PROJECT SCHEDULING:IMPROVED APPROACH TO INCORPORATEUNCERTAINTY USINC BAYESIAN NETWORKS

VAHID KHODAKARAMI, Queen Mary University of London, United KingdomNORMAN FENTON, Queen Mary University of London, United KingdomMARTIN NEIL, Queen Mary University of London, United Kingdom

Introduction

[^BSTRACf]project scheduling inevitably involvesuncertainty. The basic inputs (i.e., time,cost, and resources for each activity) arenot deterministic and are affected by var-ious sources of uncertainty. Moreover,there is a causal relationship betweenthese uncertainty sources and projectparameters; this causality is not modeledin current state-of-the-art project plan-ning techniques {such as simulation tech-niques). This paper introduces anapproach, using Bayesian network mod-eling, that addresses both uncertaintyand causality in project scheduling.Bayesian networks have been widely usedin a range of decision-support applica-tions, but the application to project man-agement is novel. The model presentedempowers the traditional critical pathmethod (CPM) to handle uncertainty andalso provides explanatory analysis to elic-it, represent, and manage differentsources of uncertainty in project planning.

Keywords: project scheduling;uncertainty; Bayesian networks;critical path method; CPM

O1007 by the Prolett Management InstftuteVol. 3B. No. 2, 39-49, ISSN 87J6-9728/D3

Project scheduling is difficult because it inevitably involves uncertainty.Uncertainty in real-world projects arises from the following characteristics;

• Uniqueness (no similar experience)• Variability (trade-off between performance measures like time, cost, and quality)• Ambiguity (lack of clarity, lack of data, lack of structure, and bias in estimates).

Matiy different techniques and tools have been developed to support betterproject scheduling, and these tools are used seriously by a large majority of proj-ect managers (Fox & Spence, 1998; Pollack-lohnson, 1998). Yet, quantifyinguncertainty is rarely prominent in these approaches.

This paper focuses especially on the problem of handling uncertainty in proj-ect scheduling. The next section elaborates on the nature of uncertainty in projectscheduling and summarizes the current state of the art. The proposed approach isto adapt one of the best-used scheduling techniques, critical path method (CPM)(Kelly, 1961), and incorporate it into an explicit uncertainty model {usingBayesian networks). The paper summarizes the basic CPM methodology and nota-tion, presents a brief introduction to Bayesian networks, and describes how theCPM approach can be incorporated (using a simple illustrative example). Also dis-cussed is a mechanism to implement the model in real-world projects, and sug-gestions on how to move forward and possible future modifications are presented.

The Nature of Uncertainty in Project SchedulingA Guide to the Project Management Body of Knowledge (PMBOK'^ Cuide)—'['hkd edi-tion (PMI, 2004) identifies risk management as a key area of project management:

"Project risk management includes the processes concerned with conductingrisk management planning, identification, analysis, response, and monitoringand control on a project."

Central to risk management is the issue of handling uncertainty. Ward andChapman (2003) argued that current project risk management processes induce arestricted focus on managing project uncertainty. They believe it is because theterm "risk" has become associated witb "events" rather than more general sourcesof significant uncertainty.

MANAGtMENT JOLIRNM 39

In different project managementprocesses there are different aspects ofuncertainty. The focus of this paper is onuncertainty in projert scheduling. Themost obvious area of uncertainty here isin estimating duration for a partiailaractivity. Difficulty in this estimation canarise from a lack of knowledge of what isinvolved as well as from the uncertainconsequences of potential threats oropportunities. This uncertainty arisesfrom one or more of the following:

• Level of available and requiredresources

• Trade-off between resources and time• Possible occurrence of uncertain

events (i.e., risks)• Causal factors and interdependencies

including common casual factorsthat affect more than one activity(such as organizational issues)

• Lack of previous experience and use ofsubjective rather than objective data

• Incomplete or imprecise data or lackof data at all

• Uncertainty about the basis of subjec-tive estimation (i.e., bias in estimation).

The best-known technique to sup-port project scheduling is CPM. Thistechnique, which is adapted by themost widely used project managementsoftware tools, is purely deterministic.It makes no attempt to handle or quan-tify uncertainty. However, a number oftechniques, such as program evaluationand review technique (PERT), criticalchain scheduling (CCS) and MonteCarlo simulation (MCS), do try to han-dle uncertainty, as follows:

• PERT (Malcom, Roseboom, Clark, &I-azer, 1959; Miller, 1962; Moder,1988) incorporates uncertainty in arestricted sense by using a probabil-ity distribution for each task.Instead of having a single determin-istic value, three different estimates(pessimistic, optimistic, and mostlikely) are approximated. Then the"critical path" and the start and fin-ish date are calculated by the use ofdistributions' means and applyingprobability rules. Results in PERTare more realistic than CPM, butPERT does not address explicitly anyof the sources of uncertainty previ-ously listed.

Critical chain (CC) scheduling isbased on Coldratt's theory of con-straints (Coldratt, 1997). Hor mini-mizing the impact of Parkinson'sLaw (jobs expand to fill the allocat-ed time), CC uses a 50% confidenceinterval for each task in projectscheduling. The safety time (remain-ing 50%) associated with each taskis shifted to the end of the criticalchain (longest chain) to form theproject buffer. Although it is claimedthat the CC approach is the mostimportant breakthrough in projectmanagement history, its oversim-plicity is a concern for many compa-nies that do not understand both thestrength and weakness of CC andapply it regardless of their particularand unique circumstances (Pinto,1999). The assumption that all taskdurations are overestimated by a cer-tain factor is questionable. The mainissue is: How does the project man-ager determine the safety time? (Raz,Barnes, & Dvir, 2003). CC relies ona fixed, right-skewed probability foractivities, which may be inappropri-ate (Herroelen & Leus, 2001), and asound estimation of project andactivity duration (and consequentlythe buffer size) is still essential(Trietsch, 2005).

Monte Carlo simulation (MCS) wasfirst proposed for project schedulingin the early 1960s (Van Slyke, 1963)and implemented in the 1980s(Fishman, 1986). In the 1990s,because of improvements in comput-er technoiogy, MCS rapidly becamethe dominant technique for han-dling uncertainty in project schedul-ing (Cook, 2001). A survey by theProject Management Institute (PMI,1999) showed that nearly 20% ofproject management software pack-ages support MCS. For example,PertMaster (PertMaster, 2006)accepts scheduling data from toolslike MS-Project and Primavera andincorporates MCS to provide projectrisk analysis in time and cost.However, the Monte Carlo approachhas attracted some criticism. VanDorp and Duffey (1999) explainedthe weakness of Monte Carlo simula-tion In assuming statistical inde-

pendence of activity duration in aproject network. Moreover, beingevent-oriented (assutning projectrisks as "independent events"),MCS and the tools that implementIt do not identify the sources ofuncertainty.

As argued by Ward and Chapman(2003), managing uncertainty in proj-ects is not just about managing per-ceived threats, opportunities, and theirimplication. A proper uncertaintymanagement provides for identifyingvarious sources of uncertainty, under-standing the origins of them, and thenmanaging them to deal with desirableor undesirable implications.

Capturing uncertainty in proj-ects "needs to go beyond variabilityand available data. It needs toaddress ambiguity and incorporatestructure and knowledge" (Chapman& Ward, 2000). In order to measureand analyze uncertainty properly, weneed to model relations betweentrigger (source), and risk and impacts(consequences). Because projects areusually one-off experiences, theiruncertainty is epistemic (i.e., relatedto a lack of complete knowledge)rather than aleatoric (i.e., related torandomness). The duration of a taskis uncertain because there is no sim-ilar experience before, so data isincomplete and suffers from impreci-sion and inaccuracy. The estimationof this sort of uncertainty is mostlysubjective and based on estimatorjudgment. Any estimation is condi-tionally dependent on some assump-tions and conditions—even ifthey are not mentioned explicitly.These assumptions and conditionsare major sources of uncertaintyand need to be addressed and han-dled explicitly.

The most well-establishedapproach to handling uncertainty inthese circumstances is the Bayesianapproach (Efron, 2004; Goldstein,2006). Where complex causal rela-tionships are involved, the Bayesianapproach is extended by usingBayesian networks. The challenge isto incorporate the CPM approachinto Bayesian networks.

40 1007 MANAGEMENT JOURNAL

CPM Methodology and Notation

CPM (Moder, 1988) is a deterministictechnique that, by use of a network ofdependencies between tasks and givendeterministic values for task durations,calfulatt's the longest path in the net-work called the "critical path." Thelength of the "critical path" is the earli-est time for project cotiipletion. Thecritical path can be identified by deter-mining the following parameters foreach activity:

D—durationES—earliest start timeEF—earliest finish timeLS—latest start timeLF—latest finish time.

The earliest start and finish timesof each activity are determined byworking forward through the networkand determining the earliest time atwhich an activity can start and finish,considering its predecessor activities.For each activity j\

I'S, = MaxjESi + Di ;over predecessor activities i\

EFj = ESj- Dj

The latest start and finish times arethe latest times that an activity canstart and finish without delaying theproject and are found by workingbackward through the network. Foreach activity i:

LFj = Min |LFj- D,;over successor activities ;'|

The activity's "total float" (TF)(i.e., the amount that the activity'sduration can be increased withoutincreasing the overall project comple-tion time) is the difference in the latestand earliest finish times of each activi-ty. A critical activity is one with no TFand should receive special attention(delay in a critical activity will delaythe entire project). The critical paththen is the path(s) throiLgh the net-work whose activities have minimal TK

The CPM approach is very simpleand provides very useful and funda-mental information about a projectand its aaivities' schedule. However,because of its single-point estimateassumption, it is loo simplistic to beused in complex projects. The chal-lenge is to incorporate the inevitableuncertainty.

Proposed BN SolutionBayesian Networks (BNs) are recog-nized as a mature formalism for han-dling causality and uncertainty(Heckerman, Mamdani, & Wellman,1995). This section provides a briefoverview of RNs and describes a newapproach for scheduling project activi-ties in which CPM parameters (i.e., BS,EF, LS, and LF) are determined in a BN.

Bayesian Networks: An OverviewBayesian networks (also known asbelief networks, causal probabilisticnetworks, causal nets, graphical proba-bility networks, probabilistic cause-

effect tnodels, and probabilistic infiu-ence diagrams) provide decision sup-port for a wide range of problemsinvolving uncertainty and probabilisticreasoning. Examples of real-worldapplications can be found inHeckerman et al. (1995), Fenton,Krause, and Neil (2002), and Neil,I enton, Forey, and I larris (2001). A BNis a directed graph, together with anassociated set of probability tables.The graph consists of nodes and arcs.Figure 1 shows a simple BN that mod-els the cause of delay in a particulartask in a project. The nodes representuncertain variables, which may or maynot be observable. Each node has a setof states (e.g. "on time" atid "late" for"Subcontract" node). The arcs repre-sent causal or infiuential reiationshipsbetween variables, (e.g., "subcontract"and "staff experience" may cause a"delay in task"). There is a probabilitytable for each node, providing theprobabilities of each state of the vari-able. For variables without parents(called "prior" nodes), the table justcontains the marginal probabilities(e.g., for the subcontract" node P(on-time)=0.95 and P(iate)=0,03). Ihis isalso caiied "prior distribution" thatrepresents the prior belief (state ofknowledge) about the variable. Foreach variable with parents, the proba-bility table has conditional probabili-ties for each combination of theparents' states (see, for example, theprobability table for a "delay in task"

On Time

Late

0.95

0.05Subcontract

Delay in Task

Staff ExperienceHigh

Low

0.7

0.3

Subcontract

Staff Experience

DelayYes

No

On Time

High

0.95

0.05

Low

0.7

0,3

Late

High

0.7

0.3

Low

0.01

0.99

Figure 1: A Bayesian network contains nodes, arcs and probability table

JUNE J007 MANAUKMENT JOURNAL 41

in Figure 1). This is also called the"likelihood function" that representsthe likelihood of a state of a variablegiven a particular state of its parent.

The main use of BNs is in situa-tions that require statistical inference.In addition to statements about theprobabilities of events, users havesome evidence (i.e., some variablestates or events that have actually beenobserved), and can infer the probabili-ties of other variables, which have notas yet been observed. These observedvalues represent a posterior probabili-ty, and by applying Bayesean rules ineach affected node, users can infiuenceother BN nodes via propagation, mod-ifying the probability distributions. Forexample, the probability that the taskfinishes on time, with no observation,is 0.855 (see Figure 2a). However if weknow that the subcontractor failed todeliver on time, this probabilityupdates to 0.49 (see Figure 2b),

The key benefits of BNs that makethem highly suitable for the projectplanning domain are that they:• Explicitly quantify uncertainty and model

the causal relation between variables• Enable reasoning from effert to cause as

well as from cause to effect (propaga-tion is both "forward" and "backward")

• Make It possible to overturn previ-ous beliefs in the light of new data

• Make predictions with incomplete data• Combine subjective and objertive data• Enable users to arrive at decisions

that are based on visible auditablereasoning.

BNs, as a tool for decision support,have been deployed in domains rang-ing from medicine to politics. BNspotentially address many of the "uncer-tainty" issues previously discussed. Inparticular, incorporating CPM-stylescheduling into a BN framework makesit possible to properly handle uncer-tainty in project scheduling.

There are numerous commercialtools that enable users to build BNmodels and run the propagation calcu-lations. With such tools it is possible toperform fast propagation in large BNs(with hundreds of nodes). In thispaper, AgenaRisk (2006) was used,since It can model continuous vari-ables (as opposed to just discrete).

BN for Activity DurationFigure 3 shows a prototype BN that theauthors have built to model uncertain-ty sources and their afferts on durationof a particular activity. The model con-tains variables that capture the uncer-tain nature of activity duration. "Initialduration estimation" is the first esti-mation of the artivity's duration; it isestimated based on historical data,previous experience, or simply expertjudgment. "Resources" incorporate anyaffeaing factor that can increase ordecrease the activity duration. It is aranked node, which for simplicity hereis restricted to three levels: low, aver-age, and high. The level of resourcescan be inferred from so-called "indica-tor" nodes. Hence, the causal link isfrom the "resources" directly to observ-

able indicator values like the "cost,"the experience of available "people"and the level of available "technology."There are many alternative indicators.An important and novel aspect of thisapproach is to allow the model to beadapted to use whichever indicatorsare available.

The power of this model is betterunderstood by showing the results ofrunning it under various scenarios. It ispossible to enter observations any-where in the model to perform not justpredictions but also many types oftrade-off and explanatory analysis. So,for example, observations for the ini-tial duration estimation and resourcescan be entered and the model willshow the distributions for duration.Figure 4 shows how the distribution ofthe activity duration in which the ini-tial estimation is five days changeswhen the level of its availableresources goes from low to high. (Allthe subsequent figures are outputsfrom the AgenaRisk software.)

Another possible analysis in thismodel is the trade-off analysis betweenduration and resources when there is atime constraint for activity durationand it is interesting to know about thelevel of required resource. For example,consider an activity in which the initialduration is estimated as five days butmust be finished in three days. Figure 5shows the probability distribution ofrequired resources to meet this dura-tion constraint. Note how it is skewedtoward high.

Subcontract Staff Experience

Delay in Task

0.64

0.32

0.0Yes No

(a) P(Task =on fime)=0.855

Staff Experience

Delay in Task

0.32

0.16

0.0Yes No

P(Task =on time}=0,49

Figure 2: New evidence updates the probability

JUNH 2007 PRCIIHCT MANAI IHMENT JOURNAI

Figure 3: Bayesian network for activity duration

Figure 4: Probability distribution for "duration" (days) cfianges when the level of "resources" changes

Low Medium High

Figure 5: Level of required "Resources" when there is a constraint on "Duration"

Mapping CPM to BNThe main components of CPM net-works are activities. Activities are linkedtogether to represent dependencies, lnorder to map a CPM network to a BN,it is necessary to first map a singleactivity. Each of the activity parametersare represented as a variable (node) inthe BN.

Figure 6 shows a schematic modelof the BN fragment associated with anactivity. It clearly shows tbe relationbetween the activity parameters andalso the relation with predecessor andsuccessor activities.

The next step is to define the con-necting link between depetident activi-ties. The forward pass in CPM ismapped as a link between the EF ofeach activity to the ES of the successoractivities. The backward-pass in CPM ismapped as a link between the LS ofeach activity to the LF of the predeces-sor activities.

ExtwipleThe following illustrates this mappingprocess. The example is deliberatelyvery simple to avoid extra complexityin the BN. How the approach can beused in real-size projects is discussedlater in the paper.

Consider a small project with fiveactivities—A, B, C, D, and E. The activ-ity on arc (AOA) network of the projeais shown in Figure 7,

The results of the CPM calculationare summarized in Table 1. ActivitiesA, C, and E with 1 F=0 are critical andthe overall project takes 20 days (i.e.,earliest finish of activity E).

Figure 8 shows the full BN repre-sentation of the previous example.Each activity has five associated nodes.Forward pass calculation of CPM isdone through the connection betweenthe ES and EF. Activity A, the first activ-ity of the project, has no predecessor,so its ES is set to zero. Activity A ispredecessor for activities B and C sothe EF of activity A is linked to the ESof activities B and C. The EF of activityB is linked to the ES of its successor,activity D. And finally, the EF of activi-ties C and D are connected to the ES ofactivity E. In fact, the ES of activity E isthe maximum of the EF of activities C

JUNK 2007 MANM^FMHNT JOURN,AI.

PredecessorPredecessor

Activities

SuccessorActivities

Successor

Figure 6: Schematic of BN for an activity

Figure 7: CPM network

and D. The EF of activity E is the earli-est time for project completion time.

The same approach is used forbackward CPM calculations connectingthe LF and LS, Activity E is the last activ-ity of the project and has no successor,so its EF is set to EF. Activity E is succes-sor of activities C and D so the ES ofactivity E is linked to the LF of activitiesC and D. The LS of activity D is linkedto the LF of its predecessor activity B.And finally, the LS of activities B and Care linked to the LFofactivityA. TheLFof activity A is the minimum of the LSof activities B and C.

For simplicity in this example, it isassumed that activities A and E aremore risky and need more detailedanalysis. For all other activities theuncertainty about duration is expressedsimply by a normal distribution.

ResultsThis section explores different scenar-ios of the BN model in Figure 8. Themain objective is to predict the proj-

ect completion time (i.e., the earliestfinish of E) in such a way that it fullycharacterizes uncertainty.

Suppose the initial estimationof activities' duration is the same asin Table 1. Suppose the resourcelevel for activities A and E is medi-um. If the earliest start of activity Ais set to zero, the distribution forproject completion is shown inFigure 9a. The distribution's mean is20 days as was expected from theCPM analysis. However, unlikeCPM, the prediction is not a singlepoint and Its variance is 4. Figure 9biliustrates the cumulative distribu-tion of finishing time, which showsthe probability of completing theproject before a given time. Forexample, with a probability of 90%the project will finish in 22 days.

In addition to this baseline sce-nario, by entering various evidence(observations) to the model, it is pos-sible to analyze the project schedulefrom different aspects. For example.

one scenario is to see how changingthe resource level affects the projectcompletion time.

Figure 10 compares the distribu-tions for project completion time asthe level of people's experiencechanges. When people's experiencechanges from low to high, the meanof finishing time changes from 22.7days to 19.5 days and the 90% confi-dence interval changes from 26.3days to 22.9 days.

Another usefui analysis is whenthere is a constraint on tbe projectcompletion time and we want toknow how many resources are need-ed. Figure II illustrates this trade-offbetween project time and requiredresources. If the project needs to becompleted in 18 days (instead of thebaseline 20 days) then the resourcerequired for activity A most likelymust be high; if the project comple-tion is set to 22, the resource level foractivity A moves significantly in thedirection of low.

The next scenario investigates theimpact of risk in activity A on theproject completion time as it isshown in Figure 12. When there is arisk in activity A, the mean of distri-bution for the project completiontime changes from 19.9 days to 22.6days and the 90% confidence intervalchanges from 22.5 days to 25.3 days.

One important advantage ofBNs is their potential for parameterlearning, which is shown in thenext scenario. Imagine activity Aactually finishes in seven days,even though it was originally esti-mated as five days. Because activityA has taken more time than wasexpected, the level of resources hasprobably not been sufficient.

By entering this observation themodel gives the resource probabilityfor activity A as illustrated in Figure13. This can update the analyst'sbelief about the actual level of avail-able resources.

Assuming both activities A and Euse the same resources (e.g., people),the updated knowledge about thelevel of available resources fromactivity A (which is finished) can beentered as evidence In the resources

PRIMHCT JOURNAL

DURATiONSUBNET

C Risk

/ ini t ia l DuratioriN ^ ^ ^( Resources > •^(^ Peopie J)

LS A D A ES A

LF A EF A

LS ES

LF_B

D

L F D J1 ^

DtJRATIONSUBNET_E

\

Figure 8: Overview of BN for example (1)

for activity E (which is not startedyet) and consequently updates theproject completion time. Figure 14shows the distributions of comple-tion time when the level of availableresource of activity E is learned fromthe actual duration of activity A.

Another application of parameterlearning in these models is the abilityto incorporate and learn about bias inestimation. So, if there are severalobservations in which actual taskcompletion times are underestimated,the model learns that this may be due

to bias rather than unforeseen risks,and this information will inform sub-sequent predictions. Work on this typeof application (caiied dynamic learn-itig), is still in progress and can be apossible way of extending the BN ver-sion of CPM.

JUNE 2007 PnnitrT MANAUEMtNT JOURNAL 45

Activity

A

B

C

D

LJJ

D

5

4

10

2

5

ES

0

5

5

9

15

EF

CJI

9

15

11

20

LS

0

9

5

13

15

LF

5

13

15

15

20

TF

0

4

0

4

0

Table 1: Activities' time (days) and summary of CPM calculations

Object-Oriented BayesianNetwork (OOBN)II is clear from Figure 8 that even simpleCPM networks lead to fairly large BNs.In real-sized projects with several aaivi-ties, constmaing the network needs ahuge effort, which is not effective espe-

daliy for users without much experiencein BNs. However, this complexity can behandled using the so-called object-ori-ented Bayesian network (OOBN)approach (Roller & Pfeffer, 1997). Thisapproach, analogous to the object-ori-ented programming languages, supports

a natural framework for abstraction andrefinement, which allows complexdomains to be described in terms ofinterrelated objects.

The basic element in OOBN is anobject; an entity with an identity, state,and behavior. An object has a set ofattributes each of which is an obiect.Each object is assigned to a class.Classes provide the ability to describe ageneral, reusable network that can beused in different instances. A class inOOBN is a BN fragment.

The proposed mode! has a highlyrepetitive structure and fits the object-oriented framework perfectly. Theinternal parts of the activity subnet(see Figure 6) are encapsulated withintbe activity class as shown in Figure 15.

D Baseline Scenario0.18

0.16

0.14

0.12

0.10.080.060.040.02

0.010.0 12.0 14.0 16.0 18.0 20.0 22.0 24.0 26.0

(a) Probability Distribution

D Baseiine Scenario

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

10.0 12.0 14.0 16.0 18.0 20,0 22.0

(b) Cumulative Distribution

24.0

Figure 9: Distribution of project completion (days) for main scenario in example (1)

0.16 -

0.14

0.12

0.1

0.08

0.06

0.04

0.02

-o- High-Quality Peopie

-o- Low-Quality People

10.0 12.0 14.0 16.0 18.0 20.0 22.0 24.0 26.0 28.0

(a) Probability

1.0 -

0.9

0.8

0.7

0.6

0.5

0.40.30.20.1

-o- High-Quality People

-o- Low-Quaiity Peopie

10.0 12.0 14.0 16.0 18.0 20.0 22.0 24.0 26.0

{b} Cumulative

Figure 10: Change in project time distribution (days) when level of people's experience changes

JUNE 2007 PROIECT MANAGEMENT JOURNAL

^ Complete in 18 days-o- Complete in 22 days

0.48

0.4

0.32

0,24

0.16

0.08

0.0Low Medium High

Figure 11 : Probability of required resource changes when the time constraint changes

Classes can be used as librariesand combined into a model as needed.By connecting interrelated objects,complex networks with several dozennodes can be constructed easily. I'igure16 shows the OOBN model for theexample previously presented.

The OOBN approach can also sig-nificantly improve the performance ofinference in the model. Although a fulldiscussion of the OOBN approach tothis particular problem is beyond thescope of this paper, the key point tonote is that there is an existing mecha-nism (and implementation of it) thatenables the proposed solution to begenuinely "scaled-up" to real-worldprojects. Moreover, research is emerg-

0.2 -

0,18

0.16

0.14

0.12

0.1

0.08

0.06

0.04

0.02

0.0

-•> No Risk in A. -o- Risk in A

/

/

/

/

/

// /

/ /

1

IVI A

\ \

1

1

\

\

i\\

\ \\

10.0 12.0 14.0 16.0 18.0 20.0 22.0 24.0 26.0 28.0

(a) Probability

1.0 n

0.9

-a- No Risk in A-o- Risk in A

10.0 12.0 14,0 16.0 18.0 20.0 22.0 24.0 26.0

(b) Cumulative

Figure 12: The impact of occurring risk in activity A on the project completion time

0.8 -

0.72 -

0.64 -

0.56 -

0.48 -

0.4 •

0.32 -

0.24 -

0.16 -

0.08 -

0.0 -1Low Medium High

ing to develop the new generation ofBNs tools and algorithms that supportOOBN concept both in constructinglarge-scale models and also in propa-gation aspeas.

Conclusions and How to Move ForwardHandling risk and uncertainty isincreasingly seen as a crucial compo-nent of project management and plan-ning. One classic problem is how toincorporate uncertainty in projectscheduling. Despite the availability ofdifferent approaches and tools, thedilemma is still challenging. Most cur-rent techniques for handling risk anduncertainty in projea scheduling (sim-ulation-based techniques) are often

Figure 13: Learnt probability distribution "resource" when the actual duration is seven days

JUSiK looy P R O I E T T M - \ N M " 5 F M F N T 47

0.2 -

0.18

0.04

0.02

0.0

<i- Learned Scenario

- o Baseline Scenario

12.0 14.0 16.0 18.0 20.0 22.0 24.0 26.0 28.0

(a) Probability Graph

1.0 n

- o Learned Scenario-o- Baseline Scenario

12.0 14.0 16.0 18.0 20.0 22,0 24.0

(b) Cumuiative Graph

26.0

Figure 14: completion time (days) based on learned parameters compare with baseline scenario

Figure 15: Activity class encapsulates internal parts of network

A

ESLF

EF

BESLFLSEF

t—1

•-••

CESLFLSEF

DESLFLSEF

>-

E

LFLS

EF )

Figure 15 :00 model for the presented example

event-oriented and try to model theimpact of possible "threats" on projectperformance. They ignore the sourceof uncertainty and the causa! relationsbetween project parameters. Moreadvanced techniques are required tocapture different aspects of uncertaintyin projects.

This paper has proposed a newapproach that makes it possible to

incorporate risk, uncertainty, andcausality in project scheduling.Specifically, the authors have shownhow a Bayesian network model canbe generated from a project's CPMnetwork. Part of this process is auto-matic and part involves identifyingspecific risks {which may be commonto many activities) and resource indi-cators. The approach brings the full

weight and power of BN analysis tobear on the problem of project sched-uling. This makes ii possible to:• Capture different sources of uncer-

tainty and use them to inform proj-ect scheduling

• Express uncertainty about comple-tion time for each activity and thewhole project with full probabilitydistributions

• Model the trade-off between timeand resources in project activities

• Use "what-if?" analysis• Ixarn from data so that predictions

become more relevant and accurate.

The application of the approachwas explained by use of a simpleexample. In order to upscale this toreal projects with many activities theapproach must be extended to usethe so-called object-oriented BNs.There is ongoing work to accommo-date such object-oriented modelingso that building a BN version of aCPM is just as simple as building abasic CPM model.

Other extensions to the workdescribed here include:• Incorporating additional uncertainty

sources in the duration network• Handling dynamic parameter learn-

ing as more information becomesavailable when the project progresses

• Handling common causal risks thataffea more than one aaivity

• Handling management action whenthe project is behind its plan.

48 I PROJECT MANAGEMENT JOURNAL

ReferencesAgfiiaRisk. (2006). Available al

www.agenarisk.comChapman, C, & Ward, S. (2000).

I^stimation and evaluation of uncer-lainty: A minimalist first passapproach. International lournal ofProject Management, i8. 369-383.

Cook M. S. (2001), Real-worldMonte Carlo analysis. Pioceeding ofPMI Annual Seminars and Symposium,Nashville, TN.

Efron, B. (2004). Bayesians, fre-quentists, and scientists, loumal of theAmerican Statistical Association,100(46")), 1-5.

lemon, N, E., Krause, P., & Neil,M. (2002). Software measurement:Uncertainty and causal modeling. IEEESoftware 10(4), 116-122.

Fishman, C. S. (1986). A MonteCarlo sampling plan for estimatingnetwork reliability. Operations Research,34(4), 581-594.

Fox, T. I.., & Spence, I. W. (1998).Tools of the trade: A survey of projectmanagement tools. Project Managementloumal, 29, 20-28.

Goldstein, M. (2006). SubjectiveBayesian analysis: Principle and prac-tice2. Bayesian Analysis, 1(3), 403-420.

Goldratt, E. M. (1997). Criticalchain. Great Barrington: The NorthRiver Press Publishing Corporation.

Heckerman, D., Mamdani, A., &Wellman, M. (1995). Real-world appli-

cations of Bayesian networks. ConimACM, .JS(3), 25-26.

Herroelen, W., & Leus, R. (2001).On the merits and pitfalls of criticalchain scheduling, loumai of OperationsManagement, 19,559-577.

Kelly, I. F.. (1961), Critical pathplanning and scheduling mathematicalbases. Operations Research, 9, 246-320.

Roller, D., & Pfeffer, A. (1997).Object-oriented Bayesian networks. InD. Geiger & P. Shenoy, (Bds.),Proceedings of the Thirteenth annualConference on Uncertainty in ArtificialIntelligence (UAl-97) (pp. 302-313),San Francisco, CA.

Malcolm, D. G., Roseboom, |. IL,Giark, C. E., & Fazer, W. (1959).Application of a technique for researchand development program evaluation.Operations Research, 7(5), 646-669.

Miller R. W. (1962). How to planand control with PERT. HarvardBusiness Review, 93-104.

Moder, |. (1988). Network tech-niques in project management. ProjeclManagement Handbook, 2nd edition(pp. 324-373). New York: VanNostrand Reinhold.

Neil, M., Fenton, N., Forey, S. &Harris, R. (2001). Lising Bayesian beliefnetworks to predict the reliability ofmilitary vehicles. lEE Computing andControl Engineering loumal / 2 ( 1 ) , 11-20.

Pinto, ). K. (1999). Some con-straints on the theory of constraints;

Taking a rritical look ai the criticalchain. PM Network l-i{S), 49-51.

Pertmaster (2006). Project risk soft-ware. Available at www.pertmastfr.com

PMI. (1999). Project managementsoftware survey, Newtown Square, PA:Project Management Institute.

PML (2004). A gutde to the pn)jecttnanagement body of knowledge [PMBOK^guide- 3rd edition). Newtown Square, PA:Project Management Institute.

Pollack-Iohnson, B. (1998).Projecl management software usagepatterns and suggested research direc-tions for future development, l^ojectManagement journal, 29, 19-29.

Raz, L, Barnes, R., & Dvir, D.(2003). A critical look at critical chainproject management. ProjectManagement loumal, 34, 24-32.

Trietsch, D. (2005). Why aitical pathby any other name would smell less sweet?l^oject Matuigement loumal, 36, 27-36.

Van Dorp, I, R., & Duffey, M. R.(1999). Modelling statistical depend-ence in risk analysis for project net-works. International lournat ofProduction Economics, 58, 17-29.

Van Slyke, R. M. (1963). MonteCarlo methods and the PliRT problem.Operations Research, !?(5), 839-860.

Ward, S., & Chapman, C. (2003).Transforming project risk managementinto project uncertainty management.International lournal of ProjectManagement, 21, 97-105.

VAHID KHODAKARAMI is a PhD student at RADARgroup at Queen Mary University ot London. Heearned a BSc in industrial engineering fromTehran Polytechnic and an MSc in industrialengineering from 5harif University of Technologyin Iran. He has more than 10 years experience inboth academia and industry. He has alsoconsulted for several companies in projectmanagement and system design. He earned hissecond MSc in information technology fromQueen Mary. His research interests include projectmanagenfient, project risk management, decision-making and Bayesian networks.

MARTIN NEIL is a reader in "systems risk" at the Department of ComputerScience, Queen Mary, University of London, where he teaches decision andrisk analysis and software engineering. He is also a joint founder and chieftechnology officer of Agena Ltd., which developed and distributes AgenaRisk,a software product for modeling risk and uncertainty. His interests coverBayesian modeling and/or risk quantification in diverse areas: operational riskin finance, systems and design reliability, project risk, decision support,simulation, artificial intelligence and personalization, and statistical learning.He earned a BSc in mathematics, a PhD in statistics and software metrics andis a chartered engineer.

NORMAN FENTON Is a professor of computing atQueen Mary (London University) and is also chiefexecutive officer of Agena, a company thatspecializes in risk management for critical systems.At Queen Mary he is the computer sciencedepartment director of research and he is the headof the Risk Assessment and Decision AnalysisResearch Group (RADAR). His books andpublications on software metrics, formal methods,and risk analysis are widely known in the softwareengineering community. His recent work hasfocused on causal models IBayesian nets) for riskassessment in a wide range of applicationdomains such as vehicle reliability, embeddedsoftware, transport systems. TV personalizationand financial services. He is a chartered engineerand chartered mathematician and is a fellow of theBritish Computer Society. He is a member of theeditorial board of the Saftuiare equality loumal.

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