a dependent project evaluation and review technique, a ...dorpjr/beyondbeta/6... · a dependent...

BEYOND BETA SHORT COURSE: La Sapienza J.R. van Dorp; [email protected] - Page 1

A Dependent Project Evaluation and Review Technique,A Bayesian Network Approach

"Presentation Short Course: Beyond Beta and Applications"November 20th, 2018, La Sapienza

J. René van Dorp‡‡ Department of Engineering Management and Systems Engineering,

School of Engineering and Applied Science,The George Washington University,800 22nd Street, N.W. Suite 2800,ß

Washington D.C. 20052. E-mail: [email protected]://www.seas.gwu.edu/~dorpjr

SHORT COURSE "BEYOND BETA" at LA SAPIENZA J.R. van Dorp; [email protected] - Page 1

OUTLINE

1. INTRODUCTION

2. ACTIVITY DURATION UNCERTAINTY MODEL

3. BAYES NETWORK STATISTICAL DEPENDENCEMODEL

4. BAYES NETWORK DEGREE OF DEPENDENCEANALYSIS

5. DEPENDENT PERT EXAMPLE

6. SUMMARY AND CONCLUSIONS

7. SELECTED REFERENCES


1. INTRODUCTION... Project Network Example

1514

1312

111097

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4

5

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8

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foundationi.b. main enginebottom shellbottom shell engine test

assemble erect erect

install

install finish

install

boiler

i.b. piping

final

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layout

PROJECT NETWORK EXAMPLE:SHIPBUILDING CONSTRUCTION

Critical Path Method (CPM) developedin the late 1950’s - DeterministicProgram Evaluation and ReviewTechnique (PERT) developed by Malcolm et al. (1959) – Stochastic CPM


1. INTRODUCTION... AOA Project Network.

1514

1312

111097

6

4

5

2

8

1

3

B K L

H

P Q

N

R

D

A


ACTIVITY ON ARC REPRESENTATION


1. INTRODUCTION... AON Project Network

A B K L P Q R

E

C

I

F G

D

H

J

M N O


ACTIVITY ON NODE REPRESENTATION


1. INTRODUCTION... PERT Mean and Variance

• in Malcolm's et al. (1959) PERT is An activity's duration uncertainty\ abeta distribution with the following PERT mean and variance restriction

IÒ\l+ ß7 ß ,+ %7 ,

B B BB B B

Ó œ Z +




36)(VARIANCE PERT

2ab −=

36)()(VARIANCE PERT MOD.

2abC −×= δ

)1(7

1675)(, δδδδ −×+=

−−= Cabam

ID a m b δ C(δ) PERT Variance Modified PERT VarianceA 22 25 30 0.375 1.250 1.778 2.222B 35 37 43 0.250 1.143 1.778 2.032C 19 22 29 0.300 1.194 2.778 3.317D 4 5 10 0.167 1.032 1.000 1.032E 23 26 31 0.375 1.250 1.778 2.222F 16 18 24 0.250 1.143 1.778 2.032G 11 14 20 0.333 1.222 2.250 2.750H 6 7 12 0.167 1.032 1.000 1.032I 25 28 33 0.375 1.250 1.778 2.222J 33 35 40 0.286 1.181 1.361 1.607K 27 30 37 0.300 1.194 2.778 3.317L 6 7 11 0.200 1.080 0.694 0.750M 4 5 9 0.200 1.080 0.694 0.750N 6 7 10 0.250 1.143 0.444 0.508O 9 10 15 0.167 1.032 1.000 1.032P 6 7 12 0.167 1.032 1.000 1.032Q 17 20 26 0.333 1.222 2.250 2.750R 13 15 20 0.286 1.181 1.361 1.607Average Variance 1.7906

4MEAN PERT bma ++=

Malcolm et al. 1959 Herreriaset al. 2011Traditional Activity Estimates



• The second source of criticism relates to the statistical independenceassumption between the activity uncertainty distributions.

• Mathematical convenience was the prime motivation for the assumptionof statistical independence in PERT, despite it the commonnot matchingphenomenon in practice that multiple activities in a project network maybe delayed.

• Its mathematical convenience is derived from allowing:

(a) for the separate sampling from the activity marginal uncertaintydistributions, and

(b) evaluation of remaining project uncertainty by simply substitutingobserved completion times of activities in the project networkstructure while not having to modify the remaining activityuncertainty distributions in the project network.


1. INTRODUCTION... Statistical Independence...

• Suggestions for haverelaxing the statistical independence assumptionsbeen made by, e.g., Jenzarli, 1994; Covaliu & Soyer, 1997; Van Dorp &Duffey, 1999; Virto et al., 2002; Cho & Covaliu, 2003; Van Dorp, 2005;Khodakarami et al., 2007; Cho, 2009; Fang & Marle, 2012.

• These have thus farsuggestions for incorporating statistical dependencesuffered from either:

(i) having to be specified,too many dependence parameters(ii) not allowing for the coherent monitoring of remaining project

completion time uncertainty as the project progresses,(iii) if (ii) is not in effect, forrelying on complex numerical analysis

that coherent monitoring.

• The purpose of this paper/presentation is to relax that independenceassumption in that builds a “pragmatic” manner on Malcolm’s et al.(1959) work while attempting to address (i), (ii) and (iii) above.


OUTLINE

1. INTRODUCTION








2. ACTIVITY DURATION... Uncertainty Model Ð\ l‡ Q ÑB

• Construct that has Aim: a three-parameter uncertainty model the betadistibution and the TSP distributions as members and has infinitly manymembers that can match the PERT mean and PERT variance.

• Duration \‡ with , given lower and upper bounds + ß ,B B modeQB− Ð+ ß , ÑB B , and Assume:shape parameter ."B !

\ µ F/>+Ð+ ßQ ß , Ñß‡ B B B B, "

• Standardizing \‡ one obtains the random variable with support via\ Ò!ß "Óthe linear transformation with \ œ Ð\ + ÑÎÐ, + Ñ‡ B B B probabilitydensity function (pdf) where and! "ß !α "B B

1ÐBl ß Ñ œ ‚ B Ð" BÑÐ Ñ

Ð Ñ Ð Ð" ÑÑα "

> "

> " α > " αB B

B

B B B B

" Ð" Ñ"" α " αB B B B ,

• From it immediately follows that:1ÐBl ß Ñα "B B

IÒ\l ß Ó œ ß Z Ò\l ß Ó œÐ" Ñ

"α " α α "

α α

"B B B B B

B B

B


2. ACTIVITY DURATION... Uncertainty Model Ð\l?BÑ • Requiring for to be unimodal" α "B B B # 1ÐBl ß Ñ and introducing the

mode relative distance of ?B BQ given the support one obtainsÒ+ ß , Óß ÀB B? ?B B B B B Bœ ÐQ + ÑÎÐ, + Ñß ! " .

• Now reparameterizing 1ÐBl ß Ñα "B B in terms of its relative mode location?B

"#œ

α ""B B

B and shape parameter # "B Bœ # ! yields

1ÐBl ß Ñ œ ‚ B Ð" BÑÐ #Ñ

Ð "Ñ Ð Ð" Ñ "Ñ? #

> #

> # ? > # ?B B

B

B B B B

Ð" Ñ# ? # ?B B B B .

• From it immediately follows that:1ÐBl ß Ñ? #B B

IÒ\l ß Ó œ ß "

#

Z Ò\l ß Ó œ Þ " Ð" Ñ

Ð $ÑÐ #Ñ

# ?# ?

#

# ?# # ? ?

# #

B BB B

B

B BB B BB

#

B B#


2. ACTIVITY DURATION... Uncertainty Model?B

• Model uncertainty in the mode relative distance such that ?B µXWTÐ ß 8 Ñ ! "ß 8 !$ $B B B B, , with pdf:

0Ð?l ß 8 Ñ œ 8 ‚Ð Ñ ß ! Ÿ ? Ÿ ß

Ð Ñ ß Ÿ ? Ÿ "Þ$

$

$B B B

? 8 "B

"?"

8 "B

$$

B

B

B

B

• The cumulative distribution function (cdf) for the pdf above is

JÐ?l ß 8 Ñ œÐ Ñ ß ! Ÿ ? Ÿ ß

" Ð" ÑÐ Ñ ß Ÿ ? Ÿ "ß$

$ $

$ $B B

B B? 8

B B"?"

8 $$

B

B

B

B

• The quantile function for the pdf above is

J ÐCl ß 8 Ñ œÐ Ñ ß ! Ÿ C Ÿ ß

" Ð" ÑÐ Ñ ß Ÿ C Ÿ ""

B BB B

C "Î8

B B"C"

"Î8$

$ $

$ $ $ $BB

B

B .

• Observe from that:the cdf and the qf

J ÐCl ß 8 Ñ œ JÐCl ß "Î8 Ñ" B B B B$ $ .


2. ACTIVITY DURATION Mean and Variance of ÞÞÞ ?B

• From it immediately follows that:0Ð?l ß 8 Ñ$B B

IÒ l ß 8 Ó œÐ8 "Ñ "

8 "

Z Ò l ß 8 Ó œ8 #Ð8 "Ñ Ð" Ñ

Ð8 #ÑÐ8 "Ñ

? $$

? $$ $

B B BB B

B

B B BB B B B

B B#

,

.

• Given that a quantile level in the quantile function completelyCdetermines the value for ?B one may write:

IÒ lCß ß 8 Ó œ JÐCl ß "Î8 ÑÞ? $ $B B B B B

• Given for 1ÐBl ß Ñ Ð\l Ñ? # ?B B B and ,the prior pdf for 0Ð?l ß 8 Ñ$ ?B B Bclosed form expression for and the predictive mean predictive variance forthe activity duration \ can be derived by applying repeatedly the law oftotal expectation ÐPSXIÑ I Ò\Ó œ I \ ] I Ò\l] Ó\l]


2. ACTIVITY DURATION Predictive Mean of ÞÞÞ \

• With ? $B B B"œ J Ð] l ß 8 Ñß ] µ Y Ò!ß "Ówhere and applying thePSXI it follows with that:J ÐCl ß 8 Ñ œ JÐCl ß "Î8 Ñ" B B B B$ $

IÒ\l ß ß 8 Ó œ

œ IÒ\l ß Ó.C œ IÒ\l ß Ó.C

# $

# #

B B B

Cœ! Cœ!

" "

B B

I] I Ò\l ß œ J Ð] l ß 8 ÑÓ\l] B B B B"# ? $ J ÐCl ß 8 Ñ J ÐCl ß "Î8 Ñ" B B B B$ $ .

• Substitution of yieldsIÒ\l ß Ó œ Ð "ÑÎÐ #Ñ \# ? # ? #B B B B B for

IÒ\l ß ß 8 Ó œJÐCl ß "Î8 Ñ.C "

## $

# $

#B B B

B B BCœ!

"

B

Þ

• Utilizing yields] µ YÒ!ß "Óß JÐCl ß "Î8 Ñ œ IÒ lCß ß 8 Ó$ ? $B B B B B

IÒ\l ß ß 8 Ó œ "

#

œ œ I \l ßIÒ l ß 8 Ó Þ "

#

# $#

#

#

## ? $

B B B

B

B

B

BB B B B

Cœ!

"B B B

B B B

IÒ lCß ß 8 Ó.C

IÒ l ß 8 Ó

? $

? $


2. ACTIVITY DURATION Predictive Variance of \

• Analogously, using similar algebraic manipulations it follows

Z +


2. ACTIVITY DURATION... Predictive Distribution of \

• a three parameter uncertainty model Summarizing, 2ÐBl ß 8 ß Ñ$ #B B B for theactivity duration uncertainty has been constructed \ combining a betadistribution and .a TSP distribution

• Given a specified value for #B, values for and can be solved such that:8B B$

IÒ\l$ # $ #B B B B B BB

#

ß 8 ß Ó œ ß Z Ò\l ß 8 ß Ó œ! % ‚7 " Ð" !Ñ

' $'

• and using theFor this given value of and #B solved values for and 8B B$ , linear transformation \ œ Ð, + Ñ\ +‡ B B B it follows that:

IÒ\l$ # $ #B B B B B BB B B B

‡ #Bß 8 ß Ó œ ß Z Ò\l ß 8 ß Ó œ

+ % ‚7 , Ð, + Ñ

' $'

• an uncertainty duration model has been constructed whereConclusion:infinitly many of its members can match the PERT mean and PERTvariance. Recall, only and only one single beta member distribution onesingle TSP member distribution can match the PERT mean and PERTvariance.



0Ð?l ß 8 Ñ 2ÐBl ß 8 ß Ñ$ $ #B B B B B

$ #B B Bœ !Þ$"')ß8 Ä ∞ œ &Þ$'

• in Figure B All "three" probability density functions match the PERTmean and PERT variance. Pdf in ( ) in Figure B is the blue betagreenPERT ( ) distributionTSP PERT with + œ !ß7 œ !Þ$ß , œ "ÞB B B




$ #B B Bœ !Þ#'"$ß8 œ &Þ'#*& œ )

• in Figure B All three probability density functions match the PERT meanand PERT variance. Pdf in ( ) in Figure B is the blue beta PERTgreen( ) distributionTSP PERT with + œ !ß7 œ !Þ$ß , œ "ÞB B B




$ #B B Bœ !Þ#%#'ß8 œ $Þ(*$* œ "'





$ #B B Bœ !Þ#$)#ß8 œ $Þ$&*) œ $#





$ #B B Bœ !Þ#$($ß8 œ $Þ"*'! œ '%





$ #B B Bœ !Þ#$("ß8 œ $Þ"#%! œ "#)





$ #B B Bœ !Þ#$("ß8 œ $Þ!'"* Ä ∞



OUTLINE

1. INTRODUCTION








3. BAYES NETWORK... Statistical Dependence Model

• Let be in a projectE ß F ß‡ ‡ two activities as described previously\‡network and let and be their standardized versions with support .ß E F Ò!ß "Ó

• The suggested Bayesian Network (BN) model to the right is a screen shotfrom the Bayes Network software AgenaRisk®.

Common Mode Quantile Levelfor and :E F

] µ YÒ!ß "Ó

Ð? $

? $

+ + +"

, , ,"

l] Ñ œ J Ð] l ß 8 Ñß

Ð l] Ñ œ J Ð] l ß 8 ÑÞ

ÐEl Ñ µ 1Ð † l ß Ñß

ÐFl Ñ µ 1Ð † l ß ÑÞ

? ? #

? ? #+ + +

, , ,


3. BAYES NETWORK... Statistical Dependence Model

• This BN model .naturally extends to more than two activities

• Observe from figure that the random variables and? $+ + +µ XWTÐ ß8 Ñ? $, , ,µ XWTÐ ß8 Ñ are .conditionally independent given ] µ YÒ!ß "Ó

• Observe from figure above that arethe random variables and E Fconditionally independent or, alternatively, given ] the random variablesE Fand are conditionally independent given Ð ß Ñ? ?+ , Þ

• The node is the common quantile level node] µ YÒ!ß "Ó for the moderelative distances .Ð ß Ñ? ?+ , Thus the modes of the activity durations Eand "F move-in-sync" as per their common quantile level ] .

• Thus given a ( common quantile level, high lowÑ both uncertaintydistributions for and will be skewedE F towards the ( ). As aright leftresult activity durations .E Fand will be positively dependent

• Their degree of dependence as measured by the correlation between Eand will be a function of F the relative mode location parameters $B Bß 8and where the activity duration parameters #Bß B − Ö+ß ,×Þ


3. BAYES NETWORK... Statistical Dependence Model...

Prior Distribution Posterior Distribution

No Instantiation of A or B Activity A instantiated with +!Þ*&

• performed by software Posterior updating AgenaRisk®. Note the shifttowards the right for and . Also, Fß ß ]? ?+ , IÒ] l+ Ó ¸ !Þ*&Þ!Þ*&


OUTLINE

1. INTRODUCTION








4. BAYES NETWORK... Degree of Dependence Analysis

• Utilizing conditional independence of and given ? ?+ , ] µ Y Ò!ß "Óand the law of total expectation it can be shown:

G9@Ò ß l ß ß 8 ß 8 Ó œ " " Ð8 "Ñ " Ð8 "Ñ

Ð8 "Ñ 8 "

Ð Ñ8 8 8 "

8 8 8 8 8 "

Ð8 8 8 "

8 8 8 8 8 " "

? ? $ $$ $

$ $$ $

$

$

+ , + , + ,, , + +

, +

+ , , +

+ , + , , + ,+ ,

#

+ , +

+ , + , + ,

8,

" ÑÐ" Ñ ‚"

"

Ð" Ñ

" ‚

‚ÒFÐ" l"Î8 "ß "Î8 "Ñ FÐ" l"Î8 "ß "Î8 "ÑÓ

Ð"Î8 "Î8 #ÑÎ Ð"Î8 "Ñ Ð"Î8 "Ñ

$ $$

$

$ $

$ $

$ $

> > >

+ ,# ,

+

+ ,

+ ,

, + , + + ,

+ , + ,

8+

8+ ,8

• G9@Ò ß l ß ß 8 ß 8 Ó Æ ! 8 Ä ∞ß 8 Ä ∞? ? $ $+ , + , + , + + as . This alsofollows from and the observation that when BN Structure 8 Ä ∞ß+8 Ä ∞ XTW+ + + the pdfs for and converge to point masses at? ?$ $+ ,ß .



• Utilizing andconditional independence of and given E F ] µ YÒ!ß "Óthe law of total expectation it can also be shown that:

G9@ÐEßFl Ó œ ‚ G9@Ð ß l ß ß 8 ß 8 ÑßÐ #ÑÐ #Ñ

@ ? ? $ $# #

# #+ ,

+ ,+ , + , + ,

• Under the condition that activity and activity adhere to the PERTE Fmean and PERT variance, one has whichZ +


4. BAYES NETWORK... Degree of Dependence Example

Step 1:

8 œ &Þ'#*ß œ !Þ#'"ß+ +$ for for ??+,8 œ $Þ(*%ß œ !Þ#%$ß, ,$Ì

G9@Ò ß l ß ß 8 ß 8 Ó ¸ !Þ!"&%Þ? ? $ $+ , + , + ,

Step 2:Z Ò l ß 8 Ó ¸ !Þ!""&ß? $+ + + Z Ò l ß 8 Ó ¸ !Þ!#!)ß? $, , ,

G9@Ò ß l ß ß 8 ß 8 Ó ¸ !Þ!"&%? ? $ $+ , + , + ,Ì

3 ? ? $ $Ò ß l ß ß 8 ß 8 Ó ¸ !Þ**)'+ , + , + , .

Step 3:G9@Ò ß l ß ß 8 ß 8 Ó ¸ !Þ!"&%? ? $ $+ , + , + , ,

#+ œ ) for activity E#, œ "' for activityF

ÌG9@ÐEßFl Ñ ¸ !Þ!""!@ .

Step 4:G9@ÐEßFl Ñ ¸ !Þ!""!@ .

Z +


4. BAYES NETWORK... Degree of Dependence Example

Behavior of for the random variables and 3 @ÐEßFl Ñ E F in the BayesNetwork as a function of the dependence parameters and # #+ , with the

random variables E Fand possessing the PERT mean and PERT variance givensupports and most likely estimates .Ò!ß "Ó 7 œ 7 œ !Þ$+ ,


4. BAYES NETWORK... Degree of Dependence Examples

Predictive joint pdf can be approximated as the univariate ones.2Ð+ß ,l Ñ@

Examples of the joint pdf with most likely estimates 7 œ 7 œ !Þ$+ , with marginal pdfsfor and possessing PERT mean and PERT varianceE F but different correlations:

A: 3 @ÐEßFl Ñ ¸ !ß œ &Þ$'ß œ &Þ$'ß# #+ ,8 Ä ∞ß œ !Þ$"(ß 8 Ä ∞ß œ !Þ$"(+ + , ,$ $ ;

B: 3 @ÐEßFl Ñ ¸ !Þ$*ß œ )ß œ "'ß# #+ ,8 œ &Þ'#*ß œ !Þ#'"ß 8 œ $Þ(*%ß œ !Þ#%$+ + , ,$ $ ;



Predictive joint pdf can be approximated as the univariate ones.2Ð+ß ,l Ñ@

Examples of the joint pdf with most likely estimates 7 œ 7 œ !Þ$+ , with marginal pdfsfor and possessing PERT mean and PERT varianceE F but different correlations:

C: 3 @ÐEßFl Ñ ¸ !Þ&"ß œ )ß Ä ∞ß# #+ ,8 œ &Þ'#*ß œ !Þ#'"ß 8 œ $Þ!&*ß œ !Þ#$(+ + , ,$ $ ;

D: 3 @ÐEßFl Ñ ¸ !Þ)&ß œ œ %*Þ*!)ß Þ# #+ , 8 œ 8 œ $Þ#$*ß œ œ !Þ#$(+ , + ,$ $


4. BAYES NETWORK... Summary

• itWith and adhering to the PERT Mean and PERT Variance,E Ffollows that when the is one of8 Ä ∞ß 8 Ä ∞+ , joint predictive pdfindependent beta PERT distributions with .3ÐEßFÑ œ !

• itWith and adhering to the PERT Mean and PERT Variance,E Ffollows that when the is one of# #+ ,Ä ∞ß Ä ∞ joint predictive pdfdependent TSP PERT distributions with .3ÐEßFÑ œ 1

• With and adhering to the PERT Mean and PERT Variance,E F a jointpredictive pdf can be solved for with .! ÐEßFÑ "3

• does for This joint statistical dependence model not allow the modelingof negative dependence.

• For given values of # #+ ,ß , the PERT mean and PERT Variancerestriction determine the parameter values for 8 8+ , + ,, and , .$ $

• Conclusion: # #+ ,ß are ofdependence parameters the Bayes Network.


OUTLINE

1. INTRODUCTION





6. SUMMARY AND CONCLUSUONS



5. BAYES NETWORK... Dependent PERT Example

0

1

2

3

4

5

6

(1, 5, 12)

(4, 5, 7)

(2, 4, 7)

(8, 9, 10)

(8, 10, 15)

(2, 6, 10)

(1, 3, 6)

(1, 3, 7)

(3, 10, 20)

(5, 6, 9)

A

B

D

C

F

G

H

I

K

JE(6, 10, 14) (2, 5, 10)

L

0.5

0.5

0.5

0.5 0.5

0.5

0.50.5

0.5

0.5

0.5

0.5

Example 6-node project network consisting of 12 activities.

• When using the most likely values 7B for the activities, the critical pathhas a length of #) Þand is indicated using red arrows

• The same critical path emerges when using (the lower bounds +B thelower bounds ,B) with a length length #! %%( ).



To specify parameters for activities in the projectÐ ß ß 8 Ñ 3 œ "ßá ß \# $3 3 3 38above, it is proposed to amongselect a set of correlations 8 ß 4 Á 333 ß4 5 55 5paired activity durations to be set equal to a "project-degree of statisticaldependence" 3 œ !Þ& and solve À

minf

3 @ 3

? $#

#

$? $

À Ð Ð\ ß\ l Ð3 ß 4 ÑÑ Ñ

À IÒ l ß 8 Ó œ ß 5 œ "ßá ß ßÐ #Ñ + ß7 ß , "

œ ß 5 œ "ßá ß ßÐ8 "ÑIÒ l ß 8 Ó "

Ð8 "Ñ

Z

5œ"

3 4 5 5#

5 5 55 5 5 5

5

55 5 5 5

5

8

5 5

subject to IÒ\l Ó

8

8

Ò\l ß ß 8 Ó œ + ß7 ß , 5 œ "ßá ß 8ß# $5 5 5 5 5 5Z Ò\l Óß

• is the PERT Mean is the PERT variance.IÒ\l Ó ß Z Ò\l Ó+ ß7 ß , + ß7 ß ,5 5 5 5 5 5

• @ # $ # $ f # #Ð:ß ;Ñ œ Ð ß ß 8 ß ß ß 8 Ñß œ Ð ß 8 8 ß 5 œ "ßá8Ñ: : : ; ; ; 5 5‡ ‡5 5



Table 1. and for the PERTActivity input parameters Ð+ ß7 ß , ÑB B B 12 specified correlationsnetwork with resulting parameter solutions that Ð ß 8 ß Ñ# $B B B match the PERT Mean and PERT

Variance and are solved for using .the constrained optimization problem above

# X ax mx bx Pairs ρ(X,Y) γx nx δx x0.951 A 1 5 12 (A,B) 0.50 15.995 3.689 0.322 8.6712 B 2 4 7 (B,F) 0.50 11.947 4.123 0.372 5.5903 C 4 5 7 (C,E) 0.50 11.516 4.256 0.289 6.0414 D 5 6 9 (D,G) 0.50 13.804 4.054 0.184 7.5375 E 6 10 14 (D,H) 0.50 17.406 3.489 0.500 12.1976 F 3 10 20 (E,J) 0.50 16.369 3.603 0.383 15.3187 G 8 9 10 (F,J) 0.50 12.526 3.981 0.500 9.5498 H 8 10 15 (G,K) 0.50 11.189 4.358 0.231 12.5779 I 2 6 10 (H,I) 0.50 17.406 3.489 0.500 8.19710 J 1 3 7 (H,K) 0.50 11.516 4.256 0.289 5.08211 K 1 3 6 (I,J) 0.50 15.994 3.643 0.368 4.59012 L 2 5 10 (J,L) 0.50 15.950 3.677 0.336 7.630

• When using the -th percentiles *& B!Þ*& for the activities, the critical pathhas length of $'Þ$% Þand is indicated using the same red arrows



Correlation Matrix for the 12 activities in 6-node project network

A B C D E F G H I J K LA 1.000 0.500 0.502 0.562 0.573 0.571 0.501 0.505 0.573 0.502 0.568 0.570B 0.500 1.000 0.438 0.489 0.504 0.500 0.440 0.441 0.504 0.438 0.497 0.499C 0.502 0.438 1.000 0.494 0.500 0.500 0.437 0.445 0.500 0.441 0.497 0.500D 0.562 0.489 0.494 1.000 0.554 0.558 0.500 0.500 0.554 0.494 0.555 0.559E 0.573 0.504 0.500 0.554 1.000 0.576 0.514 0.500 0.588 0.500 0.572 0.572F 0.571 0.500 0.500 0.558 0.576 1.000 0.504 0.502 0.576 0.500 0.568 0.569G 0.501 0.440 0.437 0.500 0.514 0.504 1.000 0.437 0.514 0.437 0.500 0.500H 0.505 0.441 0.445 0.500 0.500 0.502 0.437 1.000 0.500 0.445 0.500 0.503I 0.573 0.504 0.500 0.554 0.588 0.576 0.514 0.500 1.000 0.500 0.572 0.572J 0.502 0.438 0.441 0.494 0.500 0.500 0.437 0.445 0.500 1.000 0.497 0.500K 0.568 0.497 0.497 0.555 0.572 0.568 0.500 0.500 0.572 0.497 1.000 0.566L 0.570 0.499 0.500 0.559 0.572 0.569 0.500 0.503 0.572 0.500 0.566 1.000

• The correlations in above the diagonal and in below the diagonalred greenequate to .the "project degree of dependence" 3 œ !Þ&

• The Bayes Network dependence model has 12 dependence parameters#B, whereas the correlation matrix contains "## "# œ 54 correlations.

• Thus, the Bayes Network dependence model is restricted in its flexibilitysince, for example, the correlation 3ÐFßGÑ ¸ !Þ%$) follows.



Screenshot software AgenaRisk®.• The green dashed arcs leadfrom the common quantile node to]the branch probability nodes and themode relative distance nodes .?B

• The red dashed arcs lead fromthe mode relative distance nodes to?Bthe activity duration nodes .\

• The black solid arcs in Figure 6point to finishing time nodes "F_X"that contain the usual CPMexpressions.

• Finally, inthe "FINISH" nodeevaluates the project completiontime uncertainty.



0

1

2

3

4

5

6

(1, 5, 12)

(4, 5, 7)

(2, 4, 7)

(8, 9, 10)

(8, 10, 15)

(2, 6, 10)

(1, 3, 6)

(1, 3, 7)

(3, 10, 20)

(5, 6, 9)

A

B

D

C

F

G

H

I

K

JE(6, 10, 14) (2, 5, 10)

L

0.5

0.5

0.5

0.5 0.5

0.5

0.50.5

0.5

0.5

0.5

0.5

• Evaluate under prior project completion time distribution dependenceand independence.



0

1

2

3

4

5

6

(8, 9, 10)

(8, 10, 15)

(2, 6, 10)

(1, 3, 6)

(1, 3, 7)

(3, 10, 20)

(5, 6, 9)

A = 8.671

B =6.041

D

C = 5.590

F

G

H

I

K

JE(6, 10, 14) (2, 5, 10)

L

0.5

0.5

0.5

0.5 0.5

0.5

0.50.5

0.5

0.5

0.5

0.5

• Evaluate under andposterior project completion time distr. dependenceindependence given completion A, B and C at their -th percentiles.*&



Prior project completion time distribution results comparison for the projectnetwork and comparison posterior completion time distribution given the

completion of activities A, B and C at and quantile levels.+ ß , -!Þ*& !Þ*& !Þ*&


5. BAYES NETWORK... PERT Example Observations

• In comparing the prior project completion time distributions in red and greenin, one observes a larger uncertainty apriori in the statistical dependencecase (in red) which is (see, e.g. Van Dorp, 2005).a known result

• under both scenarios are evaluated The mean prior project completion at¸ #*Þ%( ¸ #*Þ&% and for the dependence and independence scenario,

respectively, which is slightly larger than the critical path length when#)using the most likely values 7B for the activity durations.

• Given the completion of in thethe first three activities , and E F Gproject network in Figure 6 at their 95 percentile values and+ ß ,!Þ*& !Þ*&-!Þ*&, the posterior mean project completion time under statisticalindependence is evaluated by AgenaRisk® at $"Þ)( and understatistical dependence evaluated at .$'Þ&& The posterior mean of thecommon quantile node is evaluated by AgenaRisk® at ] ¸ !Þ*(Þ

• It is worthwhile to note here that using the length of the critical path all the95 percentile values was evaluated at .¸ $'Þ$%


5. BAYES NETWORK... PERT Example Observations

• In other words, the posterior mean project completion time forecast of$'Þ&&, given the completion times for activities , and , E F G is moreaccurate under statistical dependence than the posterior mean projectcompletion time forecast of under statistical independence$"Þ)( .

• Comparing the posterior project completion time distributions understatistical independence and under statistical dependence, a lesseruncertainty in the project completion time distribution under statisticaldependence whereas the converse was true apriori.

• While one also observes a reduction in uncertainty from the apriori statisticalindependence case to the aposteriori case, the reduction in uncertainty ismore pronounced in the statistical dependence case going from the reddistribution to the dark blue one.


OUTLINE

1. INTRODUCTION









• It would appear that one forecasts more accurately in terms of the projectcompletion time point estimate under statistical dependence.

• in terms of One also learns faster under statistical dependence areduction of uncertainty in the project completion time distribution.

• Heuristically and pragmatically, perhaps a better updated projectcompletion time point estimate can be obtained by averaging thequantile levels of the activities that have completed and utilizing thataverage quantile level for the remaining activities that have notcompleted in a single CPM evaluation step, rather than evaluating theremaining project completion time uncertainty and its point estimateunder an assumption of statistical independence.

• That being said, to evaluate the remaining uncertainty in the projectcompletion time under statistical dependence in a coherent mannerfollowing the completion of some activities, a Bayesian inferenceprocedure is required.


7. REFERENCES... PERT Mean and Variance

Gallagher, C. (1987). A note on PERT assumptions, Management Science, 33 (10), 1360.Golenko-Ginzburg, D. (1988). On the distribution of activity time in PERT, Journal of

Operations Research, 39, 767-771.Hahn, E. D. (2008). Mixture densities for project management activity times: A robust

approach to PERT. , 188(2), 450-459.European Journal of Operational ResearchHerrerias-Pleguezuelo, R., Garcia-Perez, J. & Cruz-Rumbaud, S. (2003). A note on the

reasonableness of PERT hypothesis, Operations Research Letters, 31 (1), 60-62Herrerias-Velasco, J. M., Herrerias-Pleguezuelo, R., & Van Dorp, J. R. (2011). Revisiting the

PERT mean and variance. , 210, 448-451.European Journal of Operational ResearchJenzarli, A. (1994). PERT belief networks, . The University of Tampa, FL.Report 535Kamburowski, J. (1997). New validations of PERT times, Omega, 25 (3), 323-328.Littlefield, T.J. & Randolph, P. (1987). An answer to Sasieni's question on PERT Times,

Management Science, 33 (10), 1357-1359.Malcolm, D., Roseboom, J., Clark, C., & Fazar, W. (1959). Application of a technique for

research and development program application. , 7 (5), 646-669.Operations ResearchSasieni, M.W. (1986). A note on PERT Times, , 32 (12), 1652-1653. Management ScienceVan Dorp, J. R., & Kotz, S. (2002). A Novel Extension of the Triangular Distribution and its

Parameter Estimation. , 51, 63-79.The Statistician


7. REFERENCES... PERT Statistical Dependence

Cho, S. (2009). A linear Bayesian stochastic approximation to update project duration estimates.European Journal of Operations Research, 196, 585-593.

Cho, S., & Covaliu, Z. (2003). Sequential estimation and crashing PERT networks with statisticaldependence. , 10, 391-399.International Journal of Industrial Engineering

Covaliu, Z., & Soyer, R. (1997). Bayesian Learning in project management networks. American StatisticalAssociation Proceedings, Section on Bayesian Statistical Science, (pp. 257-260).

Fang, C., & Marle, F. (2012). A simulation-based risk network model for decision support in project riskmanagement. , 52, 635-644.Decision Support Systems

Nduka, I. & Van Dorp, J.R. (2015). Monitoring Uncertainty in Project Completion Times: A BayesianNetwork Approach, Unpublished Manuscript, https://www2.seas.gwu.edu/~dorpjr/Publications/JournalPapers/Nduka%20and%20Van%20Dorp%20-%20Unpublished%20Manuscript.pdf , last accessed: 8/19/2018.

Neil, M., Tailor, M., & Marquez, D. (2007). Inference in Hybrid Bayesian Networks using DynamicDiscretization. , 17(3), 219-233.Statistics and Computing

Khodakarami, V., Fenton, N., & Neil, M. (2007). Project Scheduling: Improved Approach toIncorporate Uncertainty Using Bayesian Networks. , 38, 39-49.Project Management Journal

Van Dorp, J. R. (2005). Statistical Dependence through Common Risk Factors: With Applications inUncertainty Analysis. , 161 (1), 240-255.European Journal of Operational Research

Van Dorp, J.R., & Duffey, M. (1999). Statistical dependence in risk analysis for project networks usingMonte Carlo methods. , 58, 17-29.International Journal of Production Economics

Virto, M., Martin, J., & Insua, D. (2002). Approximate solutions of complex influence diagrams throughMCMC methods. In S. Gamez (Ed.), , (pp. 169-First European Workshop on Probabilistic Graphical Models175).


7. REFERENCES... Software

Microsoft Project (2018). Microsoft Project, https://en.wikipedia.org/wiki/®Microsoft_Project#Project_2016, last accessed: 8/19/2018.

Primavera (2018). Primavera (Software), https://en.wikipedia.org/wiki/ Primavera_%28software%29,®last accessed: 8/19/2018.

AgenaRisk® (2018). Agena, Bayesian Network and Simulation Software for Decision Support,http://www.agenarisk.com/, last accessed 8/19/2018.

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