onsider this scenario: You are finalizing the planning of your company’s next major maintenance outage or capital project and are concerned the timeline is too tight and your team is small. You communicate this to management and recommend increasing the size of the contracted teams by 20 percent to secure the deadline. The manager then asks, “But what is the real chance of delay? And what will be the chance of delay after the change? If I invest in increasing the team, what will my return really be?”

Instead of answering that “the risk of delay will decrease,” what if you could answer: “Currently, the probability of not meeting the deadline is 50 percent. With the suggested recommendations, the probability of delay falls to less than 15 percent.” Much more convincing, don’t you think?

Monte Carlo simulations for schedule risk management enable this type of response, as well as give detailed information on the success and delay of the various tasks, allowing for better decision-making and, consequently, increasing the gains.

Risk Management Methods

Major maintenance shutdowns (e.g., large production units once every five years) resemble large projects, with a dedicated staff and often a Project Manager, as well as procurement, planning and scheduling, and other staff.1 Smaller turnarounds (e.g., machine stops every four months) are usually already planned and performed by day-to-day maintenance personnel. In both cases, good schedule risk management helps to avoid delays.

Given the similarity between turnarounds and projects, good project management practices also apply to outages.

According to the Project Management Institute, risk management is one of the 10 areas of knowledge that constitute effective project management.2 The methods used may be qualitative or quantitative. In qualitative, risks are described in terms like very low, low, medium, high and very high. Quantitative methods, however, involve the calculation of probabilities and the estimation of the impact of different scenarios. In this way, it is possible to “identify the risks that require the most attention because of the impact on the final outcomes of the project” and “it is possible to identify realistic and achievable scope and chronogram objectives, timeframes and timelines.”3 The main quantitative method is Monte Carlo.

The Monte Carlo simulation method was initially proposed by mathematician Stanislaw Ulam4 in the 1940s and was used in the first electronic numerical integrator and computer (ENIAC) during the Manhattan Project, which developed the first atomic bombs. Nowadays, any personal computer has the ability to process the method using specific software.

How Do Monte Carlo Simulations Work?

The best way to explain Monte Carlo is with an example, as shown in Figure 1.5 The schedule shows four tasks. The most likely duration estimates are four, five, six and four days, respectively. Therefore, the total duration of the project would be 4 + 5 + 6 = 15 days (Task 4 occurs in parallel with Task 3, and since it has a shorter duration, it does not interfere with the total duration).

Figure 1: A Project schedule with four tasks

But, the duration estimates are, as the name says, estimates, and the task will hardly end at this time. A more accurate alternative is to estimate using three points, with minimum, most probable and maximum durations.

How long is the project now? With only four tasks, it is no longer possible to calculate manually. To know the answer, you need to do a simulation.

In proposing Figure 1, Kailash Awati, consultant and senior lecturer in the Faculty of Transdisciplinary Innovation at the University of Technology Sydney, explains, “The idea behind Monte Carlo is to simulate the entire project (i.e., all four tasks) using a large number of N times (10,000, for example) and get N durations for the project.” In each of the N trials, a probabilistic mathematical method establishes a duration for each task, according to the minimum, most probable and maximum estimates. In the case of Task 1, for example, the method will “sort” a duration between two and eight days, with four days being the most likely. Then the “draw” durations of all tasks are summed up to get the total duration of the project. At the end, you will have N total durations for the project, ranging from the minimum possible and the maximum possible. That is, you will obtain the probability distribution of the duration of the project, represented by Figures 2 and 3.

Figure 2: Distribution of possible execution times5

Figure 3: Deadline x likelihood to fulfill5

Thus, the probability of completing this project or turnaround by up to 15 days (sum of the most probable times of each task) is only 28 percent, or a 72 percent chance of taking more time. What would you do if you knew your project or turnaround has a 72 percent chance of being late? You would probably act to prevent this from happening.

On the other hand, there is only a 10 percent probability of termination in more than 19.5 days.

The good part of all this is if the duration estimates are updated, for example, due to a change in team size, the simulation can be easily updated and the new result can be compared to the previous one.

It is interesting to note that the sum of the maximum durations is 27 days, but according to the simulation, the maximum duration of the project is 24.5 days (that is when the probability reaches 100 percent). Why is this? Because probabilities do not add up; probabilities combine through union and intersection. So, simply adding the durations can lead to big mistakes.

Obviously, the simulation of a real project involves a much larger number of tasks and the complexity of the simulation is much greater. But the principle does not change.


In addition to the time management application, Monte Carlo simulation can be applied in any situation that involves probabilities and uncertainties, including:

  • Project cost management;
  • Analysis of the need for equipment redundancy;
  • Reliability and availability analysis.

For companies that already have a good maturity in planning for maintenance projects and outages, the Monte Carlo method helps drive performance to even higher levels through more accurate information that leads to better decision-making.


  1. Palmer, Richard. Maintenance Planning & Scheduling Handbook, Third Edition. New York City: McGraw-Hill Education, 2012; p. 394
  2. Project Management Institute. A Guide to the Project Management Body of Knowledge, (PMBOK®), Sixth Edition. Newtown Square: Project Management Institute, 2017.
  3. Rossi, Paolo. “How to link the qualitative and the quantitative risk assessment.” Paper presented at PMI® Global Congress 2007 – EMEA, Budapest, Hungary. Newtown Square, PA: Project Management Institute, 2007. https://www.pmi.org/learning/library/link- qualitative-quantitative-risk-assessment-7375
  4. Metropolis, N. and Ulam, S. “The Monte Carlo Method.” Journal of the American Statistical Association , Vol. 44. No. 247, Sept. 1949, pp. 335-341.
  5. Awati, K. “A gentle introduction to Monte Carlo simulation for project managers.” Eight to Late, 2018. https://eight2late.wordpress.com/2018/03/27/a-gentle-introduction-to-monte-carlo-simulation-for-project-managers/

Luiz Alberto Verri

Luiz Alberto Verri, is the owner of Verri Veritatis Consultoria Ltda. Luiz worked for Petrobras, a Brazilian oil company, where he held the positions of maintenance manager and general plant manager until his retirement in 2007. Luiz has authored several books. www.verriveritatis.com.br

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