criticality Index for a PathDefinition I (“Naïve Definition):Proportion of Runs in which the Activity is in the Critical Path (see Slide # 60)
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Criticality Index for a PathDefinition I (“Naïve Definition):Proportion of Runs in which the Activity is in the Critical Path (see Slide # 60)
Activity DurationSummary of Simulation Runs for Example Project
RunNumber A B C D E F G
CriticalPath
CompletionTime
Summary of Simulation Runs for Example Project
RunNumber A B C D E F G
CriticalPath
CompletionTime
1 6.3 2.2 8.8 6.6 7.6 5.7 4.6 A-C-F-G 25.42 2.1 1.8 7.4 8.0 6.6 2.7 4.6 A-D-F-G 17.41 6.3 2.2 8.8 6.6 7.6 5.7 4.6 A-C-F-G 25.42 2.1 1.8 7.4 8.0 6.6 2.7 4.6 A-D-F-G 17.43 7.8 4.9 8.8 7.0 6.7 5.0 4.9 A-C-F-G 26.54 5.3 2.3 8.9 9.5 6.2 4.8 5.4 A-D-F-G 25.03 7.8 4.9 8.8 7.0 6.7 5.0 4.9 A-C-F-G 26.54 5.3 2.3 8.9 9.5 6.2 4.8 5.4 A-D-F-G 25.05 4.5 2.6 7.6 7.2 7.2 5.3 5.6 A-C-F-G 23.06 7.1 0.4 7.2 5.8 6.1 2.8 5.2 A-C-F-G 22.35 4.5 2.6 7.6 7.2 7.2 5.3 5.6 A-C-F-G 23.06 7.1 0.4 7.2 5.8 6.1 2.8 5.2 A-C-F-G 22.37 5.2 4.7 8.9 6.6 7.3 4.6 5.5 A-C-F-G 24.28 6.2 4.4 8.9 4.0 6.7 3.0 4.0 A-C-F-G 22.17 5.2 4.7 8.9 6.6 7.3 4.6 5.5 A-C-F-G 24.28 6.2 4.4 8.9 4.0 6.7 3.0 4.0 A-C-F-G 22.19 2.7 1.1 7.4 5.9 7.9 2.9 5.9 A-C-F-G 18.910 4.0 3.6 8.3 4.3 7.1 3.1 4.3 A-C-F-G 19.79 2.7 1.1 7.4 5.9 7.9 2.9 5.9 A-C-F-G 18.910 4.0 3.6 8.3 4.3 7.1 3.1 4.3 A-C-F-G 19.7
Frequency as PATH a Critical Path %A-C-F-G 8 80%A-D-F-G 2 20%E-F-G 0 0%
10
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Criticality Index for a Path
Frequency as PATH a Critical Path %A-C-F-G 8 80%A-D-F-G 2 20%E-F-G 0 0%
10
80%
20%
0%0%
20%
40%
60%
80%
100%
A-C-F-G A-D-F-G E-F-G
Path
Freq
uenc
y as
a
Crit
ical
Pat
h
Definition I:Proportion of Runs in which the Activity is in the Critical Path (see Slide # 60)
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Criticality Index for a PathDefinition II:How much slack exists in that path.
Less Slack Higher criticalityMore Slack Lower criticality
= minimum total float
= maximum total float
= total float or slack in current path
Using the index, we can rank project paths from most critical to least critical
( )100%mm inax ααβαλ
−−
=max
minα
axmα
β
Ranges from 0% to 100%
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Criticality Index for a PathDefinition II:
See Example In Excel File
(Path Criticality Slide)
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ReferencesKerzner, Harold, “Project Management: A Systems Approach to Planning, Scheduling, and Controlling”, John Wiley and Sons, New York, 2000,7th Edition.
Meredith, Jack, R., and Samuel J. Mantel, Jr., “Project Management: A Managerial Approach,” John Wiley and Sons, New York, 2000, 4th Edition.
Halpin, Daniel, W., and Woodhead, Ronald, W., “Construction Management’, John Wiley and Sons, New York, 2000, 2nd Edition.
Schtub, Avraham, Bard, Johnson and Globerson, Schlomo. “Project Management: Engineering Technology, and Implementation”, Prentice Hall, New Jersey, 1994.
Callahan, Quackenbush, and Rowings. “Construction Project Scheduling,” McGraw-Hill, New York 1992.Pritsker, A. Alan, C. Elliot Sigal, “Management decision Making,” Prentice Hall, Englewood Cliffs, NJ, 1983.Thamhain, Hans, “Engineering Management,” John Wiley and Sons, New York, 1992.
Software Designed for Project Control Class 1.432, MIT by Roger Evje, Dan Lindholm, S. Rony Mukhopadhyay, & Chun-Yi Wang. Fall Semester Term Project, 1996. Albano (1992), An Axiomatic Approach to Performance-Based Design, Doctoral Thesis, MIT
Eppinger, S.D (1994). A Model-Based Method for Organizing Tasks in Product Development. Research in Engineering Design 6: 1-13, Springer-Verlag London Limited. Eppinger, S.D. (1997) Three Concurrent Engineering Problems in Product Development Seminar, MIT Sloan School of Management.
Moder, Phillips and Davis (1983). Project Management with CPM, PERT and Precedence Diagramming,Third Edition. Van Nostrand Reinhold Company.
Naylor, T (1995). Construction Project Management: Planning and Scheduling. Delmar Publishers, New York.Suh N.P. (1990). The Principle of Design. Oxford University Press, New York. Badiru and Pulat (1995) Comprehensive Project Management, Prentice Hall, New Jersey.Hendrickson and Au (1989) Project Management for Construction, Prentice Hall, New Jersey.Pritsker, Sigal, and Hammesfahr (1994). SLAM II: Network Models for Decision SupportTaylor and Moore (1980). R&D Project Planning with Q-GERT Network Modeling and Simulation, Institute of Management Sciences.
http://www.enr.com/