Unformatted text preview: Probabilistic Planning 2
Nathaniel Osgood 3272004 Topics
PERT (Cont'd)
Review Merge node bias PNet refinement Monte Carlo Simulation approaches
General Demo Process Interaction Activity Scanning PERT Basics
Expresses uncertainty in activity duration
Beta distribution assumed for activities Assume normally distributed project duration
Project Duration Tends to be Normally Distributed (approx. sum of random variables) Assumes Independent Activity Durations  Not Always Satisfied Stochastic Approach
Optimistic Most Likely (mode not mean) Pessimistic Expected Duration Variance Standard Deviation
d=
_ a m b
1 1 a + 4m + b 2m + (a + b ) = 3 2 6 ba s= 6 v=s 2 Recall: Steps in PERT Analysis
For each activity k
Obtain ak, mk (mode) and bk Compute expected activity duration (mean) dk=te Compute activity variance vk=s2 Compute expected project duration D=Te using standard CPM algorithm Compute Project Variance V=S2 as sum of critical path activity variance (this assumes independence!)
In case of multiple critical paths use the one with the largest variance Compute probability complete project by time t
Assuming project duration normally distributed PERT Example
Calculated
Activity Predecessor A B C D A E C F A G B,D,E a 1 5 2 1 4 3 1 m 2 6 4 3 5 4 2 b 4 7 5 4 7 5 3 d 2.17 6.00 3.83 2.83 5.17 4.00 2.00 v 0.25 0.11 0.25 0.25 0.25 0.11 0.11 Activity on Node Example Forward Pass Backward Pass PERT ExampleStandard Deviation Te = 11 S = V [C ] + V [ E ] + V [G ]
2 = 0.25 + 0.25 + 01111 . = 0.6111 S = 0.6111 = 0.7817 PERT AnalysisProbability of Ending before 10 (Critical Path Only)
P(T Td ) = P(T 10) 10  Te = P z S 10  11 = P z 0.7817 = P( z 12793) . = 1  P( z 12793) . = 1  0.8997 = 01003 . = 10% PERT Analysis  Probability of Ending before 13 (Critical Path Only) 13  11 P( T 13) = P z 07817 . = P( z 25585) . = 09948 . PERT Analysis  Probability of Ending between 9 and 11.5(CP Only)
P (TL T TU ) = P ( 9 < T 11.5 ) = P (T 11.5 )  P (T 9 ) 11.5  11 9  11 = P z  P z 0.7817 0.7817 = P ( z 0.6396 )  P ( z 2.5585 ) = P ( z 0.6396 )  1  P ( z 2.5585 ) = 0.7389  [1  0.9948] = 0.7389  0.0052 = 0.7337 Topics
PERT (Cont'd)
Review Merge node bias PNet refinement Monte Carlo Simulation approaches
General Demo Process Interaction Activity Scanning Merge Node Bias
Misleading to consider only variance from single predecessor for each node on critical path
Early start of node depends on maximum of finish (or start) times of predecessors including noncritical! Basically ES = RV that is max of (noniid) RVs Effect stronger if have
More predecessors Predecessors with almost equal timing Less dependency among predecessors Consequence: Unrealistic optimism with respect to expected completion times, but especially variance Example Merge Node
ES(C)=Max(EF(A), EF(B)) =10.777 =1.55 C A Late Finish: N(10,1) B Late Finish: N(9,3) Sample Problem Derived Parameters
MEAN AND STANDARD DEVIATION OF THE CRITICAL AND NEAR CRITICAL PATHS FOR NETWORK
TIME ESTIMATES PATH 0378
(Critical Path) PATH 03458 ACTIVITY 03 37 78 34 45 58 a
1 6 3.5 1 2 2 m
2 8 5 4 4 3 b
3 10 6.5 13 6 4 MEAN 2 8 5 VARIANCE 0.39 1.56 0.88 MEAN 2 VARIANCE 0.39 5 4 3 14.0 14.06 1.56 0.39 16.40 4.05 15.0 2.83 1.68 TOTALS* STANDARD DEVIATION   * The mean and variance of the duration of a path is merely the sum of the means and variances of the
activities along the path in question; the standard deviation of the path duration is then obtained as the square root of its variance. Impact of Multiple Paths
Log Scale (maximum of times of both paths) Nave Approach
Consider variance from all paths entering a merge node Assume Probability EF(i)<T = jPaths To(i) P(EF(j)<T) Recall PERT Analysis  ADG Path Te = 7 S = V [ A] + V [ D] + V [G ] = 0.25 + 0.25 + 011 .
2 = 0.6111 S = 0.6111 = 0.7817 PERT Analysis  ADG Path Probability of Ending before 10 10  7 P( T 10) = P z 0.7817 = P( z 38378) . = 0.9999 PERT Analysis  BG Path Te = 8 S = V [ B] + V [G ]
2 = 01111 + 01111 . . = 0.2222 S = 0.2222 = 0.4714 PERT Analysis  BG Path Probability of Ending before 10 10  8 P( T 10) = P z 0.4714 = P( z 4.2429) = 0.9999 PERT Analysis  ADG , BG and CEG Paths Probability of Ending before 10 P ( T 10) = P(TCEG 10) P(TADG 10) P(TBG 10) c = (01003)(09999)(09999) . . . = 01003 . = 10% PERT (cont):
For the G finish within 10 days, all 3 paths must finish in 10 days or less (i.e. ADG and CEG and BG) Calculated as:
P(T10)=P(ADG10)*P(CEG10)*P(BG10) What is wrong with this equation? The equation assumes the path durations are independent! This cannot be if there are shared activities between the paths. Example of Multiple Paths Dependent and Independent Activities with duration 2 have =.707 Activities with duration 4 have =1.414 PERT (cont):
A Solution: Use either
PNet Monte Carlo simulation PNet
Aims at addressing merge node bias Basically works by
Enumerate all paths P s.t. Dur(P)> Dur(crit path) Rank paths by decreasing duration (by decreasing naivelyestimated variance for ties) Compute linear correlation coefficient between paths Enter paths, eliminating any path whose correlation coefficient with a previouslyentered path is > .5 P(T a ) = # remaining paths i =1 P ( pi T ) Validity of Beta distribution for activity durations Validity of central limit theorem for project duration
Activity durations are not independent! PERT Disadvantages Take into consideration only critical path
Not just sum of random variables  have max. at joins Leads to overoptimism & underestimation of duration Multiple time estimates required to calibrate
Can be time consuming Topics
PERT (Cont'd)
Review Merge node bias PNet refinement Monte Carlo Simulation approaches
General Demo Process Interaction Activity Scanning Monte Carlo Simulation Characteristics
Replaces analytic solution with raw computing power
Avoids need to simplify to get analytic solution No need to assume functional form of activity/project distributions Used by Van Slyke (1963) Aimed at solving the merge bias problem in PERT Allows determining the criticality index of an activity (Proportion of runs in which the activity was in the critical path) Hundreds to thousands of simulations needed Monte Carlo Simulation Process
Set the duration distribution for each activity
No functional form of distribution assumed Could be joint distribution for multiple activities Iterate: for each "trial" ("realization")
Sample random duration from each distributions Find critical path & durations with standard CPM
Record these results Report recorded results
Duration distribution Pernode criticality index (% runs where critical) Network Monte Carlo Simulation Example
Optimistic Time, a 2 1 7 4 6 2 4 Statistics for E xample Activities Most Likely Pessimistic Expected Time, Time, Val u e, m b d 5 8 5 3 5 3 8 9 8 7 10 7 7 8 7 4 6 4 5 6 5 Stan d ard Deviation , s 1 0.66 0.33 1 0.33 0.66 0.33 Activity A B C D E F G Monte Carlo Simulation Example
Run Number 1 2 3 4 5 6 7 8 9 10 Summary of Simulation Runs for Example Project Activity Duration Critical Completion B C D E F G Path Time 2.2 8.8 6.6 7.6 5.7 4.6 ACFG 25.4 1.8 7.4 8.0 6.6 2.7 4.6 ADFG 17.4 4.9 8.8 7.0 6.7 5.0 4.9 ACFG 26.5 2.3 8.9 9.5 6.2 4.8 5.4 ADFG 25.0 2.6 7.6 7.2 7.2 5.3 5.6 ACFG 23.0 0.4 7.2 5.8 6.1 2.8 5.2 ACFG 22.3 4.7 8.9 6.6 7.3 4.6 5.5 ACFG 24.2 4.4 8.9 4.0 6.7 3.0 4.0 ACFG 22.1 1.1 7.4 5.9 7.9 2.9 5.9 ACFG 18.9 3.6 8.3 4.3 7.1 3.1 4.3 ACFG 19.7 A 6.3 2.1 7.8 5.3 4.5 7.1 5.2 6.2 2.7 4.0 Project Duration Distribution Probability
P( X ) =
Number of Times Project Finished in Less Than or Equal to t weeks Total Number of Replications The Probability that the project ends in 20 weeks or less is
P (X 20 )= 13 / 50 = 26 % Criticality Index Definition: Proportion of runs in which the activity was in the critical path PERT, CPM assume binary (either 100% or 0%) Helpful for prioritizing effort in Monitoring Controlling How Many Runs are Needed? Criticality Index p (particular node)
Originally very conservative (10K runs) Empirical tests suggest 1000 runs adequate Estimate of confidence interval for criticality
^ (1 ) confidence interval=symmetric interval around p such that P(true value p is within that interval) is (1)% ^ p  Z 2 ^ ^ p (1  p) ^ , p + Z 2 n ^ ^ p (1  p) n Consider a 95% confidence interval with 10% p 90%, ^ 400n 1000. Then with 95% confidence, p will be within 2%5% of p How Many Runs are Needed? Mean Project Duration
Must make assumptions regarding coefficient of variation = / (i.e. Std Dev/Mean) Z
^ u Basic formula Error % 100
2 n For Empirical range of CoV (5%..15%)
Sample size 400: within .5% to 1.5% of true value Sample size 1000: within .3% to 1% of true value Note inverseroot relationship: Halving error requires increasing # of trials by a factor of 4! How Many Runs are Needed? Project Duration Standard Deviation
Basic formula Error % 100 Z 2 2n
^ Sample size 400: within 7% of true value ^ Sample size 1000: within 4.38% of true value Inverseroot relationship again present Monte Carlo: Summary
Conceptually simple
Standard CPM used No need for special assumptions about functional form of distributions Provides criticality index (valuable prioritization) Scalable analysis quality (albeit with superlinear effort required to reduce error) Computationally expensive Estimation of duration distributions can be expensive Topics
PERT (Cont'd)
Review Merge node bias PNet refinement Monte Carlo Simulation approaches
General Demo Process Interaction Activity Scanning (Dynamic) Simulation Approach
CPMBased methods use simple representations
Onepass: No iteration Represented uncertainty only with respect to duration Explicitly representing process brings benefits
Reasoning about process design Identifying emergent behavior (e.g. dynamic bottleneck) Simpler estimation of some uncertainties Must be clear about whether representations are just processlevel or also projectlevel Detailed Representation
Repetitive processes for which aggregate representation is not desirable Processes where static planning is not possible
Repetitive processes for which # cycles unknown Scheduling and coordinating complex interactions (Large #s of brief interactions, dependent on timing) Cases where timing uncertainties change schedule Cases where individual timing component can be estimated, but where aggregate stats not known Examples of Repetitive Processes
Earth moving Tunneling Hotel/Apartment/Dormitory construction Road/Bridge construction Plumbing and glazing in highrise Topics
PERT (Cont'd)
Review Merge node bias PNet refinement Monte Carlo Simulation approaches
General Demo Process Interaction Activity Scanning Simulation Example: Excavation and Transporting
Given
Frontend loader
Output: ofrontend loader Instantaneous time between loads Trucks
n vehicles nc ncsl se otrucks = = Capacity c d d d ( se + sl ) + Load time tl sl se Instantaneous dump time Fully loaded speed sl , empty speed se Distance to dumpsite d Nave productivity: min(ofrontend loader, otrucks) ...
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 Spring '04
 DR.NATHANIELOSGOOD
 Normal Distribution, Standard Deviation, Variance, PERT Analysis

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