# Prof. Olivier de Weck

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1 ESD.36 System Project Management + Lecture 9 Probabilistic Scheduling Instructor(s) Prof. Olivier de Weck Dr. James Lyneis October 4, 2012

2 Today s Agenda Probabilistic Task Times PERT (Program Evaluation and Review Technique) Monte Carlo Simulation Signal Flow Graph Method System Dynamics 2

3 Task Representations Tasks as Nodes of a Graph Circles Boxes A,5 Tasks as Arcs of a Graph Tasks are unidirectional arrows ID:A Nodes now represent states of a project KelleyWalker form ES LS EF Dur:5 LF broken A, 5 fixed 3

4 Task Times Detail Task i ES(i) LS(i) Duration t(i) Total Slack TS(i) FS(i) Free Slack EF(i) Duration t(i) ES(j) LF(i) j is the immediate successor of i with the smallest ES j>i Free Slack (FS) is the amount a job can be delayed without delaying the Early Start (ES) of any other job. FS<=TS always 4

5 How long does a task take? Conduct a small inclass experiment Fold MIT paper airplane Have sheet & paper clip ready in front of you Paper airplane type will be announced, e.g. A1B1C1D1 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 5

6 MIT Paper Airplane Courtesy of Steven D. Eppinger. Used with permission. Credit: Steve Eppinger 6

7 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 7

8 MIT Class Results (2008) Beta Distribution parameters: a=2.27, b=5.26 8

9 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 15 Frequency completion time [days] 9

10 Today s Agenda Probabilistic Task Times PERT (Program Evaluation and Review Technique) Monte Carlo Simulation Signal Flow Graph Method System Dynamics 10

11 PERT PERT invented in 1958 for U.S Navy Polaris Project (BAH) Similar to CPM Treats task times probabilistically 10 B t=4 mo A t=3 mo t=1 mo t=3 mo Original PERT chart used activityonarc convention C E t=2 mo D F t=3 mo Image by MIT OpenCourseWare. 11

12 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 12

13 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 13

14 AMB Time Estimates Project Task i Range of Possible Times 15 Frequency times completion time [days] Time A Time M Time B Assume a Betadistribution 14

15 BetaDistribution All values are enclosed within interval As classes get finer arrive at bdistribution Statistical distribution pdf: t A, B Beta function: x 0,1 15

16 MIT Class Results (2008) Beta Distribution parameters: a=2.27, b=

17 Expected Time & Variance Estimated Based on A, M & B Mean expected Time (TE) TE Time Variance (TV) 2 B A TV t 6 Early Finish (EF) and Late Finish (LF) computed as for CPM with TE Set T=F for the end of the project A 4 M B Example: A=3 weeks, B=7 weeks, M=5 weeks > then TE=5 weeks

18 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 18

19 Probability Distribution for Finish Date PERT treats task times as probabilistic Individual task durations are bdistributed 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 19

20 Probability of meeting target? Many Projects have target completion dates, T Interplanetary mission launch windows 34 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 20

21 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 21

22 Experiences with PERT? 22

23 Today s Agenda Probabilistic Task Times PERT (Program Evaluation and Review Technique) Monte Carlo Simulation Signal Flow Graph Method System Dynamics 23

24 Project Simulation (from Lecture 5) Panel Design Panel Data Die Design Image removed due to copyright restrictions. Die Geometry Manufacturability Evaluation Verification Data Analysis Results Analysis Results Surface Data Prototype Die Production Die Die Geometry Verification Data Surface Data Surface Data Surface Modeling 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 ,

25 Signal Flow Graph Model: Stamping Die Development 25

26 Computed Distribution of Die Development Timing Simulation: 100 Runs Estimate likely completion time 20 Occurences What else can we do with the simulation? Project Duration [days] 26

27 Process Redesign/Refinement Die Prelim. M fg. Die Prelim. M fg. D ie Start 0.75 z 2 z 2 z 1 z 3 Design1 z 3 Eval.1 Design2 Eval.2 Design z z 2 Init. Surf. M odeling z z 1 1 z Most likely path 0.9 z Final Surf. M odeling z Final M fg. Eval. Finish 27

28 Whatif 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? 28

29 New Project Duration 500 Simulation: 1000 Runs Occurences New distribution of Finish time Project Duration [days] 29

30 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 30

31 Start Simulation Application to HumLog DC APlanning (14) Simplify project into 7 major steps 5 8 DStaffing (7) FCommissioning (5) BContracting (9) CConstruction (11) EInstallation (5) 31

32 Signal Flow Graph (with iterations) Add uncertain rework loops replanning loop Construction 0.2z 3 Installation 0.25z z 11 Start A B z 9 4 z C 11 5 z z D 5 7 fix construction errors loop z D E Planning Contracting 0.5z 7 Staffing End Commissioning sequential staffing state in red 32

33 HumLog Simulation Results Expected Duration: 56 weeks Shortest Duration: 44 weeks In some cases duration can exceed 100 weeks! 33

34 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. 34

35 System Dynamics Simulation Can carry out MonteCarlo 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 35

36 Example Model without project control from last class and HW#3 Probabilistic Simulations: Normal Productivity uniform distribution ( ) Normal Fraction Correct and Complete uniform distribution ( ) 36

37 Results for project finish Individual simulations Confidence bounds Class3 Q on Q Probabilistic Work Done 1,000 Class3 Q on Q Probabilistic 50% 75% 95% 100% Work Done 1, Time (Month) Time (Month) Distribution skewed to later finish 37

38 Results for project cost Individual simulations Confidence bounds Class3 Q on Q Probabilistic Cumulative Effort Expended 4,000 Class3 Q on Q Probabilistic 50% 75% 95% 100% Cumulative Effort Expended 4,000 3,000 3,000 2,000 2,000 1,000 1, Time (Month) Time (Month) 38

39 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 Using fitconstrained 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) 39

40 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 40

41 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 , For this lecture, please read: Chapter Network Planning Techniques Second Half of Chapter 41

42 MIT OpenCourseWare ESD. Fall 2012 For inforation about citing these materials or our Terms of Use, visit:

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