ESD.36 System Project Management
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Lecture 9
Probabilistic Scheduling
Instructor(s)
Prof. Olivier de Weck
Dr. James Lyneis
October 4, 2012
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Today’s Agenda
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Probabilistic Task Times
PERT (Program Evaluation and
Review Technique)
Monte Carlo Simulation
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Signal Flow Graph Method
System Dynamics
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Task Representations
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Tasks as Nodes of a Graph
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Circles
Boxes
A,5
Tasks as Arcs of a Graph
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ES
ID:A
LS
EF
Dur:5
LF
Tasks are uni-directional arrows
Nodes now represent “states” of a project
Kelley-Walker form
broken
A, 5
fixed
3
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Task Times Detail - Task i
LS(i)
ES(i)
Duration t(i)
Total Slack
TS(i)
FS(i)
LF(i)
EF(i)
Duration t(i)
j is the immediate
successor of i with
the smallest ES
ES(j)
j>i
Free Slack
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Free Slack (FS) is the amount a job can be
delayed without delaying the Early Start (ES)
of any other job.
FS<=TS always
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How long does a task take?
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Conduct a small in-class experiment
Fold MIT paper airplane
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Have sheet & paper clip ready in front of you
Paper airplane type will be announced, e.g.
A1-B1-C1-D1
Build plane, focus on quality rather than speed
Note the completion time in seconds +/- 5 [sec]
Plot results for class and discuss
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Submit your task time online , e.g. 120 sec
We will build a histogram and show results
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MIT Paper Airplane
Courtesy of Steven D. Eppinger. Used with permission.
Credit: Steve Eppinger
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Concept Question 1
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How long did it take you to complete your
paper airplane (round up or down)?
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25 sec
50 sec
75 sec
100 sec
125 sec
150 sec
175 sec
200 sec
225 sec
> 225 sec
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MIT Class Results (2008)
Beta Distribution parameters: a=2.27, b=5.26
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Discussion Point 1
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Job task durations are stochastic in reality
Actual duration affected by
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Individual skills
Learning curves … what else?
Why is the distribution not symmetric (Gaussian)?
Project Task i
15
Frequency
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10
5
0
10
15
20
25
30
35
40
com pletion tim e [days]
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Today’s Agenda



Probabilistic Task Times
PERT (Program Evaluation and
Review Technique)
Monte Carlo Simulation


Signal Flow Graph Method
System Dynamics
10
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PERT
PERT invented in
1958 for U.S
Navy Polaris
Project (BAH)
Similar to CPM
Treats task times
probabilistically
20
10
A
t=3 mo
t=
3
o
m
4
B
t=
C
30
m
o
E
t=2 mo
t=1 mo D
F
50
t=3 mo
40
Image by MIT OpenCourseWare.
Original PERT chart
used “activity-on-arc”
convention
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CPM vs PERT
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Difference how “task duration” is treated:
 CPM assumes time estimates are deterministic
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Obtain task duration from previous projects
Suitable for “implementation”-type projects
PERT treats durations as probabilistic
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PERT = CPM + probabilistic task times
Better for “uncertain” and new projects
Limited previous data to estimate time durations
Captures schedule (and implicitly some cost) risk
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PERT -- Task time durations are
treated as uncertain
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A - optimistic time estimate
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M - most likely task duration
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minimum time in which the task could be completed
everything has to go right
task duration under “normal” working conditions
most frequent task duration based on past experience
B - pessimistic time estimate
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time required under particularly “bad” circumstances
most difficult to estimate, includes unexpected delays
should be exceeded no more than 1% of the time
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A-M-B Time Estimates
Project Task i
Range of Possible Times
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Frequency
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10
5
0
10
15
20
25
30
35
40
times
com pletion tim e [days]
Time A
Time M
Time B
Assume a
Beta-distribution
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Beta-Distribution
All values are enclosed within interval t   A , B 
As classes get finer - arrive at b-distribution
Statistical distribution
pdf:
Beta function:
x   0,1 
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MIT Class Results (2008)
Beta Distribution parameters: a=2.27, b=5.26
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Expected Time & Variance
Estimated Based on A, M & B
Mean expected Time (TE)
TE 
A  4M  B
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2
BA
TV    

6
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Early Finish (EF) and Late Finish (LF) computed as
for CPM with TE
Set T=F for the end of the project
Example: A=3 weeks, B=7 weeks, M=5 weeks -->
then TE=5 weeks
Time Variance (TV)
2
t
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What Can We Do With This?
Compute probability distribution for
project finish
Determine likelihood of making a
specific target date
Identify paths for buffers and reserves
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Probability Distribution for Finish Date
PERT treats task times as probabilistic
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Individual task durations are b-distributed
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Simplify by estimating A, B and C times
Sums of multiple tasks are normally distributed
Normal Distribution for F
See draft textbook
Chapter 2, second half, for
details of calculations.
Time
E[EF(n)]=F
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Many Projects have target completion dates, T
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Probability of meeting target ?
Interplanetary mission launch windows 3-4 days
Contractual delivery dates involving financial incentives or
penalties
Timed product releases (e.g. Holiday season)
Finish construction projects before winter starts
Analyze expected Finish F relative to T
Normal Distribution for F
Probability that
project will be
done at or before
T
T
E[EF(n)]=F
Time
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Probabilistic Slack
Target date for a tasks is not met when
SL(i)<0, i.e. negative slack occurs
Put buffers in paths with high probability of
negative slack
Normal (Gaussian)
Distribution
for Slack SL(i)
SL(i)
Time
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Experiences with PERT?
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Today’s Agenda



Probabilistic Task Times
PERT (Program Evaluation and
Review Technique)
Monte Carlo Simulation
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
Signal Flow Graph Method
System Dynamics
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Project Simulation (from Lecture 5)
Panel
Design
Panel
Data
Die
Design
Die
Geometry
Manufacturability
Evaluation
Verification
Data
Analysis
Results
Die
Geometry
Analysis Results
Surface
Data
Prototype
Die
Verification
Data
Production
Die
Image removed due to copyright restrictions.
Surface
Data
Surface
Modeling
Surface
Data
Die Design and Manufacturing
Highly Iterative Process
• how often is each task carried out ?
• how long to complete?
Steven D. Eppinger, Murthy V. Nukala, and Daniel E.
Whitney. "Generalised Models of Design Iteration Using
Signal Flow Graphs", Research in Engineering Design. vol.
9, no. 2, pp. 112-123, 1997.
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Signal Flow Graph Model:
Stamping Die Development
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Computed Distribution of Die
Development Timing
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Simulation: 100 Runs
30
25
Estimate likely
completion time
36.97
Occurences
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What else can
we do with the
simulation?
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10
5
0
0
20
40
60
80
Project Duration [days]
100
120
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Process Redesign/Refinement
1
z3
2
Prelim . M fg.
D ie
Prelim . M fg.
D ie
1
Eval.-1
D esign-2
Eval.-2
z D esign-3
2
2
z3
0.75 z
z
3
5
4
6
1
0.25 z
2
2
z
0.25 z
0 .7 5
Init. S urf.
M odeling
Most likely path
7
0 .1
1
z
0.9 z 3
Final Surf.
M odeling
z1
Start
D ie
D esign-1
0
.5
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8
z
7
9
0.5
10
Finish
Final M fg.
Eval.
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What-if analysis
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Spend more time on die design (1):
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Increase time spent on initial die design (1) from 3 to 6 days
Increase likelihood of going to Initial Surface Modeling (7) from
0.25 to 0.75
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Is this worthwhile doing?
Original E[F]=37 days
New E[F]= 37 days – no real effect ! Why?
Spend more time on final surface modeling (8):
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Increase time for that task from 7 to 10 days
Increase likelihood of Finishing from 0.5 to 0.75
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New E[F] = 30.8 days
Why is this happening?
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New Project Duration
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Simulation: 1000 Runs
500
450
400
Occurences
350
New distribution of Finish time
300
250
30.796
200
150
100
50
0
20
30
40
50
60
70
80
Project Duration [days]
90
100
110
120
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Applying Project Simulation to HumLog Distribution
Center Project (From Lecture 4)
© Source unknown. All rights reserved. This content is excluded from our Creative
Commons license. For more information, see http://ocw.mit.edu/fairuse.
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Simulation Application to HumLog DC
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B-Contracting (9)
4
0
4
1
2
0
4
Start
3
2
6
6
4
6
C-Construction (11)
14
5
14 18 18 20
8
4
6
7
4
2
20 21
8
21 23
23 25
9
1
10
2
2
32 34
2
2
23 29
A-Planning (14)
34 36
6
2
19
12
34 35
16
17
36 39
2
1
18
1
3
44 44
43 44
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21
F-Commissioning (5)
3
28 32
34 36
15
29 30
Simplify project into
7 major steps
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4
13
D-Staffing (7)
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25 28
1
39 43
20
E-Installation
(5)
4
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Signal Flow Graph
1 Start
Add uncertain
rework loops
state in red
z14
2 Planning
A
re-planning
0.25z7
loop
z9
3 Contracting
B
0.25z11
4
Construction
0.2z3
Installation
(with iterations)
6
0.5z7
5
z11
C
z5
D
fix construction
errors loop
0.8z5
D
7
E
sequential
staffing
Staffing
End
Commissioning
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HumLog Simulation Results
Expected
Duration:
56 weeks
Shortest
Duration:
44 weeks
In some cases
duration can exceed
100 weeks !
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Concept Question 2
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What is your opinion of these “long tails” in
project schedule distributions?
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I don’t think they are real. This is a simulation
artifact.
Yes, they exist. I have experienced this.
This is only relevant for projects that deal with
very new products or extreme environments.
I’m not sure.
I don’t care.
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System Dynamics Simulation
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Can carry out Monte-Carlo Simulations with
System Dynamics
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Obtain insights into potential distribution of
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Vary key model parameters such as fraction
correct and complete, productivity, rework
discovery fraction, number of staff etc…
Assume distributions for these parameters
Time to completion, required staffing levels, error
rates, project cost …
Improve planning and adaptation of projects,
and confidence in “claim” numbers
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Example
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Model without project control from last
class and HW#3
Probabilistic Simulations:
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Normal Productivity uniform distribution
(0.9 – 1.1)
Normal Fraction Correct and Complete
uniform distribution (0.75 – 0.95)
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Results for project finish …
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Individual simulations
Confidence bounds
Class3 Q on Q Probabilistic
50%
75%
95%
Work Done
1,000
Class3 Q on Q Probabilistic
Work Done
1,000
750
750
500
500
250
0
100%
250
0
15
30
Time (Month)
45
60
0
0
15
30
Time (Month)
45
60
Distribution skewed to
later finish
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Results for project cost …
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Individual simulations
Confidence bounds
Class3 Q on Q Probabilistic
50%
75%
95%
Cumulative Effort Expended
4,000
Class3 Q on Q Probabilistic
Cumulative Effort Expended
4,000
3,000
3,000
2,000
2,000
1,000
0
100%
1,000
0
15
30
Time (Month)
45
60
0
0
15
30
Time (Month)
45
60
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Monte Carlo and SD
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Use in planning and adaptation:
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Size and timing (fraction compete) of buffers
Timing of project milestones
Important use in specifying confidence bands
in a “claim” situation for distribution of cost
overrun due to client (owner)
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“Fit” constrained – only simulations which fit the
data can be accepted
Graham AK, Choi CY, Mullen TW. 2002. Using fit-constrained Monte Carlo trials to quantify confidence in
simulation model outcomes. Proceedings of the 35th Annual Hawaii Conference on Systems Sciences. IEEE:
Los Alamitos, Calif. (Available on request)
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Usefulness of PERT and Simulation
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Account for task duration uncertainties
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Optimistic Schedule
Expected Schedule
Pessimistic Schedule
Helps set time reserves (buffers)
Compute probability of meeting target dates
when talking to management, donors
Identify and carefully manage critical parts of
the schedule
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Reading List
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Project Management Book
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Systems Engineering: Principles, Methods and Tools for System
Design and Management
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Selected Article
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de Weck, Crawley and Haberfellner (eds.), Springer 2010
(draft) textbook to support SDM core classes at MIT
About 200 pages
Steven D. Eppinger, Murthy V. Nukala, and Daniel E. Whitney. "Generalised
Models of Design Iteration Using Signal Flow Graphs", Research in
Engineering Design. vol. 9, no. 2, pp. 112-123, 1997.
For this lecture, please read:
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Chapter Network Planning Techniques
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Second Half of Chapter
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MIT OpenCourseWare
http://ocw.mit.edu
ESD. 6\VWHP3URMHFW0DQDJHPHQW
Fall 2012
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