(SCHEDULE RISK ANALYSIS USING MONTE CARLO SIMULATION TECHNIQUES)

In the last issue of the Trett Digest, Mathew Joseph introduced the concept of project risk and methods of its management. In this second article, he looks in more detail at schedule risk analysis using Monte Carlo Simulation Techniques.

CRITICAL PATH METHOD

CPM (Critical Path Method) scheduling is preferred for projects subjected to nominal uncertainty in project completion times and where the activity durations are reasonably predictable. CPM calculations are performed on a deterministic schedule model. Hence, you get the same result no matter how many times you recalculate the schedule. In fact, activity durations are estimates of the actual time required, and therefore there is bound to be a significant amount of uncertainty associated with actual durations. The uncertainty of activity durations is particularly large during initial project phases, since the number of unknowns are greater at this stage. For projects subjected to uncertainty in completion times, then this uncertainty limits the usefulness of deterministic project models.

The method of introducing uncertainty into the scheduling process requires more work and a greater number of assumptions, programmers generally tend to ignore this factor while developing project schedules, and proceed with the schedule calculation using expected or most likely activity durations. The primary drawback of this method is that the use of expected activity durations typically result in overly optimistic schedules for completion. Secondly, the use of fixed and single activity durations makes the schedule model inflexible and rigid thus losing its dynamic characteristics.

Since activity durations are uncertain and can vary significantly, the schedules prepared based on fixed activity durations can become outdated quickly and as a result the project team loses confidence in the realism of such schedules. In order to accommodate the effect of the uncertainty associated with activity durations, programmers tend to include a fixed and uniform contingency allowance for all schedule activities. For example, an excavation activity with an expected duration of 20 days may be scheduled for 23 days, including a 15% schedule contingency. The uniform application of contingency to all schedule activities would only result in 15% increase in the expected time to complete the project.

PERT (PROGRAMME EVALUATION AND REVIEW TECHNIQUE)

Another method of incorporating uncertainty in the scheduling process is to apply the critical path scheduling process and then analyse the results from a probabilistic perspective, often referred to as the PERT (Programme Evaluation and Review Technique) method. Here, unlike the critical path method which uses a single time estimate for an activity, three duration estimates are provided for each activity. These durations are referred to as pessimistic, most likely and optimistic durations. A single time estimate (expected time) of the activity duration is then deduced from the three time estimates using the equation:

Expected Activity Duration =

Optimistic + 4 x Most + Pessimistic Likely
                                6

CPM analysis is then performed on the schedule model so generated using the expected activity durations, the critical path is then identified. The expected project duration is the sum of the expected durations of the activities along the critical path. Assuming that activity durations are independent random variables then the variation in the duration of the critical path will be the sum of these variances along the critical path.

The main drawback of this method is that it focuses upon a single critical path, whereas in reality many paths might become critical due to random fluctuations. The focus on only a single path means that the PERT method typically underestimates the actual project duration.

However the vast majority of activity durations are not random variables, also in reality activity durations correlate with one another. For example, if problems happen in the delivery of steel, this problem is likely to influence the expected durations of many other activities such as fabrication, erection etc.

Positive correlation of this type between activity durations implies that the PERT method underestimates the variance of the critical path and thereby produces over optimistic expectations of the probability of meeting a particular project completion deadline.

MONTE CARLO SIMULATION TECHNIQUE

An alternative to the CPM and PERT methods is the Monte Carlo simulation technique. This method was developed during the 1940s, and was named after the city of Monaco famous for its casinos and games of chance. The simulation process is performed using stochastic (pertaining to chance) techniques that are based on the use of random numbers. Inputs into the model consist of statistical distributions rather than single values. Using these random inputs, the deterministic model is converted to a stochastic model, Normal, Triangular, Beta PERT etc. are some of the commonly used distribution patterns.

When a CPM calculation is used, the critical path is computed just once, whereas when the Monte Carlo simulation is used, these calculations will be performed a number of times depending on the amount of iterations chosen. So in a single scheduling simulation, different sets of activity durations might be used for each iteration. For example, if 1000 iteration steps are selected, the schedule critical path will be computed 1000 times, using activity durations randomly chosen from the assigned distribution (Normal, Triangular etc.).

The Monte Carlo methods have been applied to diverse problem areas ranging from nuclear reactor design, Dow Jones forecasting to construction/engineering and project management. This technique offers a straightforward method of obtaining information about the distribution of possible project completion times. This technique calculates sets of realistic activity duration times, and then applies a deterministic scheduling procedure to each set of durations. Numerous calculations are required since simulated activity durations must be calculated and the scheduling procedure applied many times over. The accuracy of the results generated depends on the distribution of possible activity durations and the parameters chosen. In addition, it is pertinent to specify correlation between activity durations wherever appropriate.

Monte Carlo simulation can provide the project team with a number of useful indicators concerning the project schedules.

• An estimate of the distribution of completion times or budget, so that probability of meeting a particular completion time or budget can be predicted;
• Estimates of the expected time and variance of the project completion or project duration;
• The probability that a particular activity will lie on the critical path;

These results show that the deterministic completion date based on the CPM analysis is 10th June 2009. The confidence level associated with this date is only 4%, indicating that this date is likely to slip if there is the slightest schedule uncertainty. The histogram also shows there is an 85% chance of finishing before 23rd July 2009.

Most of the commercially available Monte Carlo simulation systems are also capable of generating various other useful results and indices to give information regarding schedule sensitivity and criticality. In a traditional CPM analysis, criticality takes no account of the uncertainty of an individual task and consequently a task with zero uncertainty can still have a 100% criticality. During a Monte Carlo simulation, tasks can join or leave the critical path. The criticality index expresses, as a percentage, how often a particular task was on the critical path during the analysis. Tasks with a high criticality index are more likely to cause delay to the project, as they are more likely to be on the critical path.

Analysing the criticality of activities this way gives the project team a better understanding of the numerous risks likely to affect the project schedule.

The outcome of any type of schedule analysis is indicative only and is not absolute. The realism of the results depends on the accuracy of the input, robustness and the suitability of the method employed and the ability to interpret the results from a practical perspective. Simulation and statistical methods like Monte Carlo techniques are proving to be invaluable tools in forecasting schedule uncertainties to a large extent, and are gaining wide spread acceptance as project management technique.

Mathew Joseph is based at Trett Consulting’s Dubai office