a monte carlo approach to COMBATING DELAYED completion of development projects


J. O. Bioku

Department of Estate Management, Federal Polytechnic, Idah





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 3-bedroom 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



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 simulation-based 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 Austin-Crowe (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.

EFTj  = ESTj + Duration                                                                                                    (1)

LSTj = LFTj – 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 k1, k2, k3, ….. kn. That is

LFTj = 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:

TFj = LSTj - ESTj                                                                                                              (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 non-critical 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 non-critical tasks capable of inducing non-viable 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® add-in 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).


Table 2: Forecast statistics for EFT of project




Forecast values























Standard Deviation



















Coeff. of Variability














Range Width





Mean Std. Error




Table 3: Percentile of forecast EFT




Forecast values






























































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 non-critical path exhibits a rank correlation coefficient of 0.01 implying that non-critical 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 correlation-based 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)











Task 3










Task 2










Task 5










Task 7










Task 6










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.





Byrne, P. J. (1996). Risk, Uncertainty and Decision-Making in Property Development. (2nd Ed.) London: E &FN Spon.


Cadman, D. and Austin-Crowe, 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: Springer-Verlag
















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 


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 Xpress-MP. U.K.: Dash Optimization Ltd.


Hargitay, S. & Yu, S-M (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 Books-Earthweb 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 Chi-Ming, T. (2002). Application Simulation related techniques to construction operations. Blackwell SciencJournal of Engineering, Construction and Architectural Management. 9(5/ pp.433–445