Schedule Risk Analysis using Monte-Carlo Simulation

Introduction

The trade-offs inherent in any project are included in the so-called “Scope Triangle”. The crucial goals for any project are to finish the project on time, within budget and with satisfactory performance or quality. Scheduling is the essential part of the construction project management that deals with time, and it consists in the determination of the timing and sequence of operations in the project and their assembly to give the overall completion date [1]. Most of the construction projects failed to finish within panned duration, and one of the reasons is because the uncertainties that may cause a delay in performing specific activities are not always considered. Hence, it is vital to develop a risk management estimation which deals with the risks of execution that affects the project duration. Schedule Risk Analysis (SRA) is a project management method to assess the risk of the baseline schedule. It permits to focus on those activities that are most likely to have a negative impact on the project deadline. When the manager notices that these activities do not go as planned, it is able to make faster and more accurate responses that positively contribute to the overall project performance [2]. The scope of this research study is to give an overview of the potential of the Monte-Carlo Schedule Analysis, trying to illustrate its main issues.

1. Schedule Risk Types

Schedule risks fall into three main broad areas [1]:

1. General duration uncertainty

General duration uncertainty is the risk resulting from an inaccurate or overly simplistic estimation of durations, or the durations are based on assumptions that are not necessarily correct or accurate. Furthermore, the critical path identified in the deterministic approach may not be the same as those in the probabilistic critical path when risks are incorporated into the schedule. A probabilistic approach takes the concept of estimating durations based on average production rates to a more refined level.

2. Specific risk events

Specific risk events are potential impacts on the schedule that may or may not occur, such as accidents and other events difficult to predict. Modeling a specific risk by creating a string of activities to represent the scope of work due to the risk (a “what-if scenario”) is one good way to attempt to determine the potential ramifications of the realization of that risk.

3. Network logic risks that exist or are increased as a result of activity relationships

Network logic risks include any risks that predominately relate to the schedule network. These risks are often discovered only through a technical analysis of the schedule’s components. A good review of the schedule should identify these risks if the reviewer is knowledgeable and is provided direction in the risk management plan.

Each of these types of risk can be analyzed differently using different tools. The use of the Monte-Carlo simulations makes general duration uncertainties relatively easy to analyze, as well as risks from activity relationships and some specific event risks.

2. Deterministic vs. Probabilistic Approach

The traditional scheduling approach consists in having one end date and one critical path in a schedule, i.e. it is a deterministic approach. Activity durations are given as solitary esteem and there is a supposition that the duration is known with some assurance. This assumption ignores the fact that the schedule is attempting to predict how long it will take to complete an activity at some unknown time in the future, using an unknown crew composition, with variable experience, and working in unknown conditions. In the real world, the timetable regularly contains significant uncertainty, particularly for unsafe tasks. The application of Monte-Carlo analysis changes the traditional deterministic approach to a probabilistic approach, with a range of end dates and potential critical paths with associated probabilities (Figure 1).

The project schedule is still driven by a single critical path, but with Monte-Carlo approach hundreds of variations of the schedule and critical paths can be simulated quickly. Through the results of all the simulations, data can be statistically analyzed to identify a realistic end date and which activities have the most likelihood of impacting the end date.

3. Advantages of Monte-Carlo Schedule Analysis

The technique known as Monte-Carlo Schedule (MCS) Analysis allows to reduce general uncertainty in the project duration by modeling the uncertainty in the duration of individual activities by means of probability distributions and by running a simulation with multiple trials to produce a probability distribution for the total project duration. The main benefit of MCS Analysis is to determine the level of risk in a project schedule. By using this method, it is possible to quantify the probability of achieving a project end date. The probability found is a measure of the risk associated with achieving the schedule based on the schedule logic. One of the main benefits of risk quantification is the ability to influence factors to reduce risk in order to improve the likelihood of succeeding.

Monte-Carlo analysis gives several benefits to project execution by [3]:

• Changing the expectations of a single project end date with an unknown probability to a range of end dates with quantified probability
• Identifying the most important activities within a schedule that may not be on the critical path
• Determining the contingency needed to reduce the overrun risk to an acceptable level
• Developing a realistic execution plan
• Aligning the stakeholders involved in a project
• Avoiding claims and potential liquidated damages
• Evaluating the effect of risk management actions

4. Application of Monte-Carlo Schedule Analysis

The four main steps for a successful schedule risk analysis, shown in Figure 2, are:

1. Creation of the CPM baseline schedule for the entire project

2. Estimation of the uncertainty in the activities’ durations

3. Execution of MSC method

4. Interpretation of the MSC results

4.1. Creation of the CPM baseline schedule for the entire project

In general, the preparation of a CPM schedule includes the following four steps [1]:

• Break down the project into work activities
• Determine activities’ durations
• Determine logical relationships
• Draw the logic network and perform the CPM calculations

In order to create an effective, accurate, and successful schedule, four additional steps are essential: review and analyze the schedule, implement the schedule, monitor and control the schedule and revise the database and record feedback. Finally, two other optional steps may be implemented for a comprehensive approach to scheduling: cost/resource allocation (or loading) and resource leveling.

The two main computational steps required for a CPM are the forward pass and the backward pass. The forward pass is the process of navigating through a network from start to end and calculating the completion date for the project and the early dates for each activity, i.e. the Early Start (ES) and the Early Finish (EF) dates. On the contrary, the backward pass is the process of navigating through a network from end to start and calculating the late dates for each activity, i.e. the Late Start (LS) and the Late Finish (LF) dates. For some activities, late and early dates are the same: these activities have strict start and finish dates. Any delay in them will result in a delay in the entire project. Therefore, these activities are called critical activities and the continuous chain of critical activities from the start to the end of the project is the critical path. The critical path is defined as the longest continuous path in a network from start to finish. Other activities have some leeway, called float.

An example of a CPM network diagram and a Gantt chart derived from the CPM schedule are represented in Figure 3 and Figure 4, respectively.

4.2. Estimation of the uncertainty in the activities’ durations

During this step, uncertainty needs to be translated into activity duration distribution profiles. Four categories of methods can be distinguished [2]:

1. Single-point estimate

This method indirectly admits that there is no risk connected to the activities, and it should only be used for activities who will have a duration that is 100% certain.

2. Three-point estimate

This method consists in defining three durations per activity: an optimistic (a), a most likely (m) and a pessimistic (b) duration. A triangular distribution is then created, and it can be symmetric, skewed to the left or skewed to the right.

3. Pre-defined risk classes

When the project dimension increases (thousands of activities), it may be an enormous task to define three-point estimates for each activity. In this case, it is faster to define risk through the so-called “risk classes”, i.e. categories with a certain risk distribution behind it, as shown in Figure 5. The percentages indicate how much the optimistic and pessimistic duration deviate from the single-point estimation, which is of course 100%. These percentages are formulated by the project manager based on his experience and historical information.

4. Statistical distributions

In this latter case, in order to reduce the general uncertainty in the project duration and to simulate the probabilistic behavior of various real world processes, the uncertainty in the duration of individual activities is modeled by means of probability distributions. Though MCS, a simulation with multiple trials is then performed and the final product will be a probability distribution for the total project duration. A great advantage is the fact that they can easily be fitted to three-point estimates.

Several different input probability distributions can be used to express the uncertainty in the duration of individual activities of the project. The modern software programs allow the user to choose the most suitable for his/her purpose. For example, Figure 6 shows some of the possibilities given by @RISK, an add-in to Microsoft Excel for Monte-Carlo simulation.

J. K. Visser [6] investigated the influence of alternative input distributions on the output of a project schedule risk simulation. The input distributions were the triangular, the normal, the lognormal, the Gumbel, the Fréchet and the Weibull distributions. The mean and standard deviation values for all input distributions were exactly the same but the skewness was not. He considered two different project activity networks (series and parallel networks) with a total of ten activities each. Monte-Carlo simulations were performed for the six different input distributions and the output distributions for the total project duration were compared. The results of the study showed that the choice of the probability distribution to reflect the duration of activities does not have a significant effect on the output distribution of a schedule simulation when the input distributions are symmetric or slightly skewed (in any direction). In particular, the difference in the P80 and P90 (80% and 90% probability) values for the different input distributions was less than 1% for the two networks.

The probability distributions often used in risk analysis are the Triangular distribution and the Beta (PERT) distribution. Both of them require three duration values as input, i.e. optimistic (a), most likely (m) and pessimistic (b) estimates. The output of a simulation for PERT input values will not provide the same output distribution as the triangular distribution for the same input values since the mean and the standard deviation values are different. Note, however, that if the same mean and standard deviation (Std Dev) values are used for both the probability distributions, the output distribution will be very similar (Figure 7).

Unlike the triangular distribution, the PERT distribution constructs a smooth curve which places more emphasis on values around the most likely value, as shown in Figure 8. PERT distribution assigns a greater weight to the most likely value in comparison with the triangular one, meaning that the estimate for the most likely value is believed to be reliable.

To assess the duration range for each activity, experience is required. In this regard, is essential to compare the estimated ranges against historical benchmark data to obtain more objectivity in the assessment project. Through the use of statistical methods (such as adhesion tests or curve fitting), the probability distribution that best represents a given history can be easily found. The best results occur when a small team of people involved in the project collaboratively assess the duration ranges. The involvement of multiple people adds validity to the date range and the team approach helps to align team members on realistic durations, logic and execution plan for the schedule. This is what has been done by K. I. Wali and S. A. Othman [4] in their research work. For each activity, the researchers collected the estimated duration ranges by an interview with 26 civil engineers. Furthermore, based on the responded results, the researchers found the average durations for each activity.

4.3. Execution of MCS method

The main idea is that each Monte-Carlo run generates a duration for each activity of the project from its distribution profile. The MCS method involves the following steps [5]:

• Each iteration starts by randomly generating a number from the interval [0;1[ and calculating a duration for each risky activity from its range and probability distribution. The process, that involves three steps, is represented in Figure 9. The random variates from the distributions can be determined from analytical expressions, as shown by J. K. Visser [6] in his article. However, the simulation add-ins or standalone software automate the simulation process and is more convenient, easier and faster to use.

• For that iteration, the total project and key milestone completion dates are calculated using the CPM method for that particular configuration of durations. Those calculated dates are only possible dates, and they are not representative of all possible solutions.
• The risk analysis iterates the simulation many times to determine the entire pattern of possible completion dates for the project and its milestones. At the end of each iteration, the possible completion dates calculated are collected and stored.
• At the end of the entire simulation, the dates computed from all iterations are arrayed in graphs and tables showing, for example, the likelihood distribution (bell-curve) and the cumulative distribution (S-curve), as in Figure 10. The latter one represents the likelihood of the project completion on or before each possible date.

How to evaluate the number of simulation runs needed? P. Deshmukh & N. R. Rajhans [8], considering two different methods, obtained the same result for the optimum number of simulation runs. The two methods considered were:

A. Application of the formula proposed by E. Buka?i & Th. Korini (2016) to directly calculate the number of runs required, as shown below:

where ?? is the number of simulation runs, ???? is the value of confidence coefficients, ?? is the error of the mean, ?? is the mean of the sample and ???? is the standard deviation of the sample.

B. Slowly increasing of the simulation runs up to a value where there is a small change or no change in the output: this value of the run is considered as the optimum value to obtain desired results.

4.4. Interpretation of the MSC results

In schedules, the logic for most projects involves multiple branches and parallel paths. Each iteration can lead to a change in the project duration, activity durations and critical path: activities can become part of the set of critical activities or may be not a critical activity anymore. At each iteration, the program records which activities were on the critical path for that iteration. Through Monte-Carlo analysis it is possible to carry out a sensitivity analysis, i.e. a modeling technique that figures out which risky activities have the most impact on the project. These sensitivity measures indicate the degree of riskiness or sensitivity of the particular activity on the project objectives [2]. By understanding what a schedule is sensitive to, it can allow changes in the execution to improve the end date.

The three main indices used in the construction industry are:

Criticality Index (CI), that measures the probability that an activity lies on the critical path. It is a percentage: higher the percentage, higher the probability that the activity is critical. Its major drawback is that it only measures probability, while risk is defined as probability times impact. The CI of activity ?? is calculated as follows:

where ?????? is the total float of activity ??.

Significance Index (SI), that represents the relative importance between activities and incorporate an estimate of the potential impact that a delay in an activity may cause on the whole project. The SI of activity ?? is calculated as follows:

where ?? is the expectation, ???? is the duration of activity ?? and ???? is the total duration of the project.

Schedule Sensitivity Index (SSI), presented by the PMBOK, combines the two dimensions of risk, namely probability and impact (????????=???????????????????????????????????). The SSI of activity ?? is calculated as follows:

where ???? is the standard deviation of the activity ??’s duration, while ???? is the standard deviation of the project duration. Alternatively, a proxy for calculating the SSI is given below [2]:

The output of the sensitivity analysis is the so-called “tornado graph” (Figure 11), a bar chart that focuses on the top activities closely correlated to the statistics of the end date. The tornado graph shows the relative sensitivity of the end date to each activity in the schedule. The exact numbers shown are not as important as the relative ranking of the activities [3]. From Figure 11 it can be noted that there is a much higher degree of correlation with the lengths of the three top activities to the end date, and they have a higher chance of impacting the end date. In this way, for example, it is possible to focus limited resources where it makes the most impact. Even if there is still one critical path in any iteration of the Monte-Carlo analysis, sensitivity analysis helps to identify other paths that may have a high probability of impacting the end date. The results of this analysis can be surprising and not always evident.

5. Conclusions

The use of MCS method for SRA is increasing nowadays due to the availability of fast computers and software freely available, primarily as add-ins for MS Excel (such as @Risk, Crystal Ball, SimVoi and RiskAmp) [4]. S. Tattoni & M. M. Schiraldi showed through an algorithm and experimental results that the computational time, which is historically the major drawback of MCS, is definitely minimum these days due to the computational power available [9].

By presenting a scientific and statistical approach instead of picking and arbitrary end date, Monte-Carlo schedule analysis can help align the two main parties (owner and contractor) to agree on what a realistic target can be. Even if a misalignment persists, the identification of risks associated with the given schedule is clearer, and both parties can go forward. Moreover, the early identification of schedule risks can help avoid claims and potential liquidated damages, permitting informed decisions on both sides.

Monte-Carlo is an extremely valuable tool for identifying and managing risks associated with the schedule. However, a good judgment and the human ability to make decisions based on experience are of utmost importance. Never take the results of Monte-Carlo schedule analysis and use them without first taking a critical look and ensuring that the results are understood [3].

References

[1] S. Mubarak, Construction Project Scheduling and Control (3rd Edition), John Wiley & Sons, Inc., 2015.

[2] L. Martens, Schedule Risk Analysis, Case Studies, Universiteit Gent, 2017.

[3] J. Verschoor, "The Benefits of MonteCarlo Schedule Analysis," in AACE International Transactions, 2005.

[4] K. I. Wali and S. A. Othman, "Schedule Risk Analysis Using Monte-Carlo Simulation for Residential Projects," ZANCO Journal of Pure and Applied Sciences, vol. 31, no. 5, pp. 90-103, 2019.

[5] D. T. Hulett, "Schedule risk analysis simplified," PM Network, vol. 10, no. 7, pp. 23-30, 1996.

[6] J. K. Visser, "Suitability of different probability distributions for performing schedule risk simulations in project management," in Portland International Conference on Management of Engineering and Technology (PICMET), Honolulu, 2016.

[7] K. B. Salling, "Risk Analysis and Monte-Carlo Simulation within Transport Appraisal," 2007.

[8] P. Deshmukh and N. Rajhans, "Comparison of Project Scheduling techniques: PERT versus Monte-Carlo simulation," Industrial Engineering Journal, vol. 11, 2018.

[9] S. Tattoni, M. M. Schiraldi and L. Laura, "Estimating projects duration in uncertain environments: Monte-Carlo simulations strike back," in 22nd IPMA World Congress “Project Management to Run”, Roma, 2008.