What is Monte Carlo Analysis?

What is Monte Carlo Analysis?

Monte Carlo analysis involves determining the impact of the identified risks by running simulations to identify the range of possible outcomes for a number of scenarios. A random sampling is performed by using uncertain risk variable inputs to generate the range of outcomes with a confidence measure for each outcome. This is typically done by establishing a mathematical model and then running simulations using this model to estimate the impact of project risks. This technique helps in forecasting the likely outcome of an event and thereby helps in making informed project decisions.

While managing a project, you would have faced numerous situations where you have a list of potential risks for the project, but you have no clue of their possible impact on the project. To solve this problem, you can consider the worst-case scenario by summing up the maximum expected values for all the variables. Similarly, you can calculate the best-case scenario. You can now use the Monte Carlo analysis and run simulations to generate the most likely outcome for the event. In most situations, you will come across a bell-shaped normal distribution pattern for the possible outcomes.

Let us try to understand this with the help of an example. Suppose you are managing a project involving creation of an eLearning module. The creation of the eLearning module comprises of three tasks: writing content, creating graphics, and integrating the multimedia elements. Based on prior experience or other expert knowledge, you determine the best case, most-likely, and worst-case estimates for each of these activities as given below:

Tasks	Best-case estimate	Most likely estimate	Worst-case estimate
Writing content	4 days	6 days	8 days
Creating graphics	5 days	7 days	9 days
Multimedia integration	2 days	4 days	6 days
Total duration	11 days	17 days	23 days

The Monte Carlo simulation randomly selects the input values for the different tasks to generate the possible outcomes. Let us assume that the simulation is run 500 times. From the above table, we can see that the project can be completed anywhere between 11 to 23 days. When the Monte Carlo simulation runs are performed, we can analyse the percentage of times each duration outcome between 11 and 23 is obtained. The following table depicts the outcome of a possible Monte Carlo simulation:

Total Project Duration	Number of times the simulation result was less than or equal to the Total Project Duration	Percentage of simulation runs where the result was less than or equal to the Total Project Duration
11	5	1%
12	20	4%
13	75	15%
14	90	18%
15	125	25%
16	140	28%
17	165	33%
18	275	55%
19	440	88%
20	475	95%
21	490	98%
22	495	99%
23	500	100%

This can be shown graphically in the following manner:

What the above table and chart suggest is, for example, that the likelihood of completing the project in 17 days or less is 33%. Similarly, the likelihood of completing the project in 19 days or less is 88%, etc. Note the importance of verifying the possibility of completing the project in 17 days, as this, according to the Most Likely estimates, was the time you would expect the project to take. Given the above analysis, it looks much more likely that the project will end up taking anywhere between 19 – 20 days.

Benefits of Using Monte Carlo Analysis

Whenever you face a complex estimation or forecasting situation that involves a high degree of complexity and uncertainty, it is best advised to use the Monte Carlo simulation to analyze the likelihood of meeting your objectives, given your project risk factors, as determined by your schedule risk profile. It is very effective as it is based on evaluation of data numerically and there is no guesswork involved. The key benefits of using the Monte Carlo analysis are listed below:

• It is an easy method for arriving at the likely outcome for an uncertain event and an associated confidence limit for the outcome. The only pre-requisites are that you should identify the range limits and the correlation with other variables.
• It is a useful technique for easing decision-making based on numerical data to back your decision.
• Monte Carlo simulations are typically useful while analyzing cost and schedule. With the help of the Monte Carlo analysis, you can add the cost and schedule risk event to your forecasting model with a greater level of confidence.
• You can also use the Monte Carlo analysis to find the likelihood of meeting your project milestones and intermediate goals.

Now that you are aware of the Monte Carlo analysis and its benefits, let us look at the steps that need to be performed while analysing data using the Monte Carlo simulation.

Monte Carlo Analysis: Steps

The series of steps followed in the Monte Carlo analysis are listed below:

1. Identify the key project risk variables.
2. Identify the range limits for these project variables.
3. Specify probability weights for this range of values.
4. Establish the relationships for the correlated variables.
5. Perform simulation runs based on the identified variables and the correlations.
6. Statistically analyze the results of the simulation run.

Each of the above listed steps of the Monte Carlo simulation is detailed below:

1. Identification of the key project risk variables: A risk variable is a parameter which is critical to the success of the project and a slight variation in its outcome might have a negative impact on the project. The project risk variables are typically isolated using the sensitivity and uncertainty analysis.
   Sensitivity analysis is used for determining the most critical variables in a project. To identify the most critical variables in the project, all the variables are subjected to a fixed deviation and the outcome is analysed. The variables that have the greatest impact on the outcome of the project are isolated as the key project risk variables. However, sensitivity analysis in itself might give some misleading results as it does not take into consideration the realistic nature of the projected change on a specific variable. Therefore it is important to perform uncertainty analysis in conjunction with the sensitivity analysis.
   Uncertainty analysis involves establishing the suitability of a result and it helps in verifying the fitness or validity of a particular variable. A project variable causing high impact on the overall project might be insignificant if the probability of its occurrence is extremely low. Therefore it is important to perform uncertainty analysis.
2. Identification of the range limits for the project variables: This process involves defining the maximum and minimum values for each identified project risk variable. If you have historical data available with you, this can be an easier task. You simply need to organize the available data in the form of a frequency distribution by grouping the number of occurrences at consecutive value intervals. In situations where you do not have exhaustive historical data, you need to rely on expert judgement to determine the most likely values.
3. Specification of probability weights for the established range of values: The next step involves allocating the probability of occurrence for the project risk variable. To do so, multi-value probability distributions are deployed. Some commonly used probability distributions for analyzing risks are normal distribution, uniform distribution, triangular distribution, and step distribution. The normal, uniform, and triangular distributions are even distributions and establish the probability symmetrically within the defined range with varying concentration towards the centre. Various types of commonly used probability distributions are depicted in the diagrams below:
   
   
4. Establishing the relationships for the correlated variables: The next step involves defining the correlation between the project risk variables. Correlation is the relationship between two or more variables wherein a change in one variable induces a simultaneous change in the other. In the Monte Carlo simulation, input values for the project risk variables are randomly selected to execute the simulation runs. Therefore, if certain risk variable inputs are generated that violate the correlation between the variables, the output is likely to be off the expected value. It is therefore very important to establish the correlation between variables and then accordingly apply constraints to the simulation runs to ensure that the random selection of the inputs does not violate the defined correlation. This is done by specifying a correlation coefficient that defines the relationship between two or more variables. When the simulation rounds are performed by the computer, the specification of a correlation coefficient ensures that the relationship specified is adhered to without any violations.
5. Performing Simulation Runs: The next step involves performing simulation runs. This is typically done using a simulation software and ideally 500 – 1000 simulation runs constitute a good sample size. While executing the simulation runs, random values of risk variables are selected with the specified probability distribution and correlations.
6. Statistical Analysis of the Simulation Results: Each simulation run represents the probability of occurrence of a risk event. A cumulative probability distribution of all the simulation runs is plotted and it can be used to interpret the probability for the result of the project being above or below a specific value. This cumulative probability distribution can be used to assess the overall project risk.