JOURNAL OF RESEARCH IN NATIONAL DEVELOPMENT VOLUME 8 NO 2, DECEMBER, 2010
a monte carlo approach to COMBATING
DELAYED completion of development projects
J. O. Bioku
Department of Estate Management, Federal
Polytechnic, Idah
Email: femibioku@yahoo.com
Abstract
The objective of this paper is to unveil the
relevance of Monte Carlo critical path analysis in
resolving problem of delays in scheduled
completion of development projects. Commencing
with deterministic network scheduling, Monte Carlo
critical path analysis was advanced by assigning
probability distributions to task times. Applying
the Monte Carlo critical path analysis model to a
3bedroom bungalow project, results indicate broad
alternatives for early completion times which can
be optimized within tolerable profile of risk and
uncertainty in tasks durations. By assisting
project planners to exercise time and resource
management skills throughout the project life
cycle, Monte Carlo critical path analysis is
adjudged a vital risk assessment and monitoring
tool for combating delays in completion of
development projects.
Keywords:
Critical path analysis, risk and uncertainty,
project life cycle, earliest finish time
Introduction
Implications of delays in project completion can
be identified to include extra development costs,
discounting of cash flows beyond the earliest
finish date of a project and the erosion of
developer’s profit beyond the expected figure.
Therefore, completing projects within a reasonable
time frame has assumed complex dimension owing to
the vagaries of risk and uncertainty. This problem
is derived from the failure of deterministic
project planning models to reflect alternative
outcomes of project decisions and corresponding
impact on task durations. Inference drawn from
Capiński, and Zastawniak, (2003) reveals that
deterministic input variables in any form of
analysis imply the application of models that are
too simple to describe reality. Extending project
time control tools by allowing random changes in
input variable entails Monte Carlo simulation task
durations in network diagrams. Viewed as a
stochastic method with numerous application
environments, the Monte Carlo technique
incorporates the use of continuous variables in a
simulationbased framework (Hargitay & Yu, 1993
and Cvitanic & Zapaterop, 2004).
While Trofin (2004) reiterated that stochastic
simulation of outcomes of network schedules could
be undertaken with the aim of achieving multiple
project objectives ranging from cost minimization,
time control, profit maximization, and production
of an optimal design among others, the focus of
this paper is on the optimization of project
completion dates to curtail the adverse occurrence
of delays.
Project life cycle
Although there is no general consensus among
scholars as to what constitutes the life cycle of
a project, the six major stages in the life cycle
for property development projects can be
identified to include conception, feasibility and
viability studies, preliminary planning, detail
planning, execution, and testing and
commissioning. These stages in project life cycle
are correlated with the property development
process which Cadman and AustinCrowe (1991)
identified as commencing with project initiation
and terminating with letting, management or
disposal.
Risk and uncertainty in project planning
According to Kerzner (2003), risks are dangerous
factors capable of increasing the inability to
meet the goals of time, cost, and performance
objectives of a project. Uncertainty on the other
hand describes an unknown outcome of project plan
and thrives in the absence of information about
future events and conditions (Uher, 2003). Kerzner
(2003) defines risk management as a holistic
approach towards planning for risk, risk
assessment, development of risk handling
strategies, and risks monitoring.
Project planning and control system
Project planning entails determining activities
that need to be performed; those responsible for
performing specific activities, the time they are
expected to implement these activities and fulfil
their responsibility (Kerzner, 2003). Project
control ensures successful attainment of
objectives that have been formulated in plans
(Uher, 2003).
Deterministic critical path analysis
Deterministic processes in any given system
exhibits constant behavioural pattern at a point
in time (Guéret et. al. 2000 and Dupacová, et. al.
2003). In consonance with this assertion,
deterministic scheduling entails the application
of average time estimates in determining duration
of project activities (Uher, 2003) and its
application to critical path analysis entails
adoption of deterministic task durations.
Methodology for critical path analysis begins with
calculation of start and finish times of each
activity in a network schedule. These times enable
the identification of specific chain of activities
on the critical path (Lewis, 1995). Scheduling
Techniques for critical path analysis (CPA)
include the activity on arrow method (AOA) and the
activity on node (AON) method otherwise referred
to as the precedence technique. Unlike the AOA
method, there are no events in the AON method as
network construction focuses on activities which
are represented graphically by squares or boxes
(Uher, 2003). With respect to AON, relationships
among network activities are defined by dependency
lines. For the purpose of this study, the
precedence network (AON method) will be adopted.
In deterministic network analysis of critical path
(CP), the forward pass was adopted in determining
earliest start times (EST) and earliest finish
times (EFT) of activities. A backward pass through
the network schedule calculates in reverse order
the latest start times (LST) and latest finish
times (LFT) of activities. Since an activity time
is known, the latest starting time (LST) can be
calculated by subtracting the activity time from
latest finishing time (LFT). Hence, the latest
finishing time (LFT) for an activity entering a
node is the earliest starting time (EST) of
activities exiting the node.
EFT_{j} = EST_{j}
+ Duration
(1)
LST_{j
}= LFT_{j} – Duration
(2)
In event of two or more tasks originating from the
same preceding task
j, the
LFT of activity
k will
be the minimum LST of the activities
k_{1},
k_{2}, k_{3}, ….. k_{n}.
That is
LFT_{j
}= MIN LST _{k1,
k2, k3, ….. kn}
(3)
Floats in network analysis can function as early
warning system for a project manager; therefore,
Total Floats (TF) for critical and non critical
tasks is computed using the model:
TF_{j}
= LST_{j}  EST_{j}
(4)
Since activities on the critical path have zero
TF, spreadsheet logic was utilized to return
values of 1 for critical tasks and 0 for
noncritical tasks. The logic is defined by:
“=IF(Float<0.0001,1,0)”
(5)
Bringing formulas (1)  (5) to bear, attention of
this paper will now be turned to stochastic
modelling of network analysis.
Stochastic critical path analysis
The unscientific characteristic of deterministic
critical path analysis renders it incapable of
modelling the natural occurrences of task duration
in construction projects. Uher (2003), Kerzner
(2003) and Trofin (2004) suggested probability
scheduling as a robust approach towards
controlling risk and uncertainty in project
durations.
The Monte Carlo approach to critical path analysis
is among the myriads of risk analysis techniques
for proactive management of earliest finish times
(EFT) of projects (Uher, 2003).
Among quantitative risk assessment and monitoring
tools, the Monte Carlo method exhibits superior
analytical coverage because it enables expected
value, normal distribution, sensitivity and
scenarios analysis techniques to be handled within
a stochastic framework.
Chen and Hong (2007) identified the three typical
stages in Monte Carlo simulation to include
generating sample paths, evaluating the payoff
along each path, and calculating an average
estimate. As a cyclical iteration process (Fig.
1), computerized iteration is sustained for as
many times as possible until a desirable
statistical distribution of computed outcomes of
earliest completion times of the project is
displayed alongside tables of vital statistics
from the simulation process.
It would be wrong to conclude that simulations do
not take into cognizance interdependencies among
input variables in project duration. With the
application of spreadsheet modelling of
deterministic outcomes of decisions, the computer
can detect and analyze the interdependencies in
the model and run stochastic processes through the
deterministic model, thereby churning out wide
range of outcomes of a decision from a single
strategy.
Implications of Monte Carlo based critical path
analysis on project planning
On completion of 10,000 iterations, the mean
stochastic EFT indicates average earliest
completion time. In addition, the standard
deviation informs the project planner of the
decision to schedule tasks and resources within
the overall risk factor. The 100% confidence
interval of minimum and maximum EFT is another
result which informs the project planner to
schedule completion of a project within the budget
and risk profile of the developer. Besides these,
results from stochastic sensitivity analysis
provide information about sensitivity of EFT to
stochastic tasks times for all activities in the
project. The essence is to identify those tasks
that contribute significantly to variance in
stochastic EFT while keeping
tab on noncritical tasks capable of inducing
nonviable completion times or otherwise.
Monte Carlo critical path analysis case study
Deterministic scheduling
The simple case study for this paper is the
construction of a three bedroom bungalow. The work
breakdown schedule (WBS) indicates site
preparation as Task 1 and painting and decoration
as Task 10 among others. Preliminary analysis with
gantt chart (Fig. 2) reveal task durations,
preceding tasks and task dependencies.
The probability of each task falling within the
critical path have been expressed in percentages
for which the deterministic critical path is
defined by shaded tasks in the network diagram
comprising Tasks 1 – 2 – 3 – 5 – 6 – 7– 8 – 10 on
the chain with an Earliest Finish Time (EFT) of
129 days.
To
pave way for the application of Monte Carlo
simulation, the deterministic task duration,
assumptions of optimistic duration (Dur. Opt.) and
pessimistic duration (Dur. Pess.) for all tasks as
drawn from the project planner’s experience in
similar projects were imputed similar projects were imputed in
Monte Carlo environment as triangular
distributions for likeliest task durations,
minimum, and maximum respectively.
While simulated task durations (Sim. Value)
correspond with the likeliest tasks durations, the
deterministic EFT of 129 days shaded in Table 1
was set as the forecast variable using Crystal
Ball 7^{®}, a Microsoft^{®} Excel^{®}
addin utilized for simulations and stochastic
analyses.
Outcomes of Simulated Earliest Finish Time (EFT)
After 10,000 iterations as indicated in Table 2,
Crystal Ball® simulation analysis revealed that
the expected mean Earliest Finish Time (EFT) for
the project is 148 days as against 129 days
produced by the deterministic modelling of
completion date. A standard deviation of 9 days
indicates the risk factor responsible for
variation in the simulated mean Earliest Finish
Time on commencement of project. Fitted with a
positively skewed t – distribution of 0.1242
degree of asymmetry, the Monte Carlo analysis of
the EFT of this project exhibits a minimum of 118
days and a maximum of 180 days. (Fig.4 and Table
2).




Variation of outcomes from both techniques can be
attributed to deterministic forward pass technique
as distinct from Monte Carlo modelling of tasks
durations; implying an extra cost of project
resources for 19 days to engender completion. The
minimum and maximum stochastic EFT suggests that
the developer can schedule and implement the
project from a minimum of 118 days to the maximum
period his budget and risk profile can
accommodate. period his budget and risk profile can
accommodate.
With respect to analysis of forecast percentiles
in Table 3, the 50th percentile indicates a 50%
probability that EFT is above or below the median
duration of 148 days. At the top end, there is
only a 10% probability that the EFT will be above
160 days which is the 90th percentile.
Stochastic sensitivity chart and determinants of
earliest finish time (EFT)
Figures 5 and 6 indicate that the stochastic EFT
is sensitive to Tasks 7, 6, 2, 3, 5, 8, 10, and 1
in descending order. While, the stochastic
sensitivity analysis indicates a positive rank
correlation between EFT and all Tasks on the
Critical path, there is no correlation between the
stochastic EFT and Task 4, which is on the non
critical path.
Furthermore, Task 9 on the noncritical
path exhibits a rank correlation coefficient of
0.01 implying that noncritical tasks can
influence the outcome of project positively or
otherwise.
It can be inferred from Fig. 6 that Task 7
(Electrical Works) contributed more to the
variation in EFT compared to Task 4 (Roofing) with
zero contribution to variation. However, Task 4
may contribute significantly to variance in the
stochastic EFT if task times are altered
accordingly.
Worst and best case scenarios of earliest finish
times
Tornado^{®} tool in Crystal Ball^{®}
7 was utilized in testing each input variable
included in the project planning model as distinct
from the correlationbased sensitivity charts. The
procedure involves analyzing “at worst” and “at
best” value for the chosen input variable and then
calculating the estimated finish times at these
values whilst freezing each of the remaining
variables at their mean.
In Table 4, the “at best task times” for each task
yields “at best EFT” which is 20.0% deviation
from the mean stochastic EFT, while the “at worst
task times” for each task churns out “at worst
EFT” which corresponds with +20.0% deviation from
the mean stochastic EFT. A corollary to the worst
and best case scenario is Fig. 7 which illustrates
the swing between the maximum and minimum EFT for
each input variable.
Table 4: Major determinants of
EFT by percentage deviation from mean stochastic
EFT 


Earliest Finish Time (EFT) 

Variable 
20.0% 
15.0% 
10.0% 
5.0% 
0.0% 
5.0% 
10.0% 
15.0% 
20.0% 
Task 3 
138 
139 
141 
143 
145 
146 
148 
150 
152 
Task 2 
138 
140 
142 
143 
145 
146 
148 
149 
151 
Task 5 
139 
141 
142 
143 
145 
146 
147 
149 
150 
Task 7 
141 
142 
143 
144 
145 
146 
146 
147 
148 
Task 6 
142 
143 
143 
144 
145 
145 
146 
147 
147 
On the downside are input variables with positive effect on EFT ( )
while the upside of the variable ( )
implies that task duration churns out EFT which is
higher than the mean EFT for that variable.
Results indicate that higher durations for all
tasks tend to increase the EFT while lower task
durations tend to reduce the EFT to a manageable
period. While certain elements of Tasks 2 such as
curing of concrete cannot be hastened, the ability
or otherwise of concluding the project within a
tolerable earliest finish time depends on the
project manager’s ability to apply Monte Carlo
scheduling techniques to time management skills as
examined in this paper.
Conclusion and recommendation
Deterministic tasks durations exhibit constant
behavioural pattern at a point in time and cannot
account for the impact of risk and uncertainty in
earliest completion times of projects. In reality,
variation in skill levels of site personnel,
varying efficiency of work time for project team,
mistakes, and misunderstandings among other
factors
account for variation in tasks durations and
constitute an impediment to early completion of
projects. To combat this anomaly, Monte Carlo
critical path analysis was examined in a project
case study and adjudged a robust prescriptive
decision models which incorporates risk and
uncertainty in task durations and helps optimize
project completion dates. Aided by Monte Carlo
critical path analysis, a project manager can
complete a project within a tolerable earliest
finish time by exercising time and resource
management skills throughout the project life
cycle.
References
Byrne, P. J. (1996).
Risk,
Uncertainty and DecisionMaking in Property
Development. (2nd Ed.) London: E &FN Spon.
Cadman, D. and AustinCrowe, L. (1991).
Property
Development. (3rd Ed.). London: Chapman and
Hall
Capiński, M. and Zastawniak, T. (2003).
Mathematics
for Finance: An Introduction to Financial
Engineering. London: SpringerVerlag
Chen, N. and Hong, J. L. (2007). Monte Carlo
Simulation in Financial Engineering. In Henderson,
S. G. et. al. (Eds). 2007
Winter
Simulation Conference Proceedings. pp.919
931. Retrieved July 3, 2008 from
http://www.ielm.ust.hk/dfaculty/hongl/papers/MCWSC2007.pdf
Cvitanić, J and Zapaterop, F. (2004).
Introduction
to the Economics and Mathematics of Financial
Markets. Massachusetts: The MIT Press
Dupačová, J., Hurt, J., and Štepán, J. (2003).
Stochastic
Modeling in Economics and Finance. New
York: Kluwer Academic Publishers
Formoso, C. T. Tzortzopoulos, P., and Liedtke, R.
(2002). A Model for Managing the Product
Development Process in House Building.
Blackwell
science Journal of Engineering,
Construction
and Architectural Management. 9 (5/6),
pp.419–432
Guéret, C., Prins, C., and Sevaux, M. (2000).
Applications
of optimization with XpressMP. U.K.: Dash
Optimization Ltd.
Hargitay, S. & Yu, SM (1993).
Property
Investment Decisions: A Quantitative Approach.
London: E & FN Spon.
Howes, N. R. (2001).
Modern
Project Management: Successfully Integrating
Project Management Knowledge Areas and Processes.
New York: AMACOM – American
Management Association.
Kerzner, H. (2003).
Project
Management: A Systems Approach to Planning,
Scheduling and Controlling. (8th ed.).New
Jersey: John Wiley and Sons Inc.
Lewis, J. P. (1995).
Fundamentals
of Project Management. New York: AMACOM
BooksEarthweb Resources
Trofin, I. (2004). “Impact of uncertainty on
Construction Project Performance using Linear
Scheduling”. Unpublished Thesis, MSc Building
Construction, University of Florida, USA.
Uher, T. E. (2003).
Programming
and Scheduling Techniques. Sydney: University
of New South Wales Press Ltd.
Wysocki, R.K. and McGary, R. (2003).
Effective
Project Management: Traditional, Adaptive, Extreme.
(3rd Ed). Indiana: Wiley Publishing, Inc.
Zhang, H., Shi, J. J., and ChiMing, T. (2002).
Application Simulation related techniques to
construction operations.
Blackwell
SciencJournal of Engineering, Construction and
Architectural Management. 9(5/ pp.433–445