ESD.36 System Project Management
Instructor(s)
+
-
Probabilistic Scheduling
Prof. Olivier de Weck
Dr. James Lyneis
Lecture 9
+
-
Today’s Agenda
Probabilistic Task Times
PERT (Program Evaluation and
Review Technique)
Monte Carlo Simulation
Signal Flow Graph Method
+
-
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+
-
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.
+
-
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
+
+
-
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
+
+
-
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
com pletion tim e [days]
F re q u e n c y
+
-
Today’s Agenda
Probabilistic Task Times
PERT (Program Evaluation and
Review Technique)
Monte Carlo Simulation
Signal Flow Graph Method
+ -
PERT
PERT invented in
1958 for U.S
Navy Polaris
Project (BAH)
Similar to CPM
Treats task times
probabilistically
Original PERT chart used “activity-on-arc” convention 50 t=3 mo t=3 mo t=3 mo t=4 mo A B C E D F t=1 mo t=2 mo 30 40 10 20
+
-
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
+
-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
+
-
A-M-B Time Estimates
Project Task i 0 5 10 15 10 15 20 25 30 35 40
com pletion tim e [days]
F re q u e n c y
Range of Possible Times
times Time A Time M Time B Assume a Beta-distribution
+
-
Beta-Distribution
All values are enclosed within interval
As classes get finer - arrive at
b
-distribution
Statistical distribution
,
t A B
0,1 x pdf: Beta function:+
+
-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 4 6 A M B T E 2 2 6 t B A T V
+
-
What Can We Do With This?
Compute probability distribution for
project finish
Determine likelihood of making a
specific target date
+
-
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
Time Normal Distribution for F See draft textbook
Chapter 2, second half, for details of calculations.
+
-
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
Normal Distribution for F
T Probability that
project will be done at or before
+ -
Probabilistic Slack
Time SL(i) Normal (Gaussian) Distributionfor 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
+
+
-
Today’s Agenda
Probabilistic Task Times
PERT (Program Evaluation and
Review Technique)
Monte Carlo Simulation
Signal Flow Graph Method
+
-
Project Simulation (from Lecture 5)
Panel Design Die Design Manufacturability Evaluation Prototype Die Production Surface Modeling Panel Data Verification Data Surface Data Analysis Results Surface Data Surface Data Analysis Results Die Geometry Die Geometry Verification Data
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 Graph Model:
Stamping Die Development
+
-Computed Distribution of Die
Development Timing
0 20 40 60 80 100 120 0 5 10 15 20 25 30Project Duration [days]
O cc ur en ce s Simulation: 100 Runs 36.97 Estimate likely completion time
What else can we do with the simulation?
+ -
Process Redesign/Refinement
. 1 2 3 4 5 6 7 8 9 10 Init. Surf. M odeling Final Surf. M odeling 0.75 z 2 z 2 z 2 0.25 z 1 z 0.25 1 z1 0.1 z 3 0.9 z7 0.5 Finish Start 0 .7 5z 2 z 0 . 5 z 3 z 3 Final M fg. Eval. Prelim . M fg. Eval.-1 Prelim . M fg. Eval.-2 Die Design-1 Die Design-2 D ie Design-3 1 Most likely path+
-
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?
+
-
New Project Duration
20 30 40 50 60 70 80 90 100 110 120 0 50 100 150 200 250 300 350 400 450 500
Project Duration [days]
O cc ur en ce s Simulation: 1000 Runs 30.796
+
-Applying Project Simulation to HumLog Distribution Center Project (From Lecture 4)
+
-Simulation Application to HumLog DC
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 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) Simplify project into
+
-
Signal Flow Graph
(with iterations)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 Add uncertain rework loops re-planning loop sequential staffing state in red 1 2 3 4 5 6 7
+
-
HumLog Simulation Results
Shortest Duration: 44 weeks Expected Duration: 56 weeks In some cases
duration can exceed 100 weeks !
+
-
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.
+
-
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,
+
-
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
+
-
Results for project finish …
Individual simulations Class3 Q on Q Probabilistic Work Done 1,000 750 500 250 0 0 15 30 45 60 Time (Month) Confidence bounds Class3 Q on Q Probabilistic 50% 75% 95% 100% Work Done 1,000 750 500 250 0 0 15 30 45 60 Time (Month) Distribution skewed to later finish
+
-
Results for project cost …
Individual simulations
Class3 Q on Q Probabilistic Cumulative Effort Expended
4,000 3,000 2,000 1,000 0 0 15 30 45 60 Time (Month) Confidence bounds Class3 Q on Q Probabilistic 50% 75% 95% 100% Cumulative Effort Expended
4,000 3,000 2,000 1,000 0 0 15 30 45 60 Time (Month)
+
-
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)
+
-
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
+
-
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
MIT OpenCourseWare
http://ocw.mit.edu
ESD.6\VWHP3URMHFW0DQDJHPHQW
Fall 2012
