Samuel Labi and Fred Moavenzadeh

1

Probabilistic Planning Part I
1.040/1.401 Project Management
March 19, 2007

• Samuel Labi and Fred Moavenzadeh
• Department of Civil and Environmental Engineering
• Massachusetts Institute of Technology

2

Recall

3
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

4

Deterministic Systems Planning and Decision-making
Influence of deterministic inputs on the outputs
of engineering systems -- the general picture

FIXED VALUES of the Outputs (system performance
criteria, etc.)
Input variables and their FIXED VALUES

Final evaluation result and decision
Deterministic Analysis of Engineering System
Examples queuing systems, network systems, etc.
Alt. 1
Alt. k
Alt. n
A SINGLE evaluation outcome
Examples costs, time duration, quality, interest
rates, etc.
VALUE OF combined output (index, utility or
value) representing multiple performance measures
of the system
5
But engineering systems are never
deterministic!Why?

• For sample, for Project Planning Systems
• Variations in planning input parameters
• Beyond control of project manager
• Categories 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)

6
Why planning is never deterministic -- II
Influence of stochastic inputs on the outputs of
engineering systems -- the general picture

Input variables and their probability
distributions
Outputs (system performance criteria, etc.) and
their probability distributions
Probability distributions for individual outputs
f(O1)
fX
Variability of final evaluation result and
decision
f(O2)
fY
Probabilistic Analysis of Engineering System
f(OM)
PAlt. i
Examples queuing systems, network systems, etc.
Alt. 1
Alt. 2
Alt. n
fZ
f(OCOMBO)
Discrete probability distribution for evaluation
outcome PAlt. i is the probability that an
alternative turns out to be most desirable or
most critical
Examples costs, time durations, quality,
interest rates, etc.
Probability distribution for combined output
(index, utility or value) representing multiple
performance measures
7


Why planning is never deterministic -- III



Probability distribution? What exactly do you
mean by that?
Input variable and its probability distribution

f
See survey for dinner durations (times) of class
members
8

Probability Distribution for your Dinner Times
9

Probability that a randomly selected student
spends less than T minutes for dinner is less
than T P(T lt T)
10


P(T)
Mean µ Std dev s
T
µ
0
T

Standard Normal Transformation
Probability that a randomly selected student
spends less than T minutes for dinner is less
than T1 P(T lt T)
f(Z)
Mean 0 Std dev 1
Z
Z
µ 0
11


P(T)
Mean µ Std dev s
T
µ
0
T

Standard Normal Transformation
Probability that a randomly selected student
spends less than T1 P(T lt T)
more than T1 P(T gt T)
f(Z)
Mean 0 Std dev 1
Z
Z
µ 0
12


P(T)
Mean µ Std dev s
T
µ
0
T

Standard Normal Transformation
Probability that a randomly selected student
spends less than T P(T lt T)
more than T P(T gt T) between T1 and T2
P(T1 lt T lt T2)
f(Z)
Mean 0 Std dev 1
Z
Z
µ 0
13

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

14

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
15

• 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.

16

• 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.

17


Influence of stochastic inputs -- the specific
picture of project planning


Input variable and its probability distribution
Outputs (system performance criteria, etc.) and
their probability distributions

Probability distributions for the output
Variability of final evaluation result and
decision
Probabilistic Analysis of Project Planning
PAlt. i
fX
f(O1)
Project planning system
Path 1
Path 2
Path n
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
Project activity durations
18

Probabilistic planning An example in everyday
life
KEY
Activity
Perfectly deterministic
Start Time
Activity Duration
Finish Time
SAM Bathing, Breakfast,, Reading, Transport to
MIT, etc.
SAM Waking up and meditating
START 8AM
Duration 5hrs
FINISH 1 PM
START 7AM
Duration 1hr
FINISH 8AM
US Meeting in Class For this Lecture
FINISH 230
START 1PM
Duration1.5hr
YOU Preparing for Classes, etc.
YOU showing up at Other Classes
YOU missing the lecture
START 8AM
Duration 5 hr
FINISH1 PM
START 7 AM
Duration 1 hr
FINISH 8AM
START 1 PM
Duration1.5hr
FINISH 230
19


Probabilistic planning An example in everyday
life
KEY
Activity
Partly deterministic, Partly probabilistic
Earliest Start
Duration of Activity
Earliest Finish
Latest Start
Latest Finish

SAM Bathing, Breakfast,, Reading, Transport to
MIT, etc.
SAM Waking up and meditating
ES 7AM
Duration 5 hrs
EF NOON
ES 6AM
Duration 1 hour
EF 7AM
US Meeting in Class For this Lecture
LS 7AM
LF 8AM
LS 8AM
LF 1 PM
ES 1255
Duration 1.5 hrs
EF 225
LS 1 PM
LF 230
YOU Preparing for Classes, etc.
YOU showing up at Other Classes
YOU missing the lecture
ES 745
Duration 5 hour
EF 1245
ES 645
Duration 1 hr
EF 745
ES 6AM
Duration 1 hour
EF 7AM
LS 7AM
LF 8AM
LS 8AM
LF 1 PM
LS 7AM
LF 8AM
20



Probabilistic planning An example in everyday
life
KEY
Partly deterministic, Partly probabilistic
Activity
Earliest Start
Duration of Activity
Earliest Finish
Latest Start
Latest Finish

SAM Bathing, Breakfast,, Reading, Transport to
MIT, etc.
SAM Waking up and meditating
ES 7AM
Duration 5 hrs
EF NOON
ES 6AM
Duration 1 hour
EF 7AM
US Meeting in Class For this Lecture
LS 7AM
LF 8AM
LS 8AM
LF 1 PM
ES 1255
Duration 1.5 hrs
EF 225
Deterministic Parts
LS 1 PM
LF 230
YOU Preparing for Classes, etc.
YOU showing up at Other Classes
YOU missing the lecture
ES 745
Duration 5 hour
EF 1245
ES 645
Duration 1 hr
EF 745
ES 6AM
Duration 1 hour
EF 7AM
LS 7AM
LF 8AM
LS 8AM
LF 1 PM
LS 7AM
LF 8AM
21




Probabilistic planning An example in everyday
life
KEY
Activity
Partly deterministic, Partly probabilistic
Earliest Start
Duration of Activity
Earliest Finish
Latest Start
Latest Finish

SAM Bathing, Breakfast,, Reading, Transport to
MIT, etc.
SAM Waking up and meditating
ES 7AM
Duration 5 hrs
EF NOON
ES 6AM
Duration 1 hour
EF 7AM
US Meeting in Class For this Lecture
LS 7AM
LF 8AM
LS 8AM
LF 1 PM
ES 1255
Duration 1.5 hrs
EF 225
Deterministic Parts
LS 1 PM
LF 230
YOU Preparing for Classes, etc.
YOU showing up at Other Classes
YOU missing the lecture
ES 745
Duration 5 hour
EF 1245
ES 645
Duration 1 hr
EF 745
ES 6AM
Duration 1 hour
EF 7AM
LS 7AM
LF 8AM
LS 8AM
LF 1 PM
LS 7AM
LF 8AM
22




Probabilistic planning An example in everyday
life
KEY
Activity
Partly deterministic, Partly probabilistic
Earliest Start
Duration of Activity
Earliest Finish
Latest Start
Latest Finish

SAM Bathing, Breakfast,, Reading, Transport to
MIT, etc.
SAM Waking up and meditating
ES 7AM
Duration 5 hrs
EF NOON
ES 6AM
Duration 1 hour
EF 7AM
US Meeting in Class For this Lecture
LS 7AM
LF 8AM
LS 8AM
LF 1 PM
ES 1255
Duration 1.5 hrs
EF 225
Deterministic Parts
LS 1 PM
LF 230
YOU Preparing for Classes, etc.
YOU showing up at Other Classes
YOU missing the lecture
ES 745
Duration 5 hour
EF 1245
ES 645
Duration 1 hr
EF 745
ES 6AM
Duration 1 hour
EF 7AM
LS 7AM
LF 8AM
LS 8AM
LF 1 PM
LS 7AM
LF 8AM
23




Probabilistic planning An example in everyday
life
KEY
Activity
Partly deterministic, Partly probabilistic
Earliest Start
Duration of Activity
Earliest Finish
Latest Start
Latest Finish

SAM Bathing, Breakfast,, Reading, Transport to
MIT, etc.
SAM Waking up and meditating
ES 7AM
Duration 5 hrs
EF NOON
ES 6AM
Duration 1 hour
EF 7AM
US Meeting in Class For this Lecture
LS 7AM
LF 8AM
LS 8AM
LF 1 PM
ES 1255
Duration 1.5 hrs
EF 225
Deterministic Parts
LS 1 PM
LF 230
YOU Preparing for Classes, etc.
YOU showing up at Other Classes
YOU missing the lecture
ES 745
Duration 5 hour
EF 1245
ES 645
Duration 1 hr
EF 745
ES 6AM
Duration 1 hour
EF 7AM
LS 7AM
LF 8AM
LS 8AM
LF 1 PM
LS 7AM
LF 8AM
24




Probabilistic planning An example in everyday
life
KEY
Partly deterministic, Partly probabilistic
Activity
Earliest Start
Duration of Activity
Earliest Finish
Latest Start
Latest Finish

SAM Bathing, Breakfast,, Reading, Transport to
MIT, etc.
SAM Waking up and meditating
ES 7AM
Duration 5 hrs
EF NOON
ES 6AM
Duration 1 hour
EF 7AM
US Meeting in Class For this Lecture
LS 7AM
LF 8AM
LS 8AM
LF 1 PM
ES 1255
Duration 1.5 hrs
EF 225
Deterministic Parts
LS 1 PM
LF 230
YOU Preparing for Classes, etc.
YOU showing up at Other Classes
YOU missing the lecture
ES 745
Duration 5 hour
EF 1245
ES 645
Duration 1 hr
EF 745
ES 6AM
Duration 1 hour
EF 7AM
LS 7AM
LF 8AM
LS 8AM
LF 1 PM
LS 7AM
LF 8AM
25




Probabilistic planning An example in everyday
life
KEY
Partly deterministic, Partly probabilistic
Activity
Earliest Start
Duration of Activity
Earliest Finish
Latest Start
Latest Finish

SAM Bathing, Breakfast,, Reading, Transport to
MIT, etc.
SAM Waking up and meditating
ES 7AM
Duration 5 hrs
EF NOON
ES 6AM
Duration 1 hour
EF 7AM
US Meeting in Class For this Lecture
LS 7AM
LF 8AM
LS 8AM
LF 1 PM
ES 1255
Duration 1.5 hrs
EF 225
Deterministic Parts
LS 1 PM
LF 230
YOU Preparing for Classes, etc.
YOU showing up at Other Classes
YOU missing the lecture
ES 745
Duration 5 hour
EF 1245
ES 645
Duration 1 hr
EF 745
ES 6AM
Duration 1 hour
EF 7AM
LS 7AM
LF 8AM
LS 8AM
LF 1 PM
LS 7AM
LF 8AM
26




Probabilistic planning An example in everyday
life
KEY
Partly deterministic, Partly probabilistic
Activity
Earliest Start
Duration of Activity
Earliest Finish
Latest Start
Latest Finish

SAM Bathing, Breakfast,, Reading, Transport to
MIT, etc.
SAM Waking up and meditating
ES 7AM
Duration 5 hrs
EF NOON
ES 6AM
Duration 1 hour
EF 7AM
US Meeting in Class For this Lecture
LS 7AM
LF 8AM
LS 8AM
LF 1 PM
ES 1255
Duration 1.5 hrs
EF 225
Deterministic Parts
LS 1 PM
LF 230
YOU Preparing for Classes, etc.
YOU showing up at Other Classes
YOU missing the lecture
ES 745
Duration 5 hour
EF 1245
ES 645
Duration 1 hr
EF 745
ES 6AM
Duration 1 hour
EF 7AM
LS 7AM
LF 8AM
LS 8AM
LF 1 PM
LS 7AM
LF 8AM
27



Probabilistic planning An example in everyday
life
KEY
Partly deterministic, Partly probabilistic
Activity
Earliest Start
Duration of Activity
Earliest Finish
Latest Start
Latest Finish

SAM Bathing, Breakfast,, Reading, Transport to
MIT, etc.
SAM Waking up and meditating
ES 7AM
Duration 5 hrs
EF NOON
ES 6AM
Duration 1 hour
EF 7AM
US Meeting in Class For this Lecture
LS 7AM
LF 8AM
LS 8AM
LF 1 PM
ES 1255
Duration 1.5 hrs
EF 225
LS 1 PM
LF 230
YOU Preparing for Classes, etc.
YOU showing up at Other Classes
YOU missing the lecture
ES 745
Duration 5 hour
EF 1245
ES 645
Duration 1 hr
EF 745
ES 6AM
Duration 1 hour
EF 7AM
LS 7AM
LF 8AM
LS 8AM
LF 1 PM
LS 7AM
LF 8AM
28



Probabilistic planning An example in everyday
life
KEY
Partly deterministic, Partly probabilistic
Activity
Earliest Start
Duration of Activity
Earliest Finish
Latest Start
Latest Finish

SAM Bathing, Breakfast,, Reading, Transport to
MIT, etc.
SAM Waking up and meditating
ES 7AM
Duration 5 hrs
EF NOON
ES 6AM
Duration 1 hour
EF 7AM
US Meeting in Class For this Lecture
LS 7AM
LF 8AM
LS 8AM
LF 1 PM
ES 1255
Duration 1.5 hrs
EF 225
LS 1 PM
LF 230
YOU Preparing for Classes, etc.
YOU showing up at Other Classes
YOU missing the lecture
ES 745
Duration 5 hour
EF 1245
ES 645
Duration 1 hr
EF 745
ES 6AM
Duration 1 hour
EF 7AM
LS 7AM
LF 8AM
LS 8AM
LF 1 PM
LS 7AM
LF 8AM
29



Probabilistic planning An example in everyday
life
KEY
Partly deterministic, Partly probabilistic
Activity
Earliest Start
Duration of Activity
Earliest Finish
Latest Start
Latest Finish

SAM Bathing, Breakfast,, Reading, Transport to
MIT, etc.
SAM Waking up and meditating
ES 7AM
Duration 5 hrs
EF NOON
ES 6AM
Duration 1 hour
EF 7AM
US Meeting in Class For this Lecture
LS 7AM
LF 8AM
LS 8AM
LF 1 PM
Probabilistic Parts
ES 1255
Duration 1.5 hrs
EF 225
LS 1 PM
LF 230
YOU Preparing for Classes, etc.
YOU showing up at Other Classes
YOU missing the lecture
ES 745
Duration 5 hour
EF 1245
ES 645
Duration 1 hr
EF 745
ES 6AM
Duration 1 hour
EF 7AM
LS 7AM
LF 8AM
LS 8AM
LF 1 PM
LS 7AM
LF 8AM
30



Probabilistic planning An example in everyday
life
KEY
Fully probabilistic
Activity
Earliest Start
Duration of Activity
Earliest Finish
Latest Start
Latest Finish

SAM Bathing, Breakfast,, Reading, Transport to
MIT, etc.
SAM Waking up and meditating
ES 7AM
Duration
EF NOON
ES 6AM
Duration
EF 7AM
US Meeting in Class For this Lecture
LS 7AM
LF 8AM
LS 8AM
LF 1 PM
µ 5 hr s 0.6hr
µ 1hr s 0.25hr
ES 1255
Duration
EF 225
LS 1 PM
LF 230
µ 1.5 hr s 0.31hr
YOU Preparing for Classes, etc.
YOU showing up at Other Classes
YOU missing the lecture
ES 745
Duration
EF 1245
ES 645
Duration
EF 745
ES 6AM
Duration
EF 7AM
LS 7AM
LF 8AM
LS 8AM
LF 1 PM
LS 7AM
LF 8AM
µ 5 hr s 0.35 hr
µ 1 hr s 0.15 hr
µ 1hr s 0.2 hr
31


Probabilistic planning An example in everyday
life
KEY
Fully probabilistic
Activity
Earliest Start
Duration of Activity
Earliest Finish
Latest Start
Latest Finish

SAM Bathing, Breakfast,, Reading, Transport to
MIT, etc.
SAM Waking up and meditating
ES 7AM
Duration
EF NOON
ES 6AM
Duration
EF 7AM
US Meeting in Class For this Lecture
LS 8AM
LF 1 PM
LS 7AM
LF 8AM
µ 5 hr s 0.6hr
µ 1hr s 0.25hr
ES 1255
Duration
EF 225
LS 1 PM
LF 230
µ 1.5 hr s 0.31hr
YOU Preparing for Classes, etc.
YOU showing up at Other Classes
YOU missing the lecture
ES 755
Duration
EF 1255
ES 655
Duration
EF 755
ES 6AM
Duration
EF 7AM
LS 7AM
LF 8AM
LS 8AM
LF 1 PM
LS 7AM
LF 8AM
µ 5 hr s 0.35 hr
µ 1 hr s 0.15 hr
µ 1hr s 0.2 hr
32
Probabilistic planning An example in Project
Management

KEY
Activity Name
Early Start
Early Finish
Duration
Late Start
Late Finish
Activity A
Month 4
Month 8
4 months
Month 15
Month 11
Activity R
Activity G
Activity J
Activity W
Month 0
Month 4
Month 4
3 months
Month 7
Month 10
5 months
Month 15
4 months
Month 15
2 months
Month 17
Month 4
Month 0
Month 10
Month 7
Month 15
Month 10
Month 17
Month 15
Activity C
Activity M
Month 4
Month 10
Month 10
Month 13
6 months
3 months
Month 10
Month 4
Month 15
Month 12
33


Probabilistic planning An example in Project
Management
KEY
Activity Name
Early Start
Early Finish
Duration (O-M-P)

Late Start
Late Finish
O Optimistic (earliest time) M Most probable
time P Pessimistic (latest time)
Activity A
Month 4
4 Months (3-4-5)
Month 8
Month 15
Month 11
Activity R
Activity G
Activity J
Activity W
Month 0
4 Months (3-4-8)
Month 4
Month 4
3 Months (2-3-5)
Month 7
Month 10
5 Months (2-5-6)
Month 15
Month 15
2 Months (1-2-4)
Month 17
Month 4
Month 0
Month 10
Month 7
Month 15
Month 10
Month 17
Month 15
Activity C
Activity M
Month 4
6 Months (3-6-7)
Month 10
Month 10
3 Months (1-3-5)
Month 13
Month 10
Month 4
Month 15
Month 12
34

Probabilistic planning involving activity
durations An Illustration

Activity A
Month 4
4 Months (3-4-5)
Month 8

Month 15
Month 11
Activity R
Activity G
Activity J
Activity W
Month 0
4 Months (3-4-8)
Month 4
Month 4
3 Months (2-3-5)
Month 7
Month 10
5 Months (2-5-6)
Month 15
Month 15
2 Months (1-2-4)
Month 17
Month 4
Month 0
Month 10
Month 7
Month 15
Month 10
Month 17
Month 15
Activity C
Activity M
Month 4
6 Months (3-6-7)
Month 10
Month 10
3 Months (1-3-5)
Month 13
Month 10
Month 4
Month 15
Month 12
35




Probabilistic planning involving activity
durations An Illustration

Activity A
Month 4
4 Months (3-4-5)
Month 8

Month 15
Month 11

Activity R
Activity G
Activity J
Activity W
Month 0
4 Months (3-4-8)
Month 4
Month 4
3 Months (2-3-5)
Month 7
Month 10
5 Months (2-5-6)
Month 15
Month 15
2 Months (1-2-4)
Month 17
Month 4
Month 0
Month 10
Month 7
Month 15
Month 10
Month 17
Month 15
Activity C
Activity M
Month 4
6 Months (3-6-7)
Month 10
Month 10
3 Months (1-3-5)
Month 13
Month 10
Month 4
Month 15
Month 12

• Lets 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)

36




Probabilistic planning involving activity
durations An Illustration

Activity A
Month 4
4 Months (3-4-5)
Month 8

Month 15
Month 11
Activity R
Activity G
Activity J
Activity W
Month 0
4 Months (3-4-8)
Month 4
Month 4
3 Months (2-3-5)
Month 7
Month 10
5 Months (2-5-6)
Month 15
Month 15
2 Months (1-2-4)
Month 17
Month 4
Month 0
Month 10
Month 7
Month 15
Month 10
Month 17
Month 15
Activity C
Activity M
Month 4
6 Months (3-6-7)
Month 10
Month 10
3 Months (1-3-5)
Month 13
Month 10
Month 4
Month 15
Month 12
37

• Calculate the Expected Duration of each path,
  and Identify the Critical Path on the basis
  of the mean only

38

• Calculate the Expected Duration of each path,
  and Identify the Critical Path on the basis
  of both the mean and the std dev

Path R-C-J-W
Path R-C-M-W
Path R-G-J-W
Path R-A-W
39

Critical Path

Path R-C-J-W
Path R-C-M-W
Path R-G-J-W
Path R-A-W
40

Critical path here can be considered as that
with - Longest duration (mean) - Greatest
variation (stdev)
41

Consider the following hypothetical project paths

Path P-H-W-R
Path P-H-M-R
Path P-Q-R
Path P-F-W-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?
42

• Better way to identify critical path is using the
  amount of slack in each path (see later slides)

43

• 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

44

Benefits of Probabilistic Project Planning
Discussed in previous slides

• 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

45

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
46

• Probabilistic planning
• Is it ever used in real-life project management?

47
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
• - RD Projects
• - Time Oriented
• - Probabilistic Times
• - Assumes that activity durations are Beta
  distributed

48
PERT Parameters

• Optimistic duration a
• Most Likely duration m
• Pessimistic duration b
• Expected duration
• Standard deviation
• Variance

49
Steps in PERT Analysis

• Obtain a, m and b for each activity
• Compute Expected Activity Duration dte
• Compute Variance vs2
• Compute Expected Project Duration DTe
• Compute Project Variance VS2 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

50
PERT Example
51
PERT ExampleFinding the Standard Deviation of
the duration of a given path comprising
Activities C, E, and G.
52
PERT AnalysisFinding probability that project
duration is less than some valueExample
probability that project ends before 10 months
53
Probability that the project will end before 13
months
54
Probability that the project will have a duration
between 9 and 11.5 months
55
PERT Advantages

• Includes Variance
• Assessment of Probability of Achieving a Goal

56
PERT Disadvantages

• Data intensive - Very Time Consuming
• Validity of Beta Distribution for Activity
  Durations
• Only one Critical Path considered
• Assumes independence between activity durations

57
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

58
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

59
Example Network
D 2.83
C
A 2.17
A
G 2
D
F 4
B 6
F
B
E 5.17
G
End
End
Start
Start
C 3.83
E
60
Monte Carlo Simulation Example
61
Monte Carlo Simulation Example
Activity Duration
62
Project Duration Distribution
10
9
8
7
6
Frequency
5
4
3
2
1
0
Project Length
29
17
23
24
25
26
27
28
18
19
20
21
22
63
Probability Computations from Monte Carlo results
Number of Times Project Finished in Less Than or
Equal to T units Total Number of Replications
Number of Times Project Finished in More Than or
Equal to T units Total Number of Replications
ETC.
64
Criticality Index for an Activity
Definition Proportion of Runs in which the
Activity is in the Critical Path

65

Criticality Index for a Path
Definition I (Naïve Definition) Proportion of
Runs in which the Activity is in the Critical
Path (see Slide 60)

66


Criticality Index for a Path
Definition I (Naïve Definition) Proportion of
Runs in which the Activity is in the Critical
Path (see Slide 60)

67


Criticality Index for a Path
Definition I Proportion of Runs in which the
Activity is in the Critical Path (see Slide 60)

68

Criticality Index for a Path
Definition II How much slack exists in that
path. Less Slack Higher criticality More
Slack Lower criticality

Ranges from 0 to 100
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
69

Criticality Index for a Path
Definition II See Example In Excel File (Path
Criticality Slide)

70
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