ESD.36 System Project Management
Instructor(s)
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- Probabilistic Scheduling
Prof. Olivier de Weck
Dr. James Lyneis
Lecture 9
October 4, 2012
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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
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Task Representations
Tasks as Nodes of a Graph Circles
Boxes
Tasks as Arcs of a Graph Tasks are uni-directional arrows
Nodes now represent “states” of a project
Kelley-Walker form
ID:A
ES EF
Dur:5
LS LF
A,5
broken A, 5
fixed
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Task Times Detail - Task i
ES(i) EF(i)
Duration t(i)
LS(i) LF(i)
Duration t(i) Total Slack TS(i)
ES(j) j>i
j is the immediate successor of i with the smallest ES
FS(i)
Free Slack
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?
Conduct a small in-class experiment
Fold MIT paper airplane 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 Submit your task time online , e.g. 120 sec
We will build a histogram and show results
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- MIT Paper Airplane
Credit: Steve Eppinger 6
Courtesy of Steven D. Eppinger. Used with permission.
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- Concept Question 1
How long did it take you to complete your paper airplane (round up or down)? 25 sec
50 sec
75 sec
100 sec
125 sec
150 sec
175 sec
200 sec
225 sec
> 225 sec
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Discussion Point 1
Job task durations are stochastic in reality
Actual duration affected by
Individual skills
Learning curves … what else?
Why is the distribution not symmetric (Gaussian)?
Project Task i
0
5
10
15
10 15 20 25 30 35 40
completion time [days]
Fre
qu
en
cy
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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
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- PERT
PERT invented in 1958 for U.S Navy Polaris Project (BAH)
Similar to CPM
Treats task times probabilistically
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Original PERT chart used “activity-on-arc” convention
50t=3 mo
t=3 mo
t=3 mo
t=4 mo
A
B C
E
D Ft=1 mo
t=2 mo30
40
10
20
Image by MIT OpenCourseWare.
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CPM vs PERT
Difference how “task duration” is treated:
CPM assumes time estimates are deterministic
Obtain task duration from previous projects
Suitable for “implementation”-type projects
PERT treats durations as probabilistic
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
A - optimistic time estimate
minimum time in which the task could be completed
everything has to go right
M - most likely task duration
task duration under “normal” working conditions
most frequent task duration based on past experience
B - pessimistic time estimate
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
0
5
10
15
10 15 20 25 30 35 40
completion time [days]
Fre
qu
en
cy
Range of Possible Times
times
Time A
Time M
Time B
Assume a Beta-distribution
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Beta-Distribution
All values are enclosed within interval
As classes get finer - arrive at b-distribution
Statistical distribution
,t A B
0,1x
pdf:
Beta function:
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Expected Time & Variance Estimated Based on A, M & B
Mean expected Time (TE)
Time Variance (TV)
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
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6
A M BTE
2
2
6t
B ATV
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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
Individual task durations are b-distributed
Simplify by estimating A, B and C times
Sums of multiple tasks are normally distributed
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Time
Normal Distribution for F
E[EF(n)]=F
See draft textbook Chapter 2, second half, for details of calculations.
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Probability of meeting target ?
Many Projects have target completion dates, T 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
Time E[EF(n)]=F
Normal Distribution for F
T Probability that project will be done at or before T
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Probabilistic Slack
Time SL(i)
Normal (Gaussian) Distribution for Slack SL(i)
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
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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
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- Project Simulation (from Lecture 5)
PanelDesign
DieDesign
ManufacturabilityEvaluation
PrototypeDie
ProductionDie
SurfaceModeling
PanelData
VerificationData
SurfaceData
Analysis Results
SurfaceData
SurfaceData
AnalysisResults
DieGeometry
DieGeometry
VerificationData
Highly Iterative Process • how often is each task carried out ? • how long to complete?
Image removed due to copyright restrictions.
Die Design and Manufacturing
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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Computed Distribution of Die Development Timing
0 20 40 60 80 100 1200
5
10
15
20
25
30
Project Duration [days]
Occ
uren
ces
Simulation: 100 Runs
36.97
Estimate likely completion time
What else can we do with the simulation?
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Process Redesign/Refinement
. 1 432 5 6
7
8 9 10
Init. Surf.M odeling
Final Surf.M odeling
0.75 z 2z 2
z 20.25
z 1
z0.25 1
z1
0.1
z 30.9
z7Finish
0.5
Start
0 .7 5z2
z0 . 5
z 3z 3
Final M fg.Eval.
Prelim . M fg.Eval.-1
Prelim . M fg.Eval.-2
D ieDesign-1
DieDesign-2
D ieDesign-3
1
Most likely path
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What-if analysis
Spend more time on die design (1):
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
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):
Increase time for that task from 7 to 10 days
Increase likelihood of Finishing from 0.5 to 0.75
New E[F] = 30.8 days
Why is this happening?
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New Project Duration
20 30 40 50 60 70 80 90 100 110 1200
50
100
150
200
250
300
350
400
450
500
Project Duration [days]
Occ
uren
ces
Simulation: 1000 Runs
30.796
New distribution of Finish time
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Applying Project Simulation to HumLog Distribution Center Project (From Lecture 4)
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Simulation Application to HumLog DC
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1 2
3
4
5
6 7 8 9 10 11
12
17 15 16
18
20 21 22
13
19
14 Start
0 4
2
2
8
4 2 1 2 2 3
4
6
2
2 2 1
3 1
4
1
0 4
4 6
4 6 6 14
14 18
44 44
23 29
34 36
32 34 28 32
25 28 20 21 18 20 21 23 23 25
34 36 34 35
36 39
39 43 43 44
29 30
A-Planning (14)
B-Contracting (9) C-Construction (11)
D-Staffing (7)
E-Installation (5)
F-Commissioning (5)
Simplify project into 7 major steps
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- Signal Flow Graph (with iterations)
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A
B
C D
D E
Start
End
z14
Planning
Contracting
z9
0.5z7
0.25z7
Staffing Construction
0.25z11
z5
Installation
0.2z3
0.8z5
z11
Commissioning
Add uncertain rework loops
re-planning loop
fix construction errors loop
sequential staffing
state in red 1
2
3
4 5
6 7
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- HumLog Simulation Results
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Shortest Duration: 44 weeks
Expected Duration: 56 weeks
In some cases duration can exceed 100 weeks !
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- Concept Question 2
What is your opinion of these “long tails” in project schedule distributions?
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
Can carry out Monte-Carlo Simulations with System Dynamics
Vary key model parameters such as fraction correct and complete, productivity, rework discovery fraction, number of staff etc…
Assume distributions for these parameters
Obtain insights into potential distribution of
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
Model without project control from last class and HW#3
Probabilistic Simulations:
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 …
Individual simulations
Class3 Q on Q ProbabilisticWork Done1,000
750
500
250
00 15 30 45 60
Time (Month)
Confidence bounds
Class3 Q on Q Probabilistic50% 75% 95% 100%Work Done1,000
750
500
250
00 15 30 45 60
Time (Month)
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Distribution skewed to later finish
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- Results for project cost …
Individual simulations
Class3 Q on Q ProbabilisticCumulative Effort Expended4,000
3,000
2,000
1,000
00 15 30 45 60
Time (Month)
Confidence bounds
Class3 Q on Q Probabilistic50% 75% 95% 100%Cumulative Effort Expended4,000
3,000
2,000
1,000
00 15 30 45 60
Time (Month)
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- Monte Carlo and SD
Use in planning and adaptation:
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)
“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
Account for task duration uncertainties
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
Project Management Book
Systems Engineering: Principles, Methods and Tools for System Design and Management
de Weck, Crawley and Haberfellner (eds.), Springer 2010
(draft) textbook to support SDM core classes at MIT
About 200 pages
Selected Article 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:
Chapter Network Planning Techniques Second Half of Chapter
MIT OpenCourseWarehttp://ocw.mit.edu
ESD.36 System Project ManagementFall 2012
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